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Global regularity and convergence to equilibrium of reaction–diffusion systems with nonlinear diffusion

  • Klemens FellnerEmail author
  • Evangelos Latos
  • Bao Quoc Tang
Open Access
Article

Abstract

We study the boundedness and convergence to equilibrium of weak solutions to reaction–diffusion systems with nonlinear diffusion. The nonlinear diffusion is of porous medium type, and the nonlinear reaction terms are assumed to grow polynomially and to dissipate (or conserve) the total mass. By utilising duality estimates, the dissipation of the total mass and the smoothing effect of the porous medium equation, we prove that if the exponents of the nonlinear diffusion terms are high enough, then weak solutions are bounded, locally Hölder continuous and their \(L^{\infty }(\Omega )\)-norm grows in time at most polynomially. In order to show convergence to equilibrium, we consider a specific class of nonlinear reaction–diffusion models, which describe a single reversible reaction with arbitrarily many chemical substances. By exploiting a generalised logarithmic Sobolev inequality, an indirect diffusion effect and the polynomial in time growth of the \(L^{\infty }(\Omega )\)-norm, we show an entropy–entropy production inequality which implies exponential convergence to equilibrium in \(L^p(\Omega )\)-norm, for any \(1\le p < \infty \), with explicit rates and constants.

Keywords

Reaction–diffusion systems Nonlinear diffusion Porous medium Convergence to equilibrium Entropy method 

Mathematics Subject Classification

35B35 35B40 35K57 35Q92 

1 Introduction and main results

In this article, we study the boundedness and convergence to equilibrium of weak solutions to reaction–diffusion systems with nonlinear diffusion
$$\begin{aligned} \left\{ \begin{array}{lll} \partial _t u_i - d_i\Delta (u_i^{m_i}) = f_i(u), &{}\quad x\in \Omega , \quad \ \; t>0, &{} \quad i=1,\ldots , S,\\ d_i\nabla (u_i^{m_i})\cdot \overrightarrow{n} = 0, &{}\quad x\in \partial \Omega , \quad t>0, &{} \quad i=1,\ldots , S,\\ u_i(x,0) = u_{i,0}(x), &{}\quad x\in \Omega , &{} \quad i=1,\ldots , S, \end{array} \right. \end{aligned}$$
(S)
with the unknown functions \(u = (u_1, \ldots , u_S)\) and \(u_i: \Omega \times {\mathbb {R}}_+ \mapsto {\mathbb {R}}\), the positive diffusion coefficients \(d_i >0\), the porous medium exponents \(m_i >1\) and where \(\Omega \subset {\mathbb {R}}^d\) denotes a bounded domain with sufficiently smooth boundary \(\partial \Omega \) (e.g. \(\partial \Omega \) is of class \(C^{2+\varrho }\) for some \(\varrho >0\)) with outward unit normal \(\overrightarrow{n}\) on \(\partial \Omega \). Moreover, the conditions imposed on the nonlinear reaction terms \(f_i(u)\) and the non-negative initial data \(u_{i,0}\) will be specified later.
The first part of this paper considers weak solutions to system (S). Our aim is to provide sufficient conditions on the porous medium exponents \(m_i\) and on the nonlinearities \(f_i(u)\), under which weak solutions are indeed bounded in \(L^{\infty }\) (and thus locally Hölder continuous) for all times and grow at most polynomially in time. More precisely, we assume the following conditions on the nonlinearities:
  1. (i)
    The nonlinearities \(f_i: {\mathbb {R}}^S \rightarrow {\mathbb {R}}\) are locally Lipschitz functions and satisfy
    $$\begin{aligned} |f_i(u)|\le C (1+|u|^\nu ),\quad \forall u=(u_1,\ldots ,u_S)\in {\mathbb {R}}^S, \quad \forall i=1,\ldots , S, \end{aligned}$$
    (G)
    where \({\mathbb {R}}\ni \nu \ge 1\) is the maximal growth exponent of the reaction terms.
     
  2. (ii)
    There exist positive constants \(\lambda _1,\ldots ,\lambda _S>0\) such that:
    $$\begin{aligned} \sum ^S_{i=1}\lambda _if_i(u)\le 0,\qquad \forall u\in {\mathbb {R}}^S, \end{aligned}$$
    (M)
    which formally implies the following mass dissipation law
    $$\begin{aligned} \frac{\hbox {d}}{\hbox {d} t}\int _\Omega \sum ^S_{i=1}\lambda _iu_i \hbox {d}x\le 0. \end{aligned}$$
     
  3. (iii)
    The nonlinearities are assumed quasi-positive, that is for all \(i=1,\ldots ,S,\) holds
    $$\begin{aligned} f(u_1,\ldots ,u_{i-1},0,u_{i+1},\ldots ,u_S)\ge 0,\qquad \forall u_1,\ldots ,u_S\ge 0. \end{aligned}$$
    (P)
    The quasi-positivity condition (P) ensures global non-negativity of solutions subject to non-negative initial data, see e.g. [26, 36].
     
The existence of global weak solutions to (S) subject to homogeneous Dirichlet boundary conditions and under the assumptions (G)–(M)–(P) was recently obtained in [26]. The proof of the following Theorem 1.1 on the existence of weak solutions to (S) subject to Neumann boundary conditions uses similar arguments to [26] and is postponed to Sect. 5.

Theorem 1.1

Assume the conditions (G), (M) and (P) and consider non-negative initial data \((u_{i,0}) \in L^2(\Omega )^S\). If
$$\begin{aligned} m_i > \max \{\nu - 1; 1\} \quad \text { for all } \quad i=1\ldots S, \end{aligned}$$
then, there exists a global weak non-negative solution to system (S) in the sense that, for all \(i=1,\ldots , S\), \(u_i \in C([0,+\infty ); L^1(\Omega ))\), \(u_i^{m_i}\in L^1(0,T;W^{1,1}(\Omega ))\), \(f_i(u)\in L^1(\Omega \times [0,T])\) and
$$\begin{aligned} -\int _{\Omega }\psi (0)u_{i,0}\hbox {d}x - \int _{0}^{T}\int _{\Omega }(u_i\partial _t\psi + d_iu_i^{m_i}\Delta \psi )\hbox {d}x\hbox {d}t = \int _0^T\int _{\Omega }\psi f_i(u)\hbox {d}x\hbox {d}t \end{aligned}$$
for all test function \(\psi \in C^{2,1}({\overline{\Omega }}\times [0,T])\) with \(\nabla \psi \cdot \overrightarrow{n} = 0\) on \(\partial \Omega \times (0,T)\) and \(\psi (\cdot , T) = 0\).
Moreover, a solution \(u = (u_1,\ldots , u_S)\) to (S) with (M) and (P) satisfy
$$\begin{aligned} \Vert u_i\Vert _{L^{m_i+1}(Q_T)} \le C \quad \text { for all } \quad T>0 \quad \text { and } \quad i = 1,\ldots , S, \end{aligned}$$
where the constant C depends on the \(L^2\)-norm of the initial data, the constants \(\lambda _i\) in (M), the diffusion coefficients \(d_i>0\) and the domain \(\Omega \).

Remark 1.1

With a more careful analysis, it seems possible to generalise Theorem 1.1 and consider initial data \(u_{i,0}\in L^1(\Omega )\). We refer the interested reader to [38] for the case of systems with quadratic nonlinearities and \(L^1\) initial data.

Given the weak solutions of Theorem 1.1, our aim is to establish their boundedness and a polynomially in time growing \(L^{\infty }\)-estimate under stronger assumptions on the porous medium exponents \(m_i\): first, we recall the a priori estimate \(u_i \in L^{m_i+1}(Q_T)\) of Theorem 1.1 and the growth condition (G) imply \(f_i(u)\in L^{1+\varrho }(Q_T)\) for some \(\varrho >0\), which also justifies the definition of weak solutions in Theorem 1.1. In fact, the \(L^{1+\varrho }\) integrability guarantees uniform integrability of nonlinearities in a suitable approximating scheme (see the proof of Theorem 1.1 in Sect. 5).

Intuitively, Theorem 1.1 states that larger exponents \(m_i\) yield higher integrability of the nonlinearities \(f_i(u)\). Moreover, the functions \(u_i\) solve a porous medium equation with the right-hand side having higher integrability. Thus, by quantifying the smoothing effect from the porous medium equation, this allows to start a bootstrap argument, which eventually leads to boundedness of \(u_i\) in \(L^{\infty }\). In particular, it is of importance that our argument allows to show that the growth in time of the \(L^{\infty }\)-norms is at most polynomial. The first main result of this article is the following theorem.

Theorem 1.2

(Global bounded weak solutions) Let \(\Omega \subset {\mathbb {R}}^d\) be bounded with sufficiently smooth boundary. Let the initial data \(0\le u_{i,0}\in L^\infty (\Omega )\), assume the conditions (G), (M) and (P) and \(m_i > \max \{\nu - 1; 1\}\) for all \(i=1\ldots S\) as required by Theorem 1.1. Finally, in dimensions \(d\ge 3\), we additionally assume
$$\begin{aligned} m_i>\nu - \frac{4}{d+2},\qquad \forall i=1\ldots S. \end{aligned}$$
(1)
Then, any weak solution of (S) obtained in Theorem 1.1 is bounded in \(L^{\infty }(\Omega )\) and grows in time at most polynomially in the sense that, for any \(T>0\),
$$\begin{aligned} \Vert u_i\Vert _{L^\infty (Q_T)}\le C_T,\quad \forall i=1\ldots S \end{aligned}$$
where \(C_T\) is a constant which depends at most polynomially on time. Consequently, these solutions are locally (in \(Q_T\)) Hölder continuous, see e.g. [43].

Remark 1.2

(Weakened assumptions on mass dissipation and initial data) If one is only interested in the boundedness of solutions but not in the polynomial growth of the \(L^{\infty }\)-norm, then the mass dissipation condition (M) can in fact be weakened to
$$\begin{aligned} \sum _{i=1}^{S}\lambda _i f_i(u) \le C_1\sum _{i=1}^{S}|u_i| + C_2 \quad \text { for all } u \in {\mathbb {R}}^S, \end{aligned}$$
for some positive constants \(C_1, C_2\).

Also the assumed initial regularity \(u_{i,0}\in L^\infty (\Omega )\) is not optimal and could be relaxed to \(L^{p}\) integrability for sufficiently large p according to the details of the proof yet at the price of the readability of the Theorem.

Remark 1.3

When \(m_i = 1\), the condition (1) becomes
$$\begin{aligned} \nu < \frac{d+6}{d+2}, \end{aligned}$$
which agrees with the results for linear diffusion systems obtained in [8, Proposition 1.4].

Theorem 1.2 contributes to the large literature on global existence and boundedness of solutions to reaction–diffusion systems, which nevertheless poses still many open questions due to the lack of a unified approach (maximum principles do not hold for general systems). The largest part of the available literature, however, considers the case of linear diffusion, i.e. \(m_i = 1\) in system (S). We refer the reader to the extensive review of Pierre [36] and the references therein, in particular [2, 4, 5, 6, 14, 22, 23, 24, 25, 31, 35, 37, 39]

The case of nonlinear diffusion, on the other hand, is much less investigated. Most of the existing results considered special systems with special structures, see e.g. [28, 30, 42]. Up to the best of our knowledge, system (S) under the general structural assumptions (G)–(M)–(P) was only studied very recently in [26], where the authors showed the global existence of weak solutions. Therefore, the present paper serves as the first result to show the boundedness of weak solutions by assuming stronger conditions on porous medium exponents. Moreover, our proof allows to estimate explicitly the growth in time of the \(L^{\infty }\)-norm, which turns out to be essential in studying the large-time behaviour of solutions in the following second part of the paper.

The second main result of this paper proves exponential convergence to equilibrium for a class of reaction–diffusion systems with porous media diffusion of the form (S), where the nonlinearities model the following reversible reaction with arbitrarily many chemical substances
$$\begin{aligned} \alpha _1{\mathcal {A}}_1+\cdots +\alpha _M{\mathcal {A}}_M \underset{k_f}{\overset{k_b}{\leftrightharpoons }} \beta _1{\mathcal {B}}_1+\cdots +\beta _N{\mathcal {B}}_N. \end{aligned}$$
(2)
Here, \(\alpha _i,\beta _i\in [1,+\infty )\) are the stoichiometric coefficients of the \(M+N\) involved substances \({\mathcal {A}}_1, \ldots , {\mathcal {A}}_M\), \({\mathcal {B}}_1, \ldots , {\mathcal {B}}_N\) and \(k_f,k_b>0\) are the forward and backward reaction rate constants. For simplicity, yet without loss of generality, we assume \(k_f=k_b=1\). By applying mass action kinetics to (2) and by using the short notation
$$\begin{aligned} a= & {} (a_1, \ldots , a_M), \quad b = (b_1, \ldots , b_N), \quad \alpha = (\alpha _1, \ldots , \alpha _M), \quad \beta = (\beta _1, \ldots , \beta _N),\\ a^\alpha= & {} \prod _{i=1}^Ma_i^{\alpha _i}, \quad \quad b^{\beta } = \prod _{j=1}^{N}b_j^{\beta _j}, \end{aligned}$$
we study the following reaction–diffusion system:
$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}\partial _t a_i - d_i\Delta (a_i^{m_i}) = f_i(a,b)\,{:}{=} -\alpha _i\left[ a^\alpha - b^\beta \right] ,\ \forall i=1,\ldots ,M &{}&{}\quad x\in \Omega , \quad \ t>0,\\ &{}\partial _t b_j - h_j\Delta (b_j^{p_j}) = g_j(a,b)\,{:}{=} \beta _j\left[ a^\alpha - b^\beta \right] ,\ \forall j=1,\ldots ,N &{}&{}\quad x\in \Omega , \quad \ \, t>0,\\ &{} d_i\nabla (a_i^{m_i})\cdot \overrightarrow{n} = 0, \quad \forall i=1,\ldots ,M,&{}&{}\quad x\in \partial \Omega , \quad t>0,\\ &{} h_j\nabla (b_j^{p_j})\cdot \overrightarrow{n} = 0, \quad \forall j=1,\ldots ,N,\quad &{}&{}\quad x\in \partial \Omega , \quad t>0,\\ &{} a_i(x,0) = a_{i,0}(x),\quad \forall i=1,\ldots ,M,&{}&{}\quad x\in \Omega ,\\ &{} b_j(x,0) = b_{j,0}(x),\quad \forall j=1,\ldots ,N, &{}&{}\quad x\in \Omega . \end{aligned} \end{array}\right. } \end{aligned}$$
(R)
Here, \(d_i, h_j >0\) are diffusion coefficients, and \(m_i, p_j>1\) are nonlinear diffusion exponents. It is clear that (R) is a special case of (S). It is also straightforward to verify condition (P), while condition (G) is satisfied by choosing,
$$\begin{aligned} \nu =\max \Biggl \{\sum _{i=1}^M\alpha _i,\sum _{j=1}^N\beta _j\Biggr \}. \end{aligned}$$
Finally condition (M) is a consequence from noting that
$$\begin{aligned} \frac{1}{M}\sum _{i=1}^M\frac{1}{\alpha _i }f_i(a,b) + \frac{1}{N}\sum _{j=1}^N\frac{1}{\beta _j }g_j(a,b)=0. \end{aligned}$$
After having the conditions (P), (G) and (M) verified, Theorem 1.1 implies the existence of global weak non-negative solutions of system (R) provided
$$\begin{aligned} m_i, p_j > \max \left\{ \nu - 1; 1\right\} \quad \text { for all } \quad i=1\ldots M, \; j=1\ldots N. \end{aligned}$$
Moreover by Theorem 1.2, these solutions are bounded in dimensions \(d=1,2\), or in dimensions \(d\ge 3\) when additionally assuming
$$\begin{aligned} m_i, p_j > \nu - \frac{4}{d+2} \quad \text { for all } \quad i=1\ldots M, \; j=1\ldots N. \end{aligned}$$
By multiplying the equations for \(a_i\) and \(b_j\) with \(\beta _j\) and \(\alpha _i\), respectively, and by adding the resulting terms, integration by parts with the homogeneous Neumann boundary conditions implies that these solutions satisfy the following mass conservation laws:
$$\begin{aligned}&\beta _j\int _\Omega a_i(x,t)\hbox {d}x+\alpha _i\int _\Omega b_j(x,t)\hbox {d}x \nonumber \\&\quad = \beta _j\int _\Omega a_{i,0}(x)\hbox {d}x +\alpha _i\int _\Omega b_{j,0}(x)\hbox {d}x =:M_{ij}>0,\qquad \forall i,j, \end{aligned}$$
(3)
amongst which exactly \(M+N-1\) linearly independent conservation laws ought to be selected and only the corresponding \(M+N-1\) components of the initial mass vector \(M_{ij}\) need to be calculated from the initial data.
System (R) possesses for each fixed positive initial mass vector \((M_{ij})\) a unique positive detailed balanced equilibrium \((a_\infty ,b_\infty ) = (a_{1,\infty }, \ldots , a_{M,\infty }, b_{1,\infty }, \ldots , b_{N,\infty }) \in (0,\infty )^{M+N}\), which is the solutions of the following equilibrium equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} \prod ^M_{i=1}a_{i\infty }^{\alpha _i}=\prod ^N_{j=1}b_{j\infty }^{\beta _j},\\ \beta _ja_{i\infty }+\alpha _ib_{j\infty }=M_{ij},\quad \forall i,j, \end{array}\right. } \end{aligned}$$
where we recall that the second line constitutes of only \(M+N-1\) linearly independent conditions.

To study the convergence to equilibrium for (R), we will use the so-called entropy method, which recently proved a highly suitable tool in the analysis of the large-time behaviour of dissipative PDE systems. With respect to reaction–diffusion systems with linear diffusion, we refer in particular to [10, 11, 12, 13, 19, 20, 33].

The key entropy functional (or in this case the free energy functional) of system (R) is defined by
$$\begin{aligned} E[a,b] = \sum _{i=1}^{M}\int _{\Omega }(a_i\ln a_i - a_i + 1)\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }(b_j\ln b_j - b_j + 1)\hbox {d}x \end{aligned}$$
which dissipates according to the non-negative entropy production functional, that is formally
$$\begin{aligned} -\frac{\hbox {d}}{\hbox {d}t}E[a,b]=: D[a,b]= & {} \sum _{i=1}^{M}d_i\int _{\Omega }\frac{|\nabla a_i|^2}{a_i^{2-m_i}}\hbox {d}x + \sum _{j=1}^{N}h_j\int _{\Omega }\frac{|\nabla b_j|^2}{b_j^{2-p_j}}\hbox {d}x \\&+ \int _{\Omega }(a^\alpha - b^\beta )\ln {\frac{a^\alpha }{b^\beta }}\hbox {d}x \ge 0. \end{aligned}$$
In the case of linear diffusion, i.e. \(m_i = p_j = 1\) for all \(i=1\ldots M, j=1\ldots N\), the convergence to equilibrium of solutions of (R) (or some special cases) was recently studied in e.g. [10, 12, 19, 33, 40].
Let us briefly review the entropy method used in the case of linear diffusion and then highlight the difficulties to be overcome in the current paper when dealing with nonlinear diffusion. In the case of linear diffusion, the entropy production writes as
$$\begin{aligned} D_{lin}[a,b]= & {} \sum _{i=1}^{M}d_i\int _{\Omega }\frac{|\nabla a_i|^2}{a_i}\hbox {d}x + \sum _{j=1}^{N}h_j\int _{\Omega }\frac{|\nabla b_j|^2}{b_j}\hbox {d}x \\&+ \int _{\Omega }(a^\alpha -b^\beta )\ln {\frac{a^\alpha }{b^\beta }}\hbox {d}x \ge 0 \end{aligned}$$
and the entropy method consists in establishing a functional inequality of the form
$$\begin{aligned} D_{lin}[a,b] \ge \lambda (E[a,b] - E[a_{\infty },b_{\infty }]) \end{aligned}$$
(4)
for all functions \(a = (a_i)\), \(b= (b_j)\) satisfying the conservation laws (3). In order to do that, one first uses an additivity property of the relative entropy to calculate
$$\begin{aligned} \begin{aligned} E[a,b] - E[a_{\infty },b_{\infty }]&= \left[ \sum _{i=1}^{M}\int _{\Omega }a_i\log {\frac{a_i}{\overline{a}_i}}\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }b_j\log {\frac{b_j}{\overline{b}_j}}\hbox {d}x\right] \\&\quad + \left[ \sum _{i=1}^{M}(\overline{a}_i\log {\frac{\overline{a}_i}{a_{i,\infty }}} - \overline{a}_i + a_{i,\infty }) \right. \\&\qquad \left. + \sum _{j=1}^{N}(\overline{b}_j\log {\frac{\overline{b}_j}{b_{j,\infty }}} - \overline{b}_j + b_{j,\infty }) \right] \\&=: I_1 + I_2. \end{aligned} \end{aligned}$$
The term \(I_1\) is controlled in terms of the entropy production \(D_{lin}[a,b]\) thanks to the logarithmic Sobolev inequality (LSI)
$$\begin{aligned} \int _{\Omega }\frac{|\nabla f|^2}{f}\hbox {d}x \ge C_{\mathrm {LSI}}\int _{\Omega }f\log \frac{f}{\overline{f}}\hbox {d}x \quad \text { for all } \quad 0 \le f \in H^1(\Omega ). \end{aligned}$$
(5)
The remain term \(I_2\) only involves the averages of the concentrations \(\overline{a}_i, \overline{b}_j\) and can be controlled by \(D_{lin}[a,b]\) through lengthly, technical, but constructive estimates (see e.g. [19, 40] for more details). Note that this entropy approach applies successfully to more complex chemical reaction networks than (R), see [13, 20, 32, 33]. We emphasise that the logarithmic Sobolev inequality (5) is not only used to control the term \(I_1\) but also plays an important role in the estimates controlling the term \(I_2\).
In the case of nonlinear diffusion as here considered, we need a generalisation of the LSI (5) to exponents \(m_i, p_j \ge 1\). In this paper, we utilise the following generalisation (see e.g. [34]): for any \(m > (d-2)_+/d\) with \((d-2)_+ = \max \{d-2;0\}\), there exists a constant \(C(\Omega ,m)>0\) such that
$$\begin{aligned} \int _{\Omega }\frac{|\nabla f|^2}{f^{2-m}}\hbox {d}x \ge C(\Omega ,m)\,\overline{f}^{\,m-1}\int _{\Omega }f\log \frac{f}{\overline{f}}\hbox {d}x. \end{aligned}$$
When \(m=1\), this coincides with the classical logarithmic Sobolev inequality (5). For system (R), we have in particular
$$\begin{aligned} \int _{\Omega }\frac{|\nabla a_i|^2}{a_i^{2 - m_i}}\hbox {d}x\ge & {} C(\Omega , m_i)\,\overline{a}_i^{\,m_i - 1}\int _{\Omega }a_i\log \frac{a_i}{\overline{a}_i}\hbox {d}x \quad \text { and } \nonumber \\ \int _{\Omega }\frac{|\nabla b_j|^2}{b_j^{2-p_j}}\hbox {d}x\ge & {} C(\Omega ,p_j)\,\overline{b}_j^{\,p_j - 1}\int _{\Omega }b_j\log \frac{b_j}{\overline{b}_j}\hbox {d}x. \end{aligned}$$
(6)
Note that if we assume the averages \(\overline{a}_i\) and \(\overline{b}_j\) to be bounded below by a positive constant, then one can apply the same strategy as for the linear diffusion case in order to obtain the convergence to equilibrium. However, there is no chemical/physical reason for such a lower bound to hold in the transient behaviour of system (R) subject to general initial data. There are even perfectly admissible initial conditions, where some averages are zero since the corresponding species have not yet been formed.

To overcome this difficulty, we first observe that the mass conservation laws (3) subject to a positive mass vector \(M_{i,j}>0\) imply that the averages \(\overline{a}_i\) and \(\overline{b}_j\) cannot be simultaneously small. Thus, at any fixed time, at least one of the inequalities in (6) is useful, since either \(\overline{a}_i \ge \varepsilon \) or \(\overline{b}_j \ge \varepsilon \) for some suitably chosen \(\varepsilon >0\) depending on \(M_{i,j}>0\). Secondly, we are able to compensate the still lacking lower bounds in (6) by a phenomena which can be called “indirect diffusion effect” and which means in our context that the reversible reaction (2) transfers diffusion from a species \(a_i\) (with strictly positive diffusion bound in (6) due to \(\overline{a}_i \ge \varepsilon \)) to other species \(b_j\) (with lacking positive lower diffusion bound) in terms of a functional inequality, see Lemma 3.2 below.

Examples of indirect diffusion effect inequalities were already derived in e.g. [11, 17, 18], yet typically with a proof which requires uniform in time \(L^{\infty }\)-bounds on the solutions, which is a severe technical restriction as \(L^{\infty }\)-bounds for general reaction–diffusion systems are often unknown due to the lack of comparison principles. Note that also the \(L^\infty \)-bounds of Theorem 1.1 would be insufficient since polynomially growing and not uniform in time.

In this work, we are able to prove an indirect diffusion functional inequality without using any \(L^{\infty }\)-bounds on solutions but instead by exploiting the special structure of (R), see Lemma 3.2. Nevertheless, in the remaining part of applying the entropy method, the polynomial growth in time of the \(L^{\infty }\)-norm of Theorem 1.2 is still needed in one estimate concerning the relative entropy, yet the \(L^{\infty }\)-norm appears only within a logarithm. While it is unclear to us whether this is essential or just technical necessary in our approach, it allows to derive a time-dependent entropy–entropy production inequality (as a generalisation of the functional inequality (4)) of the form
$$\begin{aligned} D[a(T),b(T)] \ge \Theta (T)(E[a(T),b(T)] - E[a_{\infty },b_{\infty }]) \quad \text { for all }\quad T>0, \end{aligned}$$
(7)
where the function \(\Theta : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) is of order \(1/\ln (1+T)\) and satisfies \(\int _0^{+\infty }\Theta (\tau )d\tau = +\infty \). Thus, a classical Gronwall argument implies explicit algebraic decay of \(E[a(T),b(T)] - E[a_{\infty },b_{\infty }]\) to zero and thus algebraic convergence to equilibrium in relative entropy.

To obtain exponential from algebraic decay, we show that after some sufficiently large time \(T_0>0\), the averages \(\overline{a}_i(T)\) and \(\overline{b}_j(T)\) are bounded below by a positive constant for all \(T\ge T_0\) (since the equilibrium \((a_\infty ,b_\infty )\) consists of positive constants). Hence, for \(T\ge T_0\), we can use the inequalities (6) like in the case for systems with linear diffusion and obtain accordingly exponential convergence to equilibrium. Finally, since \(T_0\) can be explicitly estimated, one recovers global exponential convergence to equilibrium (i.e. for all \(T\ge 0\)) at the price of a smaller, yet explicit constant. Hence, the second main result of this paper is the following theorem.

Theorem 1.3

Let \(\Omega \subset {\mathbb {R}}^d\) be bounded with sufficiently smooth boundary. Consider system (R)—which satisfies the conditions (G), (M) and (P)—subject to non-negative initial data \(a_{i,0},b_{j,0}\in L^\infty (\Omega )\). Assume for all \(i=1\ldots M, j=1\ldots N\) that
$$\begin{aligned} m_i, p_j > \max \{\nu -1;1\}, \qquad \text {where } \quad \nu =\max \Biggl \{\sum _{i=1}^M\alpha _i,\sum _{j=1}^N\beta _j\Biggr \}. \end{aligned}$$
Moreover, in dimensions \(d\ge 3\), we additionally assume
$$\begin{aligned} m_i,p_j>\nu - \frac{4}{d+2}, \qquad \text { for all } \quad i=1\ldots M,\ j=1\ldots N. \end{aligned}$$
Finally, consider a positive initial mass vector \(M_{ij}>0\), which uniquely determines a positive equilibrium \((a_{i\infty },b_{j\infty })\) of system (R).
Then, the bounded global weak solutions of Theorem 1.2 converge exponentially to \((a_\infty ,b_\infty )\) in all \(L^p\)-norms for \(1\le p<\infty ,\) that is
$$\begin{aligned} \sum ^M_{i=1}\Vert a_i(t)-a_{i\infty }\Vert _{L^p(\Omega )} + \sum ^N_{j=1}\Vert b_j(t)-b_{j\infty }\Vert _{L^p(\Omega )} \le C\, e^{-\lambda _pt} \end{aligned}$$
where the constant \(C>0\) and the convergence rate \(\lambda _p>0\) can be computed explicitly.

Remark 1.4

We remark that in Theorem 1.3, we showed the convergence to equilibrium in any \(L^p\)-norm with \(p<\infty \). In the case of linear diffusion, i.e. \(m_i = 1\) for all \(i=1,\ldots , S\), we are able to get the exponential convergence to equilibrium in \(L^\infty \)-norm thanks to the Duhamel formula for semilinear equations, see [16, Proof of Theorem 5.1] (see also [21] for local stability in \(L^\infty \)-norm). This technique is not applicable for nonlinear diffusion, and therefore, the question of global stability in \(L^\infty \)-norm for (S) remains as an interesting open problem.

Notation:
  • We denote by \(\Vert \cdot \Vert \) the usual norm of \(L^2(\Omega )\). For other \(1\le p < +\infty \), we write \(\Vert \cdot \Vert _p\) as the norm of \(L^p(\Omega )\).

  • For any \(T>0\), \(Q_T = \Omega \times (0,T)\) and \(L^p(Q_T) = :L^p(0,T;L^p(\Omega ))\). The space-time norm is defined as usual
    $$\begin{aligned} \Vert f\Vert _{L^p(Q_T)}^p = \int _{0}^T\int _{\Omega }|f(x,t)|^p\hbox {d}x\hbox {d}t. \end{aligned}$$
  • Throughout this work, we will denote by \(C_T\) a generic positive constant which depends on certain parameters, and more importantly \(C_T\) grows at most polynomially, i.e. there exists a polynomial P(x) such that \(C_T \le P(T)\) for all \(T>0\).

Organisation of the paper: Sect. 2 states the proof of Theorem 1.2. The proof of Theorem 1.3 is detailed in Sect. 3. This proof uses also a previously proven entropy–entropy production estimate for reaction–diffusion systems with linear diffusion, which is recalled in Sect. 4 for the sake of completeness. Finally, the existence of global weak solution is stated in Sect. 5.

2 Boundedness and local continuity of weak solutions

In this section, we prove for sufficiently large diffusion exponents \(m_i\) that the weak solutions obtained in Theorem 1.1 are actually bounded in \(L^{\infty }\) and thus locally Hölder continuous. In Lemma 2.1, we devise a bootstrap argument for the inhomogeneous porous media equation which proves that if the porous media exponents \(m_i\) and the initial integrability are high enough, then the weak solutions of Theorem 1.1 satisfy an improve integrability in a space \(L^{s}(Q_T)\) and the \(L^{s}\)-norm grows at most polynomially in time T.

Lemma 2.1

(Smoothing effect of porous medium equation) Suppose that \(m\ge 1\). Assume \(f\in L^{p_0}(Q_T)\) for some \(p_0>1\) with \(\Vert f\Vert _{L^{p_0}(Q_T)} \le C_T\). Let u be the weak solution to the inhomogeneous porous medium equation with positive diffusion coefficient \(\delta >0\)
$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \delta \Delta (|u|^{m-1}u) = f, &{}x\in \Omega , \qquad t>0,\\ \delta \nabla (|u|^{m-1}u)\cdot \overrightarrow{n} = 0, &{}x\in \partial \Omega , \quad \ t>0,\\ u(x,0) = u_0(x), &{}x\in \Omega , \end{array}\right. } \end{aligned}$$
(8)
and subject to initial data \(u_0 \in L^{\infty }(\Omega )\). Then, u satisfies
$$\begin{aligned} \Vert u\Vert _{L^r(Q_T)}\le C_T,\quad \forall r\in [1,s), \end{aligned}$$
where
$$\begin{aligned} s= {\left\{ \begin{array}{ll} +\infty ,\quad &{}\text {if}\quad p_0\ge \frac{d+2}{2},\\ \frac{(md+2)p_0}{d+2-2p_0},\quad &{}\text {if}\quad p_0<\frac{d+2}{2}, \end{array}\right. } \end{aligned}$$
and with a constant \(C_T\), which only depends on \(q, d, m, \Omega \) and at most polynomially on T.

Remark 2.1

In the linear case \(m=1\), Lemma 2.1 recovers the corresponding regularity estimates of the heat equation, see [8]. While the smoothing effect stated in Lemma 2.1 is certainly well known, our main contribution here lies in the polynomial growth in time of the norms, which will be crucial in Sect. 3.

Proof

The existence of the weak solution to (8) can be obtained by standard techniques [43, Chapter 11] so we omit it here. The idea of the proof of this lemma follows [8, Lemma 3.3] and is divided into several steps.

Step 1. Let \(\mu > 1\). By multiplying (8) by \(\mu |u|^{\mu -1}\mathrm {sign}(u)\) (more precisely by multiplying with a smoothed version of the modulus |u| and its derivative \(\mathrm {sign}(u)\) and letting then the smoothing tend to zero) then integrating over \(\Omega \), we obtain
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{\mu }^{\mu } -\delta \mu \int _{\Omega }\Delta (|u|^{m-1}u) |u|^{\mu -1}\mathrm {sign}(u)\hbox {d}x = \mu \int _{\Omega }f |u|^{\mu -1}\mathrm {sign}(u)\hbox {d}x. \end{aligned}$$
(9)
Integration by parts and the homogeneous Neumann boundary condition \(\nabla (|u|^{m-1}u)\cdot \overrightarrow{n} = 0\) lead to
$$\begin{aligned} \begin{aligned}&-\delta \mu \int _{\Omega }\Delta (|u|^{m-1}u) |u|^{\mu -1}\mathrm {sign}(u)\hbox {d}x\\&\quad \ge {m(\mu -1)\mu }\delta \int _{\Omega }|u|^{m+\mu -3}|\nabla u|^2\hbox {d}x + m \mu \delta \int _{\Omega } |u|^{m+\mu -2}|\nabla u|^2 \hbox {d}x\\&\quad \ge \underbrace{\frac{4m(\mu -1)\mu \delta }{(m+\mu -1)^2}}_{=:{C(\mu )}} \int _{\Omega }\left| \nabla \left( |u|^{\frac{m+\mu -1}{2}}\right) \right| ^2\hbox {d}x. \end{aligned} \end{aligned}$$
By Hölder’s inequality
$$\begin{aligned} \left| \mu \int _{\Omega }f |u|^{\mu -1}\mathrm {sign}(u)\hbox {d}x\right| \le \mu \Vert f\Vert _{p_0}\Vert u\Vert _{\frac{p_0(\mu -1)}{p_0-1}}^{\mu -1}. \end{aligned}$$
Therefore, it follows from (9) that
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{\mu }^{\mu } + C(\mu )\int _{\Omega }\left| \nabla \left( |u|^{\frac{m+\mu -1}{2}}\right) \right| ^2\hbox {d}x \le \mu \Vert f\Vert _{p_0}\Vert u\Vert _{\frac{p_0(\mu -1)}{p_0-1}}^{\mu -1}. \end{aligned}$$
(10)
Step 2. Choose \(\mu = p_0>1\) in (10), we get
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{p_0}^{p_0} + C(p_0)\int _{\Omega }\left| \nabla \left( |u|^{\frac{m+p_0-1}{2}}\right) \right| ^2\hbox {d}x \le p_0\Vert f\Vert _{p_0}\Vert u\Vert _{p_0}^{p_0-1}. \end{aligned}$$
(11)
By applying for \(r<1\) the elementary inequality
$$\begin{aligned} y' \le \alpha (t)y^{1-r} \quad \Longrightarrow \quad y(T) \le \left[ y(0)^{r}+r\int _0^T\alpha (t)\hbox {d}t\right] ^{1/r}, \end{aligned}$$
(12)
to (11) with \(r=1/p_0\) and \(y(t) = \Vert u(t)\Vert _{p_0}^{p_0}\), we obtain
$$\begin{aligned} \Vert u(T)\Vert _{p_0}^{p_0}\le & {} \left[ \Vert u_0\Vert _{p_0} + \int _0^{T}\Vert f\Vert _{p_0}\hbox {d}t \right] ^{p_0} \nonumber \\\le & {} \left[ \Vert u_0\Vert _{p_0} + \Vert f\Vert _{L^{p_0}(Q_T)}T^{(p_0-1)/p_0} \right] ^{p_0} =: C_{T,0}. \end{aligned}$$
(13)
That means
$$\begin{aligned} u \in L^{\infty }(0,T;L^{p_0}(\Omega )) \quad \text { and } \quad \Vert u(T)\Vert _{p_0}^{p_0} \le C_{T,0} \end{aligned}$$
(14)
with \(C_{T,0}\) is defined in (13) grows at most polynomially in T. By integrating (11) with respect to t on (0, T) and by using Young’s inequality and the convention \(r_0 {:}{=} m+p_0 - 1>1\), we get
$$\begin{aligned} \begin{aligned} C(p_0)\int _0^T\int _{\Omega }\left| \nabla \left( |u|^{\frac{r_0}{2}}\right) \right| ^2\hbox {d}x\hbox {d}t&\le \Vert u_0\Vert _{p_0}^{p_0} + p_0\int _0^T\Vert f\Vert _{p_0}\Vert u\Vert _{p_0}^{p_0-1}\hbox {d}t\\&\le \Vert u_0\Vert _{p_0}^{p_0} + p_0\Vert f\Vert _{L^{p_0}(Q_T)}\Vert u\Vert _{L^{p_0}(Q_T)}^{p_0-1}. \end{aligned} \end{aligned}$$
By adding \(C(p_0)\int _0^T\int _{\Omega }\left| |u|^{\frac{r_0}{2}}\right| ^2\hbox {d}x\hbox {d}t\) to both sides, we have
$$\begin{aligned} C(p_0)\int _0^T\left\| |u|^{\frac{r_0}{2}}\right\| _{H^1(\Omega )}^2\hbox {d}t&= C(p_0)\int _0^T\left[ \int _{\Omega }\left| \nabla \left( |u|^{\frac{r_0}{2}}\right) \right| ^2\hbox {d}x + \int _{\Omega }\left| |u|^{\frac{r_0}{2}} \right| ^2 \hbox {d}x\right] \hbox {d}t\nonumber \\&\le \Vert u_0\Vert _{p_0}^{p_0} + p_0\Vert f\Vert _{L^{p_0}(Q_T)}\Vert u\Vert _{L^{p_0}(Q_T)}^{p_0-1} \nonumber \\&\quad + C(p_0)\int _0^T\Vert u\Vert _{r_0}^{r_0}\hbox {d}t. \end{aligned}$$
(15)
By the Sobolev’s embedding, we have
$$\begin{aligned} C(p_0)\int _0^T\left\| |u|^{\frac{r_0}{2}}\right\| ^2_{H^1(\Omega )}\ge & {} C(p_0)\,C_S^2\int _0^T\Vert u\Vert _{s_0}^{r_0}\hbox {d}t \quad \text { with }\nonumber \\ s_0= & {} {\left\{ \begin{array}{ll}\frac{r_0d}{d-2} &{}\text { if } d \ge 3,\\ r_0< s_0 < \infty \text { arbitrary } &{}\text { if } d = 1,2.\end{array}\right. } \end{aligned}$$
(16)
On the other hand, by using the bound \(\Vert u(t)\Vert _{p_0}^{p_0} \le C_{T,0}\) in (14) and the interpolation inequality
$$\begin{aligned} \Vert u\Vert _{r_0}\le & {} \Vert u\Vert _{p_0}^{\gamma }\Vert u\Vert _{s_0}^{1-\gamma } \le C_{T,0}^{\gamma /p_0}\Vert u\Vert _{s_0}^{1-\gamma } \quad \text {with} \quad \frac{1}{r_0} = \frac{\gamma }{p_0} + \frac{1-\gamma }{s_0} \quad \text {for} \\ \gamma= & {} \frac{2p_0}{2p_0+(m-1)d}\in (0,1], \end{aligned}$$
we estimate in the cases \(m>1\) for which \(\gamma <1\)
$$\begin{aligned} C(p_0)\int _0^T\Vert u\Vert _{r_0}^{r_0}\hbox {d}t\le & {} C(p_0)\int _{0}^{T}C_{T,0}^{\gamma r_0/p_0} \Vert u\Vert _{s_0}^{(1-\gamma )r_0}\hbox {d}t \nonumber \\\le & {} \frac{C(p_0)\,C_S^2}{2}\int _0^T\Vert u\Vert _{s_0}^{r_0}\hbox {d}t + CC_{T,0}^{{r_0}/p_0}T, \end{aligned}$$
(17)
where we have used Young’s inequality (with the exponents \(1=(1-\gamma ) + \gamma \)) in the last step. Note that if \(m=1\), the bound (17) holds still true yet without the first term and with \(r_0/p_0=1\). Inserting (16) and (17) into (15) leads to
$$\begin{aligned} \int _0^T\Vert u\Vert _{s_0}^{r_0}\hbox {d}t&\le \frac{2}{C(p_0)\,C_S^2}\left[ \Vert u_0\Vert _{p_0}^{p_0} + p_0\Vert f\Vert _{L^{p_0}(Q_T)}\Vert u\Vert _{L^{p_0}(Q_T)}^{p_0-1} + CC_{T,0}^{{r_0}/p_0}T\right] \nonumber \\&\le \frac{2}{C(p_0)\,C_S^2}\left[ \Vert u_0\Vert _{p_0}^{p_0} + p_0\Vert f\Vert _{L^{p_0}(Q_T)}\left( TC_{T,0}\right) ^{\frac{p_0-1}{p_0}} + CC_{T,0}^{{r_0}/p_0}T\right] \nonumber \\&=: D_{T,0} \quad (\text {use } (14)).\end{aligned}$$
(18)
It follows that
$$\begin{aligned} u \in L^{r_0}(0,T;L^{s_0}(\Omega )) \quad \text { with } \quad {\left\{ \begin{array}{ll}s_0 = \frac{r_0d}{d-2} &{}\text { if } d\ge 3,\\ r_0< s_0 <\infty \text { arbitrary } &{}\text { if } d=1,2, \end{array}\right. } \end{aligned}$$
(19)
and
$$\begin{aligned} \int _0^T\Vert u\Vert _{s_0}^{r_0}\hbox {d}t \le D_{T,0} \end{aligned}$$
with \(D_{T,0}\) defined in (18).
Next, we construct a sequence \(p_n \ge 1\) based on the estimate (14) and (19) such that
$$\begin{aligned} \Vert u(T)\Vert _{p_n}^{p_n} \le C_{T,n} \end{aligned}$$
(20)
and
$$\begin{aligned}&\int _0^T\Vert u\Vert _{s_n}^{r_n}\hbox {d}t \le D_{T,n} \quad \text { with }\quad r_n = m + p_n - 1 \quad \text { and } \nonumber \\&\quad {\left\{ \begin{array}{ll}s_n = \frac{r_nd}{d-2} &{}\text { if } d\ge 3,\\ r_n< s_n <\infty \text { arbitrary } &{} \text { if } d=1,2, \end{array}\right. } \end{aligned}$$
(21)
in which \(C_{T,n}\) and \(D_{T,n}\) are constants growing at most polynomially in T.
Step 3 (Iteration of (20)). In (10), we set \(\mu = p_{n+1}\) for \(p_{n+1}\) to be chosen later. Thus, we have
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{{p_{n+1}}}^{{p_{n+1}}} + C({p_{n+1}})\int _{\Omega }\left| \nabla \left( |u|^{\frac{r_{n+1}}{2}}\right) \right| ^2\hbox {d}x \le p_{n+1}\Vert f\Vert _{p_0}\Vert u\Vert _{\frac{p_0({p_{n+1}}-1)}{p_0-1}}^{{p_{n+1}}-1}, \end{aligned}$$
(22)
where we recall that \(r_{n+1} = m + p_{n+1}-1\). By \(L^p\)- interpolation, we have
$$\begin{aligned} \Vert u\Vert _{\frac{p_0({p_{n+1}}-1)}{p_0-1}} \le \Vert u\Vert _{p_{n+1}}^{1-\theta }\Vert u\Vert _{s_n}^{\theta } \end{aligned}$$
and where \(p_{n+1}>1\) has to be chosen such that \(\frac{p_0(p_{n+1}-1)}{p_0-1}\in (p_{n+1},s_n)\) with \(p_{n+1}<s_n\), which entails \(\theta \in (0,1)\) in
$$\begin{aligned} \frac{p_0-1}{p_0(p_{n+1}-1)} = \frac{1-\theta }{p_{n+1}} + \frac{\theta }{s_n}. \end{aligned}$$
(23)
Note that \(\frac{p_0(p_{n+1}-1)}{p_0-1} > p_{n+1}\) is always satisfied provided that \(p_{n+1}>p_0\), i.e. that the sequence \(p_n\) is strictly monotone increasing.
It then follows from (22) (by neglecting the second term on the left-hand side) that
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\Vert u\Vert _{p_{n+1}}^{p_{n+1}} \le p_{n+1}\Vert f\Vert _{p_0}\Vert u\Vert _{s_n}^{\theta (p_{n+1}-1)} \left( \Vert u\Vert _{p_{n+1}}^{p_{n+1}}\right) ^{1 - \frac{1+\theta (p_{n+1}-1)}{p_{n+1}}}. \end{aligned}$$
By applying again the elementary inequality (12) with \(y(t) = \Vert u(t)\Vert _{p_{n+1}}^{p_{n+1}}\) and \(r = \frac{1+\theta (p_{n+1}-1)}{p_{n+1}} < 1\), it yields
$$\begin{aligned} \begin{aligned}&\Vert u(T)\Vert _{p_{n+1}}^{p_{n+1}}\\&\quad \le \left[ \Vert u_0\Vert _{p_{n+1}}^{1+\theta (p_{n+1}-1)} + (1+\theta (p_{n+1}-1))\int _0^T\Vert f\Vert _{p_0}\Vert u\Vert _{s_n}^{\theta (p_{n+1}-1)}\hbox {d}t \right] ^{\frac{p_{n+1}}{1+\theta (p_{n+1}-1)}}\\&\quad \le \left[ \Vert u_0\Vert _{p_{n+1}}^{1+\theta (p_{n+1}-1)} + (1+\theta (p_{n+1}-1))\Vert f\Vert _{L^{p_0}(Q_T)}\right. \\&\qquad \left. \biggl (\int _0^T\Vert u\Vert _{s_n}^{\theta (p_{n+1}-1) \frac{p_0}{p_0-1}}\hbox {d}t\biggr )^{\frac{p_0-1}{p_0}} \right] ^{\frac{p_{n+1}}{1+\theta (p_{n+1}-1)}}. \end{aligned} \end{aligned}$$
(24)
In order to continue estimating by using (21), we choose \(p_{n+1}\) as
$$\begin{aligned} \theta (p_{n+1}-1)\frac{p_0}{p_0-1} = r_n. \end{aligned}$$
(25)
Since \(r_n= s_n\frac{d-2}{d}\), Eq. (25) implies \( \frac{\theta }{s_n} = (1-\frac{2}{d})\frac{p_0-1}{p_0(p_{n+1}-1)}\) and thus with (23)
$$\begin{aligned} \theta = 1 - \frac{2}{d} \frac{p_0-1}{p_0}\frac{p_{n+1}}{p_{n+1}-1}<1. \end{aligned}$$
(26)
In order to verify that above choice of \(p_{n+1}\) satisfies \(\frac{p_0(p_{n+1}-1)}{p_0-1}<s_n\), we insert (26) into (25) and calculate
$$\begin{aligned}&(p_{n+1}-1)\frac{p_0}{p_0-1}-\frac{2}{d} p_{n+1} = s_n \frac{d-2}{d} \quad \\&\quad \Rightarrow \quad s_n-\frac{p_0(p_{n+1}-1)}{p_0-1} = \frac{2}{d}(s_n-p_{n+1})>0. \end{aligned}$$
Similar, by recalling \(s_n \frac{d-2}{d}=r_n=m-1+p_n\), we get the iteration
$$\begin{aligned} p_{n+1} = p_n\frac{d(p_0-1)}{p_0(d-2) + 2} + \frac{d[(m-1)(p_0-1) + p_0]}{p_0(d-2)+2}. \end{aligned}$$
(27)
Altogether, by inserting (25) into (24), we obtain thanks to (21)
$$\begin{aligned} \begin{aligned} \Vert u(T)\Vert _{p_{n+1}}^{p_{n+1}}&\le \left[ \Vert u_0\Vert _{p_{n+1}}^{^{1+\theta (p_{n+1}-1)}} + (1+\theta (p_{n+1}-1))\right. \\&\quad \left. \Vert f\Vert _{L^{p_0}(Q_T)}\left( \int _0^T\Vert u\Vert _{s_n}^{r_n} \hbox {d}t\right) ^{\frac{p_0-1}{p_0}} \right] ^{\frac{p_{n+1}}{1+\theta (p_{n+1}-1)}}\\&\le \left[ \Vert u_0\Vert _{p_{n+1}}^{^{1+\theta (p_{n+1}-1)}} + (1+\theta (p_{n+1}-1))\right. \\&\quad \left. \Vert f\Vert _{L^{p_0}(Q_T)}D_{T,n}^{\frac{p_0-1}{p_0}} \right] ^{\frac{p_{n+1}}{1+\theta (p_{n+1}-1)}} =: C_{T,n+1} \end{aligned} \end{aligned}$$
(28)
and thus
$$\begin{aligned} u \in L^{\infty }(0,T;L^{p_{n+1}}(\Omega )) \quad \text { and } \quad \Vert u(T)\Vert _{p_{n+1}}^{p_{n+1}} \le C_{T,n+1}. \end{aligned}$$
(29)
Step 4 (Iteration of (21)). We will use similar arguments to Step 2. Integrating (22) and adding \(\int _0^T\int _{\Omega }\left| |u|^{\frac{r_{n+1}}{2}}\right| ^2\hbox {d}x\hbox {d}t\) to both sides yields in particular
$$\begin{aligned}&C(p_{n+1})\int _0^T\left\| |u|^{\frac{r_{n+1}}{2}}\right\| _{H^1(\Omega )}^2\hbox {d}t = C({p_{n+1}})\int _0^T\int _{\Omega } \left[ \left| \nabla \left( |u|^{\frac{r_{n+1}}{2}}\right) \right| ^2\hbox {d}x \right. \nonumber \\&\qquad \left. + \left| |u|^{\frac{r_{n+1}}{2}}\right| ^2\hbox {d}x\right] \hbox {d}t&\nonumber \\&\quad \le \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} + p_{n+1}\int _0^T\Vert f\Vert _{p_0} \Vert u\Vert _{\frac{p_0({p_{n+1}}-1)}{p_0-1}}^{{p_{n+1}}-1}\hbox {d}t \nonumber \\&\qquad + C({p_{n+1}}) \int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t&\nonumber \\&\quad \le \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +p_{n+1}\int _0^T\Vert f\Vert _{p_0} \Vert u\Vert _{s_n}^{\theta (p_{n+1}-1)}\Vert u\Vert _{p_{n+1}}^{(1-\theta )(p_{n+1}-1)}\hbox {d}t \nonumber \\&\qquad + C({p_{n+1}})\int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t \qquad \qquad (\theta \text { in } (23))\nonumber \\&\quad \le \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +{p_{n+1}}{C_{T,n+1}^{(1-\theta ) \frac{(p_{n+1}-1)}{p_{n+1}}}}\int _0^T\Vert f\Vert _{p_0}\Vert u\Vert _{s_n}^{\theta (p_{n+1}-1)}\hbox {d}t\nonumber \\&\qquad + C({p_{n+1}})\int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t \qquad \qquad (\text {using } (29))\nonumber \\&\quad \le \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +{p_{n+1}}{C_{T,n+1}^{(1-\theta ) \frac{(p_{n+1}-1)}{p_{n+1}}}}\Vert f\Vert _{L^{p_0}(Q_T)}\biggl (\int _0^T\Vert u\Vert _{s_n}^{r_n}\hbox {d}t \biggr )^{\frac{p_0-1}{p_0}} \nonumber \\&\qquad + C({p_{n+1}})\int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t\qquad \qquad (\text {using } (25))\nonumber \\&\quad \le \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +{p_{n+1}}{C_{T,n+1}^{(1-\theta )\frac{(p_{n+1}-1)}{p_{n+1}}}}\Vert f\Vert _{L^{p_0}(Q_T)} D_{T,n}^{\frac{p_0-1}{p_0}}\nonumber \\&\qquad + C({p_{n+1}})\int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t \qquad \qquad (\text {using } (21)). \end{aligned}$$
(30)
Now by Sobolev’s embedding
$$\begin{aligned}&C(p_{n+1})\int _0^T\left\| |u|^{\frac{r_{n+1}}{2}}\right\| _{H^1(\Omega )}^2\hbox {d}t \ge C(p_{n+1})\,C_S^2\int _0^T\Vert u\Vert _{s_{n+1}}^{r_{n+1}}\hbox {d}t\nonumber \\&\quad \text { with } \quad s_{n+1} = {\left\{ \begin{array}{ll}\frac{r_{n+1}d}{d-2} &{}\text { if } d \ge 3,\\ r_{n+1}< s_{n+1} < \infty \text { arbitrary } &{}\text { if } d = 1,2.\end{array}\right. } \end{aligned}$$
(31)
By the bound \(\Vert u(t)\Vert _{p_{n+1}}^{p_{n+1}} \le C_{T,n+1}\), the interpolation inequality
$$\begin{aligned}&\Vert u\Vert _{r_{n+1}} \le \Vert u\Vert _{p_{n+1}}^{\gamma }\Vert u\Vert _{s_{n+1}}^{1-\gamma } \le C_{T,n+1}^{\gamma /p_{n+1}}\Vert u\Vert _{s_{n+1}}^{1-\gamma }\nonumber \\&\quad \text { with } \quad \frac{1}{r_{n+1}} = \frac{\gamma }{p_{n+1}} + \frac{1-\gamma }{s_{n+1}} \quad \text {for}\quad \gamma = \frac{2p_{n+1}}{2p_{n+1} + (m-1)d}\in (0,1].\quad \quad \end{aligned}$$
(32)
Like in Step 2 in case \(m>1\) and \(\gamma <1\), we have by Young’s inequality,
$$\begin{aligned} \begin{aligned} C(p_{n+1})\int _0^T\Vert u\Vert _{r_{n+1}}^{r_{n+1}}\hbox {d}t&\le C(p_{n+1})\int _{0}^{T}C_{T,n+1}^{\gamma r_{n+1}/p_{n+1}}\,\Vert u\Vert _{s_{n+1}}^{(1-\gamma )r_{n+1}}\hbox {d}t\\&\le \frac{C(p_{n+1})\,C_S^2}{2}\int _{0}^{T}\Vert u\Vert _{s_{n+1}}^{r_{n+1}}\hbox {d}t + CTC_{T,n+1}^{{r_{n+1}}/p_{n+1}} \end{aligned} \end{aligned}$$
analogue to (17) while the case \(m=1\) and \(r_{n+1}/p_{n+1}=1\) follows without interpolation and the first term on the right-hand side above. Combining (30), (31) and (32) yields
$$\begin{aligned} \frac{C(p_{n+1})\,C_S^2}{2}\int _0^T\Vert u\Vert _{s_{n+1}}^{r_{n+1}}\hbox {d}t\le & {} \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +{p_{n+1}}{C_{T,n+1}^{(1-\theta )\frac{(p_{n+1}-1)}{p_{n+1}}}}\Vert f\Vert _{L^{p_0}(Q_T)}D_{T,n}^{\frac{p_0-1}{p_0}}\\&+ CTC_{T,n+1}^{{r_{n+1}}/p_{n+1}}, \end{aligned}$$
hence
$$\begin{aligned} \int _0^T\Vert u\Vert _{s_{n+1}}^{r_{n+1}}\hbox {d}t \le D_{T,n+1} \end{aligned}$$
with
$$\begin{aligned} D_{T,n+1}&{:}{=} \frac{2}{C(p_{n+1})C_S^2}\left[ \Vert u_0\Vert _{p_{n+1}}^{p_{n+1}} +{p_{n+1}}{C_{T,n+1}^{(1-\theta )\frac{(p_{n+1}-1)}{p_{n+1}}}} \Vert f\Vert _{L^{p_0}(Q_T)}D_{T,n}^{\frac{p_0-1}{p_0}}\right. \nonumber \\&\quad \left. + CTC_{T,n+1}^{{r_{n+1}}/p_{n+1}}\right] . \end{aligned}$$
(33)
Step 5. Passing to the limit as \(n\rightarrow \infty \). Considering the iteration (27), the only possible fixed point \(p_{\infty }\) of the sequence \(p_n\) is
$$\begin{aligned} p_{\infty } = \frac{d[(m-1)(p_0-1) + p_0]}{2[\frac{d+2}{2}-p_0]}. \end{aligned}$$
Hence, \(p_{\infty }<0\) if and only if \(p_0 > \frac{d+2}{2}\). In particular, it is straightforward to check that the sequence \(p_n\) defined by (27) is strictly monotone increasing if and only if either \(p_n<p_{\infty }\) in the case \(p_0<\frac{d+2}{2}\) or \(p_n>p_{\infty }\) in the case \(p_0>\frac{d+2}{2}\) when \(p_{\infty }<0\) holds or \(p_0=\frac{d+2}{2}\) where \(p_{\infty }=+\infty \).
Therefore, we have as \(n\rightarrow \infty \)
$$\begin{aligned} p_n \longrightarrow {\left\{ \begin{array}{ll} p_{\infty } \quad \text { if } \quad p_0 < \frac{d+2}{2},\\ +\infty \quad \text { if } \quad p_0 \ge \frac{d+2}{2}. \end{array}\right. } \end{aligned}$$
Step 6 (Interpolation). From (20) and (21) and by using the interpolation
$$\begin{aligned} L^{\infty }(0,T;L^{p_n}(\Omega ))\cap L^{r_n}(0,T;L^{s_n}(\Omega )) \hookrightarrow L^{\frac{d+2}{d}p_n + m-1}(Q_T) \end{aligned}$$
we get \(u\in L^{r}(Q_T)\) for all \(r< \infty \) in the case \(p_0 \ge \frac{d+2}{2}\). In the case \(p_0 < \frac{d+2}{2}\), we obtain \(u\in L^s(Q_T)\) for all
$$\begin{aligned} s < \frac{d+2}{d}p_{\infty } + m - 1 = \frac{(md+2)p_0}{d+2-2p_0}. \end{aligned}$$
This completes the proof of Lemma 2.1. \(\square \)

Lemma 2.2

Let u be a weak solution to (S) and
$$\begin{aligned} \Vert u\Vert _{L^{q_0}(Q_T)}\le C_T,\quad \forall i=1,\ldots ,S,\quad \text {with}\quad q_0>\frac{d(\nu -m)+2(\nu -1)}{2}, \end{aligned}$$
where \(m = \min \{m_i: i=1\ldots S\}\), \(\nu \) is defined in (G), and \(C_T\) is growing at most polynomially in T.

Then, it follows that \(\Vert u_i\Vert _{L^\infty (Q_T)}\le C_T\) for all \(i=1\ldots S\).

Proof

From \(u_i \in L^{q_0}(Q_T)\) for all \(i=1,\ldots , S\), we have \(f_i(u)\in L^{q_0/\nu }(Q_T)\). Moreover, note that the quasi-positivity assumption (P) ensures non-negative solutions u for non-negative initial data \(u_{i,0}\). Hence, the concentrations \(u_i\) satisfy the (non-sign-changing) porous media equation
$$\begin{aligned} \partial _t u_i - d_i\Delta (u_i^{m_i}) = f_i(u) \in L^{q_0/\nu }(Q_T). \end{aligned}$$
Lemma 2.1 implies that if \(q_0/\nu \ge \frac{d+2}{2}\), then \(u_i\in L^{r}(Q_T)\) for all \(r<\infty \), while if \(q_0/\nu < \frac{d+2}{2}\), then
$$\begin{aligned}&u_i \in L^{s}(Q_T) \quad \text { for all } \quad s < q_1:= \frac{(md+2)q_0}{\nu (d+2)-2q_0} \\&\quad \le \frac{(m_id+2)q_0}{\nu (d+2)-2q_0}, \qquad \text {for all } i=1\ldots S, \end{aligned}$$
since \(m\le m_i\). We then construct a sequence \(q_n\) (equally for all \(i=1,\ldots ,S\)) such that
$$\begin{aligned} q_{n+1} = \frac{(md+2)q_n}{\nu (d+2)-2q_n} \quad \text { for } n\ge 0. \end{aligned}$$
(34)
It follows that
$$\begin{aligned} \frac{q_{n+1}}{q_n} = \frac{md+2}{\nu (d+2)-2q_n}. \end{aligned}$$
Therefore, as long as \(\nu (d+2)-2q_n>0 \iff q_n < \frac{(d+2)\nu }{2}\),
$$\begin{aligned} \frac{q_{n+1}}{q_n}> 1 \text { for all } n \ge 0 \quad \Longleftrightarrow \quad q_0 > \frac{d(\nu -m)+2(\nu -1)}{2}. \end{aligned}$$
Hence with \(q_0 > \frac{d(\nu -m)+2(\nu -1)}{2}\), after finitely many steps, we arrive at \(q_{n} > \frac{(d+2)\nu }{2}\). From \(u_i\in L^{s}(Q_T)\) for all \(s<q_n\), we have in particular \(u_i\in L^{\frac{(d+2)\nu }{2}}(Q_T)\), which implies \(f_i(u)\in L^{\frac{d+2}{2}}(Q_T)\) for \(i=1,\ldots , S\). By applying Lemma 2.1 once more, we obtain \(u_i \in L^r(Q_T)\) for all \(r, q < \infty \). Thus,
$$\begin{aligned} \partial _t u_i - d_i\Delta (u_i^{m_i}) = f_i(u)\in L^{r}(Q_T) \quad \text { for all } s<\infty \end{aligned}$$
with \(\Vert f_i(u)\Vert _{L^{r}(Q_T)} \le C_T\) for some \(r > \frac{d+2}{2}\). Therefore,
$$\begin{aligned} \Vert u_i\Vert _{L^\infty (Q_T)} \le C_T \quad \text { for all } \quad i=1,\ldots , S, \end{aligned}$$
thanks to the following Lemma 2.3. \(\square \)

Lemma 2.3

Let u be the solution to
$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \delta \Delta (|u|^{m-1}u) = f, &{}\quad (x,t)\in Q_T,\\ \nabla (|u|^{m-1}u) \cdot \overrightarrow{n} = 0, &{}\quad (x,t) \in \partial \Omega \times (0,T),\\ u(x,0) = u_0(x), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$
with \(u_0\in L^\infty (\Omega )\) and \(\Vert f\Vert _{L^q(Q_T)} \le C_T\) for some \(q > \frac{d+2}{2}\). Then,
$$\begin{aligned} \Vert u\Vert _{L^\infty (Q_T)} \le C_T. \end{aligned}$$
(35)

Though the boundedness result of this Lemma has been cited in many works, we are unable to find a precise reference. We therefore give in this paper a full proof based on the famous Moser iteration. Moreover, our proof shows the polynomial growth of the \(L^\infty \)-norm in (35), which is important for our sequel analysis.

To prove Lemma 2.3, we need the following two lemmas.

Lemma 2.4

[7, Lemma 2.5] Let \(\{y_n\}_{n\ge 1}\) be a sequence of positive numbers which satisfies
$$\begin{aligned} y_{n+1} \le KB^n(y_{n}^{\gamma } + y_n^{\kappa }) \end{aligned}$$
where \(K, B>0\) and \(\gamma , \kappa > 1\) are independent of n. Then there exists \(\varepsilon >0\) such that, if \(y_1 \le \varepsilon \), then
$$\begin{aligned} \lim _{n\rightarrow \infty }y_{n} = 0. \end{aligned}$$

Lemma 2.5

[27, II.§3] Define
$$\begin{aligned} W(0,T)&{:}{=} \left\{ u: Q_T \rightarrow {\mathbb {R}} \; \text { such that } \; \Vert u\Vert _{W(0,T)}^2{:}{=} \sup _{t\in (0,T)}\Vert u(t)\Vert ^2 \right. \\&\quad \left. + \int _0^T\Vert u(t)\Vert _{H^1(\Omega )}^2\hbox {d}t < +\infty \right\} . \end{aligned}$$
For pq satisfying
$$\begin{aligned} \frac{1}{p} + \frac{d}{2q} = \frac{d}{4}, \end{aligned}$$
there exists a constant C independent of T such that
$$\begin{aligned} \Vert u\Vert _{L^{p}(0,T;L^q(\Omega ))} \le C\Vert u\Vert _{W(0,T)}. \end{aligned}$$
In particular, when \(p = q = 2 +\frac{4}{d}\),
$$\begin{aligned} \Vert u\Vert _{L^{2 +\frac{4}{d}}(Q_T)} \le C\Vert u\Vert _{W(0,T)}. \end{aligned}$$

Proof of Lemma 2.3

Let \(k\ge 1\) be a constant which will be specified later. For each \(i\ge 0\), we define
$$\begin{aligned} v_i\,{:}{=} \left( u - k + \frac{k}{2^i}\right) _+ = \max \left\{ u - k + \frac{k}{2^i}; 0\right\} \end{aligned}$$
and
$$\begin{aligned} A_i\,{:}{=} \left\{ (x,t) \in Q_T: \; u(x,t) \ge k-\frac{k}{2^i} \right\} . \end{aligned}$$
The following simple observations will be helpful
$$\begin{aligned} \begin{aligned} v_{i+1}(x,t)&\le v_i(x,t)&\text { for all } (x,t) \in A_i,\\ v_i(x,t)&\ge \frac{k}{2^{i+1}}&\text { for all } (x,t) \in A_{i+1}\subset A_i. \end{aligned} \end{aligned}$$
(36)
By multiplying the equation \(\partial _t u - \delta \Delta (|u|^{m-1}u) = f\) by \(v_{i+1}\) and integrating on \(Q_T\), we have
$$\begin{aligned}&\sup _{t\in (0,T)}\Vert v_{i+1}(t)\Vert ^2 + 2\delta m\int _0^T\int _{\Omega }|u|^{m-1}|\nabla v_{i+1}|^2\hbox {d}x\hbox {d}t \nonumber \\&\quad \le \Vert v_{i+1}(0)\Vert ^2 + 2\int _0^T\int _{\Omega }fv_{i+1}\hbox {d}x\hbox {d}t. \end{aligned}$$
(37)
Note that \(u\ge k - \frac{k}{2^i} \ge \frac{k}{2}\) on \(A_i\), we have
$$\begin{aligned} \begin{aligned} 2\delta m \int _0^T\int _{\Omega }|u|^{m-1}|\nabla v_{i+1}|^2\hbox {d}x&\ge 2\delta m \iint _{A_i}|u|^{m-1}|\nabla v_{i+1}|^2\hbox {d}x \\&\ge \delta m \frac{k^{m-1}}{2^{m-2}} \iint _{A_i}|\nabla v_{i+1}|^2 \hbox {d}x\hbox {d}t\\&\ge \frac{\delta m }{2^{m-2}} \int _0^T\int _{\Omega }|\nabla v_{i+1}|^2 \hbox {d}x\hbox {d}t \end{aligned} \end{aligned}$$
thanks to \(k\ge 1\), and the fact that \(v_{i+1} \equiv 0\) on \(Q_T \backslash A_{i+1}\supset Q_T \backslash A_{i}\) since \(A_{i+1} \subset A_{i}\). By adding \(\int _0^T\Vert v_{i+1}\Vert ^2\hbox {d}t\) to both sides of (37), we get
$$\begin{aligned}&\sup _{t\in (0,T)}\Vert v_{i+1}(t)\Vert ^2 + \frac{\delta m}{2^{m-2}} \int _0^T\Vert v_{i+1}\Vert _{H^1(\Omega )}^2 \hbox {d}x\hbox {d}t \\&\quad \le C\int _0^T\Vert v_{i+1}\Vert ^2\hbox {d}t + \Vert v_{i+1}(0)\Vert ^2 + \int _0^T\int _{\Omega }fv_{i+1}\hbox {d}x\hbox {d}t. \end{aligned}$$
which yields
$$\begin{aligned} C\Vert v_{i+1}\Vert _{W(0,T)}^2 \le \int _0^T\Vert v_{i+1}\Vert ^2\hbox {d}t + \Vert v_{i+1}(0)\Vert ^2 + \int _0^T\int _{\Omega }fv_{i+1}\hbox {d}x\hbox {d}t. \end{aligned}$$
(38)
By definition,
$$\begin{aligned} \Vert v_{i+1}(0)\Vert ^2 = \left\| \left( u_0 - k + \frac{k}{2^{i+1}}\right) _+\right\| ^2 = 0 \end{aligned}$$
(39)
when we choose \(k\ge 2\Vert u_0\Vert _{L^\infty (\Omega )}\). By using (36), we have with \(1\le \frac{2^{i+1}}{k} v_i\) on \(A_{i+1}\)
$$\begin{aligned} \begin{aligned} \int _0^T\int _{\Omega }|v_{i+1}|^2\hbox {d}x\hbox {d}t&= \int _0^T\int _{\Omega }{\mathbf {1}}_{A_{i+1}}|v_{i+1}|^2\hbox {d}x\hbox {d}t\\&\le \int _0^T\int _{\Omega }{\mathbf {1}}_{A_{i+1}} |v_i|^2 \hbox {d}x\hbox {d}t\\&\le \left( \frac{2^{i+1}}{k}\right) ^{\frac{4}{d}}\int _{0}^T\int _{\Omega } {\mathbf {1}}_{A_{i+1}}|v_i|^{2+\frac{4}{d}}\hbox {d}x\hbox {d}t\\&\le C(2^{4/d})^{i}\Vert v_i\Vert _{W(0,T)}^{2+\frac{4}{d}}. \end{aligned} \end{aligned}$$
(40)
Since \(q > \frac{d+2}{2}\), we have
$$\begin{aligned} \sigma {:=}\, \frac{q-1}{q}\left( 2+\frac{4}{d}\right) > 2. \end{aligned}$$
Moreover,
$$\begin{aligned} \frac{\sigma q}{q-1} = 2 + \frac{4}{d} \end{aligned}$$
thus
$$\begin{aligned} \Vert v_i\Vert _{L^{\frac{\sigma q}{q-1} }(Q_T)} \le C\Vert v_i\Vert _{W(0,T)}. \end{aligned}$$
We now can use Hölder’s inequality to estimate with (36)
$$\begin{aligned} \int _0^T\int _{\Omega }fv_{i+1}\hbox {d}x\hbox {d}t&\le \int _0^T\int _{\Omega }fv_{i+1} \left( \frac{2^{i+1}}{k}\right) ^{\sigma -1} v_i^{\sigma -1} \hbox {d}x\hbox {d}t \nonumber \\&\le \left( \frac{2^{i+1}}{k}\right) ^{\sigma -1}\int _0^T\int _{\Omega }|f||v_i|^{\sigma }\hbox {d}x\hbox {d}t\nonumber \\&\le C(2^{\sigma -1})^i\Vert f\Vert _{L^q(Q_T)}\Vert v_i\Vert _{L^{\frac{\sigma q}{q-1}}(Q_T)}^{\sigma } \nonumber \\&\le C(2^{\sigma -1})^i\Vert f\Vert _{L^q(Q_T)}\Vert v_i\Vert _{W(0,T)}^{\sigma }. \end{aligned}$$
(41)
Inserting (38), (39) and (41) into (37) leads to
$$\begin{aligned} \Vert v_{i+1}\Vert _{W(0,T)}^2 \le C(1+\Vert f\Vert _{L^q(Q_T)})B^{i}(\Vert v_i\Vert _{W(0,T)}^{2+\frac{4}{d}} + \Vert v_i\Vert _{W(0,T)}^{\sigma }) \end{aligned}$$
(42)
for all \(i\ge 0\), where \(B = \max \{2^{4/d}; 2^{\sigma - 1}\}\). By setting \(Y_i = \Vert v_i\Vert _{W(0,T)}^2\), we obtain a sequence \(\{Y_n\}_{n\ge 1}\) satisfying the property in Lemma 2.4. It remains to show that \(Y_1\) is small enough.
We show now that for any \(\varepsilon > 0\), there exists \(k\ge \max \{1; 2\Vert u_0\Vert _{L^\infty (\Omega )}\}\) large enough such that
$$\begin{aligned} Y_1 = \Vert v_1\Vert _{W(0,T)} \le \varepsilon . \end{aligned}$$
(43)
From Step 2 in the proof of Lemma 2.1, we have
$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;L^q(\Omega ))} + \Vert u\Vert _{L^r(0,T;L^s(\Omega ))} \le C_T \end{aligned}$$
where \(r = m+q-1\ge q\) and \(s = \frac{rd}{d-2}\) if \(d\ge 2\) and \({r<}s<+\infty \) arbitrary if \(d\le 2\). By interpolation, see e.g. [15, Lemma 4.1], we see that
$$\begin{aligned} \Vert u\Vert _{L^{\tau }(Q_T)} \le C_T \quad \text { with } \quad \tau = {\left\{ \begin{array}{ll}\frac{dr+2q}{d} &{}\text { if } d\ge 3,\\ <r+q \text { arbitrary } &{}\text { if } d\le 2. \end{array}\right. } \end{aligned}$$
Direct calculations show that \(\tau > 2 + \frac{4}{d}\) if \(d\ge 2\) and \(\tau > 3\) if \(d=1\). In particular,
$$\begin{aligned} \Vert u\Vert _{L^{2+\frac{4}{d}}(Q_T)} \le C_T \text { for } d\ge 2 \quad \text { and } \quad \Vert u\Vert _{L^{3}(Q_T)} \le C_T \text { for } d=1. \end{aligned}$$
(44)
From (38),
$$\begin{aligned} C\Vert v_1\Vert _{W(0,T)}^2 \le \int _0^T\Vert v_1(t)\Vert ^2\hbox {d}t + \Vert v_1(0)\Vert ^2 + \int _0^T\int _{\Omega }fv_1\hbox {d}x\hbox {d}t. \end{aligned}$$
(45)
Since \(k\ge 2\Vert u_0\Vert _{L^\infty (\Omega )}\), \(\Vert v_1(0)\Vert ^2 = \Vert (u_0 - k/2)_+\Vert ^2 = 0\).
Consider now the case \(d\ge 2\). By using (36), it yields
$$\begin{aligned} \int _0^T\int _{\Omega }|v_1|^2\hbox {d}x\hbox {d}t= & {} \int _0^T\int _{\Omega }{\mathbf {1}}_{A_1}|v_1|^2\hbox {d}x\hbox {d}t \le \left( \frac{4}{k}\right) ^{\frac{4}{d}}\int _0^T\int _{\Omega }|v_0|^{2+\frac{4}{d}}\hbox {d}x\hbox {d}t \nonumber \\\le & {} \left( \frac{4}{k}\right) ^{\frac{4}{d}}\Vert u\Vert _{L^{2+\frac{4}{d}}}^{2+\frac{4}{d}} \le \left( \frac{4}{k}\right) ^{\frac{4}{d}}C_T, \end{aligned}$$
(46)
recalling that \(v_0 = u_+\). Similarly to (41), we get
$$\begin{aligned} \int _0^T\int _{\Omega }fv_1\hbox {d}x\hbox {d}t \le \left( \frac{4}{k}\right) ^{\sigma - 1}\Vert f\Vert _{L^q(Q_T)}\Vert u\Vert _{L^{2+\frac{4}{d}}(Q_T)}^{\sigma } \le \left( \frac{4}{k}\right) ^{\sigma - 1}C_T. \end{aligned}$$
(47)
From (42), (45) and (46), we get (43) if
$$\begin{aligned} k = 4\max \left\{ \left( \frac{C_T}{\varepsilon }\right) ^{\frac{d}{4}}; \left( \frac{C_T}{\varepsilon }\right) ^{\frac{1}{\sigma -1}}\right\} . \end{aligned}$$
Thus, with this choice of k, it follows that
$$\begin{aligned} 0 = \lim _{i\rightarrow \infty }Y_i = \Vert (u - k)_+\Vert ^2, \end{aligned}$$
and hence,
$$\begin{aligned} \Vert u\Vert _{L^\infty (Q_T)} \le k = 4\max \left\{ \left( \frac{C_T}{\varepsilon }\right) ^{\frac{d}{4}}; \left( \frac{C_T}{\varepsilon }\right) ^{\frac{1}{\sigma -1}}\right\} \end{aligned}$$
which is our desired estimate.
The proof for the case \(d=1\) is very similar using the
$$\begin{aligned} \int _0^T\int _{\Omega }|v_1|^2\hbox {d}x\hbox {d}t \le \frac{4}{k}\int _0^T\int _{\Omega }|v_0|^3\hbox {d}x\hbox {d}t \le \frac{4}{k}C_T \end{aligned}$$
and
$$\begin{aligned} \int _0^T\int _{\Omega }fv_1\hbox {d}x\hbox {d}t \le \left( \frac{4}{k}\right) ^{\frac{4\xi }{1+2\xi }}\Vert f\Vert _{L^q(Q_T)} \Vert u\Vert _{L^3(Q_T)}^{1+\frac{4\xi }{1+2\xi }} \le \left( \frac{4}{k}\right) ^{\frac{4\xi }{1+2\xi }}C_T \end{aligned}$$
where \(\xi = \frac{1}{2}(2q - 3) > 0\). We therefore omit the details. \(\square \)

Now, we are ready to prove the boundedness of solutions to (S):

Proof of Theorem 1.2

Assuming \(m_i > \nu - 1\), the existence of weak solutions follows similar to [26, 38] and is proven in Sect. 5 in detail. By the duality estimates in Lemma 5.1, we have
$$\begin{aligned} u_i \in L^{m_i+1}(Q_T) \quad \text { for all}\quad i=1,\ldots , S. \end{aligned}$$
Because \(m_i > \nu - \frac{4}{2+d}\), it follows that
$$\begin{aligned} m_i+1 > \frac{d(\nu -m_i)+2(\nu -1)}{2}. \end{aligned}$$
Therefore, Lemma 2.2 yields \(u_i \in L^{\infty }(Q_T)\) and \(\Vert u_i\Vert _{L^{\infty }(Q_T)} \le C_T\) for arbitrary \(T>0\), which shows that the weak solutions are bounded and the \(L^{\infty }(\Omega )\) norms grows at most polynomially in time.

The local Hölder continuity of the bounded weak solutions is a classical result, see e.g. [9] or [43, Theorem 7.18]. \(\square \)

3 Convergence to equilibrium

In this section, we prove exponential convergence to equilibrium of solutions to (R) by using the entropy method. We start by recalling the entropy (free energy) functional
$$\begin{aligned} E[a,b] = \sum _{i=1}^{M}\int _{\Omega }(a_i\ln {a_i} - a_i + 1)\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }(b_j\ln {b_j} - b_j + 1)\hbox {d}x \end{aligned}$$
and its non-negative entropy production (free energy dissipation) functional \(D[a,b] {:=} -\frac{\hbox {d}}{\hbox {d}t}E[a,b]\), i.e.
$$\begin{aligned} D[a,b] = \sum _{i=1}^{M}d_i\int _{\Omega }\frac{|\nabla a_i|^2}{a_i^{2-m_i}}\hbox {d}x + \sum _{j=1}^{N}h_j\int _{\Omega }\frac{|\nabla b_j|^2}{b_j^{2-p_j}}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta )\ln {\frac{a^\alpha }{b^\beta }}\hbox {d}x\ge 0, \end{aligned}$$
where we have used the short-hand notation
$$\begin{aligned} a^\alpha =\prod _{i=1}^M a_i^{\alpha _i} \quad \text {and} \quad b^\beta =\prod _{j=1}^N b_j^{\beta _j}. \end{aligned}$$
Moreover, the following additivity property of the relative entropy holds
$$\begin{aligned}&E[a,b]-E[a_\infty ,b_\infty ] \\&\quad = \sum _{i=1}^M \int _\Omega \left( a_i\ln \frac{a_i}{a_{i\infty }}-a_i+a_{i\infty }\right) \hbox {d}x + \sum _{j=1}^N\int _\Omega \left( b_j\ln \frac{b_j}{b_{j\infty }}-b_j +b_{j\infty }\right) \hbox {d}x\\&\quad = \sum _{i=1}^M \int _\Omega \left( a_i\ln \frac{a_i}{\overline{a_{i}}}\right) \hbox {d}x + \sum _{j=1}^N\int _\Omega \left( b_j\ln \frac{b_j}{\overline{b_{j}}} \right) \hbox {d}x\\&\qquad + \sum _{i=1}^M \int _\Omega \left( \overline{a_i}\ln \frac{\overline{a_i}}{a_{i\infty }} -\overline{a_i}+a_{i\infty }\right) \hbox {d}x + \sum _{j=1}^N\int _\Omega \left( \overline{b_j}\ln \frac{\overline{b_j}}{b_{j\infty }} -\overline{b_j}+b_{j\infty }\right) \hbox {d}x. \end{aligned}$$
The first Lemma 3.1 of this section states the generalisation of the logarithmic Sobolev inequality, which shall use in our approach.

Lemma 3.1

(A generalised logarithmic Sobolev inequalities, [34]) Assume that \(m \ge (d-2)_+/d\) where \((d-2)_+ = \max \{0, d-2\}\). Then, there exists a constant \(C(\Omega ,m) >0\) such that
$$\begin{aligned} \int _{\Omega }\frac{|\nabla u|^2}{u^{2-m}}\hbox {d}x \ge C(\Omega ,m)\,\overline{u}^{\,m-1}\int _{\Omega }u\ln {\frac{u}{\overline{u}}}\hbox {d}x \ge C(\Omega ,m)\,\overline{u}^{\,m-1}\Vert \sqrt{u} - \overline{\sqrt{u}}\Vert ^2 \end{aligned}$$
where \(\overline{u}=\int _\Omega u\hbox {d}x\).

Proof

The first inequality follows from [34]. The second estimate follows from an elementary inequality:
$$\begin{aligned} \int _\Omega u\ln \frac{u}{\overline{u}}\hbox {d}x = \int _\Omega \left( u\ln \frac{u}{\overline{u}}-u+\overline{u}\right) \hbox {d}x \ge \int _\Omega (\sqrt{u}-\sqrt{\overline{u}})^2\hbox {d}x. \end{aligned}$$
\(\square \)
The estimates in Lemma 3.1 constitute a generalisation of the logarithmic Sobolev inequality (5), which is recovered by setting \(m = 1\) and for which the pre-factor \(\overline{u}^{m - 1}\) vanishes. In the case of porous media diffusion \(m > 1\), the pre-factor \(\overline{u}^{m - 1}\) causes the lower bounds in Lemma 3.1 to degenerate for small spatial averages \(\overline{u}\). In particular, we have by Lemma 3.1 the following lower bound for the entropy production
$$\begin{aligned} D[a,b]&\ge \sum _{i=1}^M d_iC(\Omega ,m_i)\overline{a_i}^{\,m_i-1}\int _\Omega a_i\ln \frac{a_i}{\overline{a_i}}\hbox {d}x \nonumber \\&\quad + \sum _{j=1}^N h_jC(\Omega ,p_j)\overline{b_j}^{\,p_j-1}\int _\Omega b_j\ln \frac{b_j}{\overline{b_j}}\hbox {d}x +\int _\Omega (a^\alpha -b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x \\&\ge C_0\left[ \sum _{i=1}^M \overline{a_i}^{\,m_i-1}\int _\Omega a_i\ln \frac{a_i}{\overline{a_i}}\hbox {d}x \right. \nonumber \\&\left. \quad + \sum _{j=1}^N \overline{b_j}^{\,p_j-1}\int _\Omega b_j\ln \frac{b_j}{\overline{b_j}}\hbox {d}x +\int _\Omega (a^\alpha -b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x\right] .\nonumber \end{aligned}$$
(48)
The problem of degeneracy appears when some averages \(\overline{a_i}\) or \(\overline{b_j}\) do not satisfy a positive lower bound. To overcome this problem, we first observe that due to the mass conservation laws (3) not all spatial averages can be small at the same time. If, for instance, a particular \(\overline{a}_i\) is sufficiently small (w.r.t. \(M_{ij}\)), then another \(\overline{b}_j\) can’t be arbitrarily small because of a mass conservation law (3) connecting these two species, i.e.
$$\begin{aligned} \beta _j\overline{a_i}+\alpha _i\overline{b_j}=M_{ij}>0, \end{aligned}$$
(49)
The following crucial Lemma 3.2 shows functional inequalities, which quantity the so-called “indirect diffusion effect” and allows to compensate the lacking lower bounds for the species, whose spatial averages do not satisfy a lower bound.
We first introduce some convenient notations:
$$\begin{aligned}&A_i=\sqrt{a_i},\ A_{i\infty }=\sqrt{a_{i\infty }},&B_j=\sqrt{b_j},\ B_{j\infty }=\sqrt{b_{j\infty }},\qquad \\&\delta _i(x)=A_i(x)-\overline{A_i}, \quad \forall x\in \Omega ,&\eta _j(x)=B_j(x)-\overline{B_j},\quad \forall x\in \Omega , \end{aligned}$$
where
$$\begin{aligned} \overline{A_i}=\int _\Omega A_i\hbox {d}x \quad \text {and} \quad \overline{B_j}=\int _\Omega B_j\hbox {d}x. \end{aligned}$$
Moreover,
$$\begin{aligned} A^\alpha =\prod ^M_{i=1}A_i^{\alpha _i}\quad \text {and}\quad B^\beta =\prod ^N_{j=1}B_j^{\beta _j}. \end{aligned}$$
The conservation laws are now rewritten as
$$\begin{aligned} \beta _j \overline{A_i^2} + \alpha _i\overline{B_j^2} = M_{ij} > 0 \qquad \forall i=1\ldots M, j=1\ldots N. \end{aligned}$$
(50)

Lemma 3.2

(“Indirect diffusion transfer” functional inequality) Let \(A_i, B_j:\Omega \rightarrow {\mathbb {R}}_+\) with \(i=1\ldots M\) and \(j=1\ldots N\) be non-negative functions satisfying the conservation laws (50) and \(\varepsilon >0\) be a constant to be determined later. Assume that for some \(J\in \{1,\ldots ,N\}\),
$$\begin{aligned} \overline{B_j^2}\le \varepsilon \quad \text { for all } \quad j=1\ldots J. \end{aligned}$$
Then, there exists a constant \(K_1\) which depends on \(\varepsilon \) such that:
$$\begin{aligned} \sum _{i=1}^M\Vert \delta _i\Vert ^2 +\sum _{j=J+1}^N\Vert \eta _j\Vert ^2 +\Vert A^\alpha -B^\beta \Vert ^2\ge K_1\sum _{j=1}^J\Vert \eta _j\Vert ^2 \end{aligned}$$
(51)

Remark 3.1

Note that when the last term on the left-hand side \(\Vert A^\alpha - B^\beta \Vert ^2\) diverges, the inequality holds trivially. Therefore, in the proof, we only consider the case when it is finite.

Proof

Due to the mass conservation laws (50), we have the following natural bounds,
$$\begin{aligned} \overline{A_i^2},\overline{B_j^2}\le M_0^2,\qquad \forall i=1,\ldots ,M,\ \forall j=1,\ldots ,N \end{aligned}$$
for some constant \(M_0>0\). Therefore, by Jensen’s inequality, recalling that \(|\Omega | = 1\),
$$\begin{aligned} \overline{A_i}\le \sqrt{\overline{A_i^2}}\le M_0,\qquad \overline{B_j}\le \sqrt{\overline{B_j^2}}\le M_0,\quad \forall i,j. \end{aligned}$$
From these bounds, we get an upper bound for the right-hand side of (51)
$$\begin{aligned} \sum _{j=1}^J\Vert \eta _j\Vert ^2= \sum _{j=1}^J(\overline{B_j^2}-\overline{B_j}^2) \le \sum _{j=1}^J\overline{B_j^2}\le M_0^2J. \end{aligned}$$
We consider the following two cases.
Case 1: If there exists \(i\in \{1,\ldots ,M\}\) such that \(\Vert \delta _i\Vert ^2\ge \varepsilon \) or there exists a \(j\in \{J+1,\ldots ,N\}\) such that \(\Vert \eta _j\Vert ^2\ge \varepsilon \), we have:
$$\begin{aligned} \sum _{i=1}^M\Vert \delta _i\Vert ^2 +\sum _{j=J+1}^N\Vert \eta _j\Vert ^2 +\Vert A^\alpha -B^\beta \Vert ^2 \ge \varepsilon \ge \frac{\varepsilon }{M_0^2J}\sum _{j=1}^J\Vert \eta _j\Vert ^2 \end{aligned}$$
hence, the desired inequality (51) holds with \(K_1=\frac{\varepsilon }{M_0^2J}\).

Case 2: Assume \(\Vert \delta _i\Vert ^2\le \varepsilon \) for all \(i\in \{1,\ldots ,M\}\) and \(\Vert \eta _j\Vert ^2\le \varepsilon \) for all \(j\in \{J+1,\ldots ,N\}\), which together with the above assumption \(\overline{B_j^2}\le \varepsilon \) and \(\overline{\eta _j^2} \le \overline{B_j^2}\) for all \(j=1\ldots J\) implies \(\Vert \eta _j\Vert ^2\le \varepsilon \) for all \(j\in \{1,\ldots ,N\}\).,

Let \(\lambda >0\) and denote by
$$\begin{aligned} \Omega _{iA}=\{ x\in \Omega :|\delta _i(x)|\le \lambda \sqrt{\varepsilon } \}\quad \text {for}\ i=1,\dots ,M. \end{aligned}$$
Then
$$\begin{aligned} \varepsilon \ge \int _\Omega |\delta _i(x)|^2\hbox {d}x\ge \int _{\Omega \backslash \Omega _{iA}} |\delta _i(x)|^2\hbox {d}x\ge \lambda ^2\varepsilon |\Omega \backslash \Omega _{iA}| \end{aligned}$$
thus
$$\begin{aligned} |\Omega \backslash \Omega _{iA}|\le \frac{1}{\lambda ^2} \quad \text { which implies} \quad |\Omega _{iA}|\ge 1-\frac{1}{\lambda ^2} \end{aligned}$$
Similarly we get,
$$\begin{aligned} |\Omega _{jB}|\ge 1-\frac{1}{\lambda ^2} \quad \text { where } \Omega _{jB} = \{x\in \Omega :|\eta _j(x)| \le \lambda \sqrt{\varepsilon }\} \quad \forall j=1,\ldots ,N. \end{aligned}$$
Now choose \(\lambda ^2=2(M+N)\) and consider \(G=\cap _{i=1}^M\Omega _{iA}\cap ^N_{j=1}\Omega _{jB}\). Then, we have \(|G|\ge \frac{1}{2}\). Note that \(|\delta _i(x)|\le \lambda \sqrt{\varepsilon }\) and \(|\eta _j(x)|\le \lambda \sqrt{\varepsilon }\) for all \(x\in G\) and for all ij. Moreover, \(\forall x\in G\)
$$\begin{aligned} A_i(x)=\overline{A_i}+\delta _i(x) \le \overline{A_i}+|\delta _i(x)|\le M_0+\lambda \sqrt{\varepsilon }\le 2M_0 \end{aligned}$$
and similarly \(B_j(x)\le 2M_0,\ \forall i,j\) if we choose \(\varepsilon \) such that
$$\begin{aligned} \lambda \sqrt{\varepsilon }\le M_0. \end{aligned}$$
By Taylor’s expansion, we have
$$\begin{aligned} A^\alpha =\prod ^M_{i=1}A_i^{\alpha _i} =\prod ^M_{i=1}(\overline{A_i}+\delta _i)^{\alpha _i} =\prod ^M_{i=1}\overline{A_i}^{\alpha _i} +R(\overline{A_i},\delta _i)\sum ^M_{i=1}\delta _i \end{aligned}$$
where the remainder terms R depends polynomially on \(\overline{A_i}\) and \(\delta _i\). Note that \(|R(\overline{A}_i, \delta _i)| \le C_0(M_0)\) on G, we estimate with \((x-y)^2\ge \frac{1}{2}x^2-y^2\)
$$\begin{aligned} \Vert A^\alpha -B^\beta \Vert ^2&=\int _\Omega \left( \prod _{i=1}^MA_i^{\alpha _i}-B^\beta \right) ^2\hbox {d}x \\&\ge \int _G \left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -B^\beta +R(\overline{A_i},\delta _i)\sum _{i=1}^M\delta _i \right) ^2\hbox {d}x \\&\ge \frac{1}{2} \int _G \left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -B^\beta \right) ^2\hbox {d}x -\int _G \left| R(\overline{A_i},\delta _i)\right| ^2\left| \sum _{i=1}^M\delta _i\right| ^2 \\&\ge \frac{1}{2}\int _G \left( \prod _{i=1}^M\overline{A_i}^{\alpha _i}-B^\beta \right) ^2\hbox {d}x -C_0(M_0)^2M\int _G\sum _{i=1}^M|\delta _i|^2 \\&\ge \frac{1}{2}\int _G \left( \prod _{i=1}^M\overline{A_i}^{\alpha _i}-B^\beta \right) ^2\hbox {d}x -C_0(M_0)^2M\int _G\sum _{i=1}^M\Vert \delta _i\Vert ^2 \\&\ge \frac{1}{2}\int _G \left( \prod _{i=1}^M\overline{A_i}^{\alpha _i}-B^\beta \right) ^2\hbox {d}x -C_0(M_0)^2M^2\varepsilon \end{aligned}$$
where we used \(\Vert \delta _i\Vert ^2 \le \varepsilon \) in the last inequality.
In order to estimate further, we use again Taylor’s expansion
$$\begin{aligned} B^\beta = \prod _{j=1}^N(\overline{B_j}+\eta _j)^{\beta _j} = \prod _{j=1}^N\overline{B_j}^{\beta _j} +Q(\overline{B_j},\eta _j)\sum _{j=1}^N\eta _j \end{aligned}$$
where again, Q depends polynomially on \(\overline{B_j},\eta _j\), which implies \(|Q(\overline{B}_j, \eta _j)| \le C_1(M_0)\) on G. Therefore,
$$\begin{aligned} \int _G\left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -B^\beta \right) ^2 \hbox {d}x&= \int _G\left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -\prod _{j=1}^N\overline{B_j}^{\beta _j} -Q(\overline{B_j},\eta _j)\sum _{j=1}^N\eta _j \right) ^2 \hbox {d}x\\&\ge \frac{1}{2}\int _G\left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -\prod _{j=1}^N\overline{B_j}^{\beta _j}\right) ^2 \hbox {d}x\\&\quad -\int _G|Q(\overline{B_j},\eta _j)|^2|\sum _{j=1}^N\eta _j |^2 \hbox {d}x \\&\ge \frac{1}{2}\int _G\left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -\prod _{j=1}^N\overline{B_j}^{\beta _j}\right) ^2 \hbox {d}x -C_1(M_0)^2N^2\varepsilon \end{aligned}$$
where we used that \(\Vert \eta _j\Vert ^2\le \varepsilon \) for all \(j=1,\ldots , N\).
Combining these two estimates, we arrive at
$$\begin{aligned} \Vert A^\alpha -B^\beta \Vert ^2\ge & {} \frac{1}{4}|G|\left( \prod _{i=1}^M\overline{A_i}^{\alpha _i} -\prod _{j=1}^N\overline{B_j}^{\beta _j}\right) ^2\nonumber \\&-\varepsilon \left( \frac{1}{2}C_1(M_0)^2N^2+C_0(M_0)^2M^2\right) . \end{aligned}$$
(52)
By Jensen’s inequality and the assumption of the Lemma, we have
$$\begin{aligned} \overline{B_j}\le \sqrt{\overline{B_j^2}}\le \sqrt{\varepsilon } ,\quad \forall j=1,\ldots ,J. \end{aligned}$$
On the other hand \(\overline{B_j}\le \sqrt{\overline{B_j^2}}\le M_0 ,\ \forall j=J+1,\ldots ,N\). Thus, the conservation law (50) and \(\Vert \delta _i\Vert ^2 \le \varepsilon \) yield
$$\begin{aligned} \overline{A}_i= & {} \sqrt{\overline{A_i^2} - \Vert \delta _i\Vert ^2} = \sqrt{\frac{1}{\beta _1}(M_{i1}-\alpha _i\overline{B_1^2}) - \Vert \delta _i\Vert ^2} \\\ge & {} \sqrt{\frac{M_{i1}}{\beta _1} - \frac{\alpha _i}{\beta _1}\varepsilon - \varepsilon } \quad \forall i=1,\ldots , M. \end{aligned}$$
Hence, by using \(|G| \ge \frac{1}{2}\), we get from (52) that
$$\begin{aligned} \Vert A^\alpha - B^\beta \Vert ^2 \ge \frac{1}{8}\left[ \prod _{i=1}^M\left( \frac{M_{i1}}{\beta _1} - \frac{\alpha _i}{\beta _1}\varepsilon - \varepsilon \right) ^{\alpha _i/2} - \prod _{j=1}^{J}(\sqrt{\varepsilon })^{\beta _j}\prod _{j=J+1}^{N}M_0^{\beta _j} \right] ^2 - C_2\varepsilon . \end{aligned}$$
Because the right-hand side of the above inequality converges to \(\frac{1}{8}\prod _{i=1}^M\bigl (\frac{M_{i1}}{\beta _1}\bigr )^{\alpha _i}\) as \(\varepsilon \rightarrow 0\), we can choose \(\varepsilon >0\) small enough, but still explicit, such that
$$\begin{aligned} \Vert A^\alpha - B^\beta \Vert ^2 \ge \frac{1}{16}\prod _{i=1}^M\Bigl (\frac{M_{i1}}{\beta _1}\Bigr )^{\alpha _i} \ge \frac{1}{16M_0^2J}\prod _{i=1}^{M}\Bigl (\frac{M_{i1}}{\beta _1}\Bigr )^{\alpha _i} \sum _{j=1}^{J}\Vert \eta _j\Vert ^2, \end{aligned}$$
which implies the desired inequality (51) with the constant
$$\begin{aligned} K_1 = \frac{1}{16M_0^2J}\prod _{i=1}^M\Bigl (\frac{M_{i1}}{\beta _1}\Bigr )^{\alpha _i}. \end{aligned}$$
\(\square \)

Lemma 3.3

(A time-dependent entropy–entropy production estimate) Let \((a,b) = (a_1,\ldots , a_M, b_1,\ldots , b_N)\) with \(a_i, b_j: Q_T \rightarrow {\mathbb {R}}_+\) be non-negative functions, which satisfy the conservation laws (3). Moreover,
$$\begin{aligned} \Vert a_i\Vert _{L^{\infty }(Q_T)} \le C_T \quad \text { and } \quad \Vert b_j\Vert _{L^\infty (Q_T)} \le C_T \quad \text { for all } i, j. \end{aligned}$$
Then, there exists a constant \(K_2>0\) independent of T such that,
$$\begin{aligned} D[a(T),b(T)] \ge K_2 \frac{1}{1+\ln (1+T)} (E[a(T), b(T)] - E[a_{\infty },b_{\infty }]). \end{aligned}$$

Proof

Let \(\varepsilon >0\) be a small constant chosen in Lemma 3.2. We will consider two cases and for convenience we will drop T in \(a_i(T)\) and \(b_j(T)\) when there is no confusion.

Case 1. Assume \(\overline{a}_i \ge \varepsilon \) for all \(i=1,\ldots , M\) and \(\overline{b}_j \ge \varepsilon \) for all \(j=1,\ldots , N\). By applying (48), we have
$$\begin{aligned} \begin{aligned} D[a,b]&\ge \sum _{i=1}^{M}d_iC(\Omega ,m_i)\varepsilon ^{m_i-1} \int _{\Omega }a_i\ln {\frac{a_i}{\overline{a}_i}}\hbox {d}x \\&\quad + \sum _{j=1}^{N}h_jC(\Omega ,p_j) \varepsilon ^{p_j-1}\int _{\Omega }b_j\ln {\frac{b_j}{\overline{b}_j}}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x\\&\ge K_3\left[ \sum _{i=1}^{M}\int _{\Omega }a_i\ln {\frac{a_i}{\overline{a}_i}}\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }b_j\ln {\frac{b_j}{\overline{b}_j}}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x\right] \end{aligned} \end{aligned}$$
with
$$\begin{aligned} K_3 = \min _{i=1\ldots M; j=1\ldots N }\{d_iC(\Omega , m_i)\varepsilon ^{m_i-1}; h_jC(\Omega ,p_j)\varepsilon ^{p_j-1}; 1\}. \end{aligned}$$
Using an entropy–entropy production inequality in case of system (R) with linear diffusion, see Lemma 4.1 below, we know that
$$\begin{aligned}&\sum _{i=1}^{M}\int _{\Omega }a_i\ln {\frac{a_i}{\overline{a}_i}}\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }b_j\ln {\frac{b_j}{\overline{b}_j}}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x \\&\quad \ge K_4(E[a,b] - E[a_{\infty },b_{\infty }]) \end{aligned}$$
for an explicit constant \(K_4>0\). Therefore,
$$\begin{aligned} D[a,b] \ge K_3K_4(E[a,b] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
Case 2. Suppose either \(\overline{a}_i \le \varepsilon \) for some \(i\in \{1,\ldots , M\}\) or \(\overline{b}_j \le \varepsilon \) for some \(j=1,\ldots , N\).
Due to the mass conservation laws \(\beta _j \overline{a}_i + \alpha _i\overline{b}_j = M_{ij}\), it cannot happen that \(\overline{a}_i \le \varepsilon \) and \(\overline{b}_j \le \varepsilon \) simultaneously for a sufficiently small \(\varepsilon \), e.g. \(\varepsilon < \frac{M_{ij}}{2}\min \left\{ \frac{1}{\beta _j};\frac{1}{\alpha _i} \right\} \). Therefore, without loss of generality, we can assume that
$$\begin{aligned} \overline{b}_j \le \varepsilon \quad \forall j=1,\ldots , J \qquad \text { and } \qquad \overline{b}_j \ge \varepsilon \quad \forall j=J+1,\ldots ,N \end{aligned}$$
for some \(J \in \{1,\ldots , N\}\). Moreover, by mass conservation laws
$$\begin{aligned} \overline{a}_i = \frac{1}{\beta _1}(M_{i1}-\alpha _i\overline{b}_1) \ge \frac{1}{\beta _1}(M_{i1} - \alpha _i\varepsilon ), \qquad \text {for all } i=1,\ldots ,M. \end{aligned}$$
Thus, we can apply Lemma 3.1 to D[ab] and estimate
$$\begin{aligned} \begin{aligned} D[a,b]&\ge \sum _{i=1}^{M}d_iC(\Omega ,m_i)\left[ \frac{1}{\beta _1}(M_{i1} -\alpha _i\varepsilon )\right] ^{m_i-1}\int _{\Omega }a_i\ln \frac{a_i}{\overline{a}_i}\hbox {d}x\\&\qquad + \sum _{j=J+1}^{N}h_jC(\Omega ,p_j)\varepsilon ^{p_j-1} \int _{\Omega }b_j\ln \frac{b_j}{\overline{b}_j}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta ) \ln \frac{a^\alpha }{b^\beta }\hbox {d}x\\&\ge K_5\left[ \sum _{i=1}^M\left\| \sqrt{a_i} - \overline{\sqrt{a_i}}\right\| ^2 + \sum _{j=J+1}^N\left\| \sqrt{b_j} - \overline{\sqrt{b_j}}\right\| ^2 + \left\| A^\alpha - B^\beta \right\| ^2\right] \\&= K_5\left[ \sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=J+1}^N\Vert \eta _j\Vert ^2 + \left\| A^\alpha - B^\beta \right\| ^2\right] , \end{aligned} \end{aligned}$$
where we have used \((x-y)\ln (x/y) \ge 4(\sqrt{x} - \sqrt{y})^2\) and
$$\begin{aligned} K_5 = \min _{i=1\ldots M; j=J+1\ldots N}\left\{ d_iC(\Omega ,m_i)\left[ \frac{1}{\beta _1}(M_{i1} -\alpha _i\varepsilon )\right] ^{m_i-1}; h_jC(\Omega ,p_j)\varepsilon ^{p_j-1}; 4 \right\} . \end{aligned}$$
Applying Lemma 3.2 yields
$$\begin{aligned} D[a,b] \ge K_6\left[ \sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2 + \left\| A^\alpha - B^\beta \right\| ^2\right] \end{aligned}$$
where
$$\begin{aligned} K_6 = \frac{1}{2} \min \{K_5; K_5K_1\}. \end{aligned}$$
By using another functional inequality, which was already proven in the case of linear diffusion, see (61) in Sect. 4, we have
$$\begin{aligned} D[a,b]\ge & {} K_7\left[ \sum _{i=1}^{M}\left( \Vert \delta _i\Vert ^2 + |\sqrt{\overline{A_i^2}} - A_{i,\infty }|^2\right) \right. \nonumber \\&\left. + \sum _{j=1}^{N}\left( \Vert \eta _j\Vert ^2 + |\sqrt{\overline{B_j^2}} - B_{j,\infty }|^2\right) \right] . \end{aligned}$$
(53)
Now, we estimate \(E[a,b] - E[a_{\infty },b_{\infty }]\) from above. Consider the two variables function
$$\begin{aligned} \Phi (x,y) = \frac{x\ln (x/y) - x + y}{(\sqrt{x}- \sqrt{y})^2} \end{aligned}$$
which is continuous in \((0,\infty )^2\) and \(\Phi (\cdot , y)\) is increasing for each fixed \(y>0\). It holds that
$$\begin{aligned} \begin{aligned}&E[a,b] - E[a_{\infty },b_{\infty }]\\&\quad =\sum _{i=1}^{M}\int _{\Omega }\Phi (a_i,a_{i,\infty }) (A_i - A_{i,\infty })^2\hbox {d}x + \sum _{j=1}^N\int _{\Omega }\Phi (b_j, b_{j,\infty }) (B_j - B_{j,\infty })^2\hbox {d}x\\&\quad \le \max _{i=1\ldots M; j=1\ldots N}\{\Phi (\Vert a_i\Vert _{L^{\infty }(Q_T)}, a_{i,\infty }); \Phi (\Vert b_j\Vert _{L^{\infty }(Q_T)}, b_{j,\infty })\}\\&\qquad \left[ \sum _{i=1}^M\Vert A_i - A_{i,\infty }\Vert ^2 + \sum _{j=1}^N\Vert B_j - B_{j,\infty }\Vert ^2\right] \\&\quad \le K_8(1+\ln (1+T))\left[ \sum _{i=1}^M(\Vert \delta _i\Vert ^2 + |\overline{A}_i - A_{i,\infty }|^2) \right. \\&\qquad \left. + \sum _{j=1}^{N}(\Vert \eta _j\Vert ^2 + |\overline{B}_j - B_{j,\infty }|^2) \right] , \end{aligned} \end{aligned}$$
(54)
where in the last inequality, we have used the estimates \(\Vert a_i\Vert _{L^{\infty }(Q_T)} \le C_T\) and \(\Vert b_j\Vert _{L^{\infty }(Q_T)} \le C_T\) and that \(C_T\) is a constant growing at most polynomially w.r.t. T.
Next, from \(\Vert \delta _i\Vert ^2 = \overline{A_i^2} - \overline{A}_i^2 = (\sqrt{\overline{A_i^2}} - \overline{A}_i)(\sqrt{\overline{A_i^2}} + \overline{A}_i)\), we have
$$\begin{aligned} \overline{A}_i = \sqrt{\overline{A_i^2}} - \frac{\Vert \delta _i\Vert ^2}{\sqrt{\overline{A_i^2}} + \overline{A}_i} = \sqrt{\overline{A_i^2}} - Q_i(A_i)\Vert \delta _i\Vert \quad \text { with } \quad Q_i(A_i) = \frac{\Vert \delta _i\Vert }{\sqrt{\overline{A_i^2}} + \overline{A}_i}. \end{aligned}$$
It’s obvious that \(Q(A_i) \ge 0\) and moreover
$$\begin{aligned} Q_i(A_i)^2 = \frac{\overline{A_i^2} - \overline{A}_i^2}{(\sqrt{\overline{A_i^2}} + \overline{A}_i)^2} = \frac{\sqrt{\overline{A_i^2}} - \overline{A}_i}{\sqrt{\overline{A_i^2}} + \overline{A}_i} \le 1. \end{aligned}$$
Therefore,
$$\begin{aligned} \begin{aligned} |\overline{A}_i - A_{i,\infty }|^2&\le 2\left( |\sqrt{\overline{A_i^2}} - \overline{A}_i|^2 + |\sqrt{\overline{A_i^2}} - A_{i,\infty }|^2\right) \\&= 2\left( Q_i(A_i)^2\Vert \delta _i\Vert ^2 + |\sqrt{\overline{A_i^2}} - A_{i,\infty }|^2\right) \\&\le 2\left( \Vert \delta _i\Vert ^2 + |\sqrt{\overline{A_i^2}} - A_{i,\infty }|^2\right) \quad \text { for all } i=1\ldots M \end{aligned} \end{aligned}$$
and similarly
$$\begin{aligned} |\overline{B}_j - B_{j,\infty }|^2 \le 2\left( \Vert \eta _i\Vert ^2 + |\sqrt{\overline{B_j^2}} - B_{j,\infty }|^2\right) \quad \text { for all } j=1\ldots N. \end{aligned}$$
Hence, it follows from (54) that
$$\begin{aligned} E[a,b] - E[a_{\infty },b_{\infty }]\le & {} 3K_8(1+\ln (1+T))\left[ \sum _{i=1}^{M}(\Vert \delta _i\Vert ^2 + |\sqrt{\overline{A_i^2}} - A_{i,\infty }|^2) \right. \nonumber \\&\left. + \sum _{j=1}^{N}(\Vert \eta _j\Vert ^2 + |\sqrt{\overline{B_j^2}} - B_{j,\infty }|^2)\right] . \end{aligned}$$
(55)
A combination of (53) and (55) yields
$$\begin{aligned} D[a,b] \ge \frac{K_7}{3K_8(1+\ln (1+T))}(E[a,b] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
Finally, from Case 1 and Case 2, we can conclude the proof of Lemma 3.3 with
$$\begin{aligned} K_2 = \min \left\{ K_3K_4; \frac{K_7}{3K_8}\right\} . \end{aligned}$$
\(\square \)

Remark 3.2

The assumptions \(\Vert a_i\Vert _{L^{\infty }(Q_T)} \le C_T\) and \(\Vert b_j\Vert _{L^{\infty }(Q_T)} \le C_T\) in Lemma 3.3 are only needed to estimate \(E[a,b] - E[a_{\infty },b_{\infty }]\) above as in (54). In the case of linear diffusion, it is possible to avoid these \(L^{\infty }\)-bounds by using the additivity of the relative entropy (see also the proof of Lemma 4.1 in Sect. 4), i.e.
$$\begin{aligned} E[a,b] - E[a_{\infty },b_{\infty }] = (E[a,b] - E[\overline{a},\overline{b}]) + (E[\overline{a},\overline{b}] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
However, while for linear diffusion, the logarithmic Sobolev inequality controls to first part \(E[a,b] - E[\overline{a},\overline{b}]\le C(C_{\mathrm {LSI}}) D[a,b]\), such an estimate is unclear in the case of porous media diffusion, where the generalised logarithmic Sobolev inequality in Lemma 3.1 degenerates for states without lower bounds on the spatial averages.

We need also the following Csiszár–Kullback–Pinsker type inequality. The proof is standard and can be found in e.g. [13, 19].

Lemma 3.4

There exists a constant \(C_\mathrm{CKP}>0\) such that for any measurable non-negative functions \(a_i, b_j: \Omega \rightarrow {\mathbb {R}}_+\) satisfying the mass conservation (49), there holds
$$\begin{aligned} E[a,b] - E[a_{\infty },b_{\infty }] \ge C_\mathrm{CKP}\left( \sum _{i=1}^M\Vert a_i - a_{i,\infty }\Vert _1^2 + \sum _{j=1}^N\Vert b_j - b_{j,\infty }\Vert _1^2\right) . \end{aligned}$$

We are ready to prove Theorem 1.3.

Proof of Theorem 1.3

Due to the condition
$$\begin{aligned} m_i, p_j > \max \left\{ \nu - \min \left\{ \frac{4}{d+2}; 1\right\} ; 1\right\} \qquad \forall i=1\ldots M, j=1\ldots N, \end{aligned}$$
we can apply Theorem 1.2 to show boundedness of the weak solution (ab) to (R), i.e.
$$\begin{aligned} \Vert a_i\Vert _{L^{\infty }(Q_T)} \le C_T, \quad \Vert b_j\Vert _{L^{\infty }(Q_T)} \le C_T, \quad \forall i=1\ldots M, j=1\ldots N. \end{aligned}$$
By applying Lemma 3.3, this yields
$$\begin{aligned} D[a(T), b(T)] \ge K_2\frac{1}{1+\ln (1+T)} (E[a(T), b(T)] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
Moreover, due to the boundedness of solutions, we have the entropy–entropy production relation
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}(E[a,b] - E[a_{\infty },b_{\infty }])= & {} \frac{\hbox {d}}{\hbox {d}t}E[a,b] = -D[a,b]\\\le & {} -K_2\frac{1}{1+\ln (1+T)}(E[a,b] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
A classical Gronwall’s inequality leads to
$$\begin{aligned}&E[a(T), b(T)] - E[a_{\infty },b_{\infty }] \\&\quad \le \exp \left( -K_2\int _0^T\frac{d\tau }{1+\ln (1+\tau )}\right) (E[a_0,b_0] - E[a_{\infty },b_{\infty }]). \end{aligned}$$
By direct calculations,
$$\begin{aligned} \exp \left( -K_2\int _0^T\frac{d\tau }{1+\ln (1+\tau )}\right) \ge \exp \left( -K_2\int _0^T\frac{d\tau }{1+\tau }\right) = (1+T)^{-K_2}. \end{aligned}$$
Hence,
$$\begin{aligned} E[a(T), b(T)] - E[a_{\infty },b_{\infty }] \le (1+T)^{-K_2}(E[a_0,b_0] - E[a_{\infty },b_{\infty }]), \end{aligned}$$
(56)
and therefore thanks to the Csiszár–Kullback–Pinsker inequality in Lemma 3.4
$$\begin{aligned}&\sum _{i=1}^M\Vert a_i(T) - a_{i,\infty }\Vert _1^2 + \sum _{j=1}^N\Vert b_j(T) - b_{j,\infty }\Vert _1^2\nonumber \\&\quad \le C_\mathrm{CKP}^{-1}(1+T)^{-K_2}(E[a_0,b_0] - E[a_{\infty },b_{\infty }]) \end{aligned}$$
(57)
which implies algebraic convergence to equilibrium of solutions to (R).
We will now show that from this it is possible to recover exponential convergence. Since the right-hand side of (57) tends to zero as \(T\rightarrow \infty \), we can choose
$$\begin{aligned} T_0 = \max \left\{ 1; \left[ \frac{C_\mathrm{CKP}^{-1}(E[a_0,b_0]-E[a_{\infty },b_{\infty }])}{\frac{1}{2}\min _{i=1\ldots M;j=1\ldots N}\{a_{i,\infty }^2, b_{j,\infty }^2\}}\right] ^{1/K_2} - 1\right\} \end{aligned}$$
(58)
which implies for all \(t\ge T_0\)
$$\begin{aligned} \Vert a_{i}(t) - a_{i,\infty }\Vert _{1} \le \frac{1}{2} a_{i,\infty } \quad \text { and } \quad \Vert b_j(t) - b_{j,\infty }\Vert _1 \le \frac{1}{2} b_{j,\infty }, \end{aligned}$$
and thus,
$$\begin{aligned} \overline{a}_i(t) = \Vert a_{i}(t)\Vert _1 \ge \frac{1}{2} a_{i,\infty } \quad \text { and } \quad \overline{b}_j(t) = \Vert b_j(t)\Vert _1 \ge \frac{1}{2} b_{j,\infty } \quad \text { for all } \quad t \ge T_0. \end{aligned}$$
Therefore, for all \(t\ge T_0\), we can apply these lower bounds on the spatial averages bounds and Lemma 3.1 to estimate the entropy–entropy production as follows
$$\begin{aligned} D[a(t),b(t)]\ge & {} C_1\left[ \sum _{i=1}^M \int _\Omega a_i\ln \frac{a_i}{\overline{a_i}}\hbox {d}x + \sum _{j=1}^N \int _\Omega b_j\ln \frac{b_j}{\overline{b_j}}\hbox {d}x \right. \nonumber \\&\left. +\int _\Omega (a^\alpha -b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x \right] \quad \text { for all } t\ge T_0, \end{aligned}$$
with
$$\begin{aligned} C_1 = \min _{i=1\ldots M; j=1\ldots N}\left\{ d_iC(\Omega ,m_i)\left( \frac{1}{2}a_{i,\infty }\right) ^{m_i-1}; h_jC(\Omega ,p_j)\left( \frac{1}{2} b_{j,\infty }\right) ^{p_j-1}; 1 \right\} . \end{aligned}$$
By applying again Lemma 4.1, we obtain
$$\begin{aligned} D[a(t),b(t)] \ge C_1\lambda (E[a(t),b(t)] - E[a_{\infty },b_{\infty }]) \quad \text { for all } \quad t\ge T_0, \end{aligned}$$
which in a combination with the classical Gronwall’s inequality yields for all \(t\ge T_0\),
$$\begin{aligned} \begin{aligned} E[a(t),b(t)] - E[a_{\infty },b_{\infty }]&\le e^{-\lambda C_1(t-T_0)}(E[a(T_0),b(T_0)] - E[a_{\infty },b_{\infty }])\\&\le e^{-\lambda C_1 t}e^{\lambda C_1 T_0}(1+T_0)^{-K_2}(E[a_0, b_0] - E[a_{\infty },b_{\infty }])\\&\le e^{-\lambda C_1 t}e^{\lambda C_1 T_0}(E[a_0, b_0] - E[a_{\infty },b_{\infty }]) \end{aligned} \end{aligned}$$
where we used (56) for the second inequality. On the other hand, it follows from (56) that for all \(0\le t< T_0\),
$$\begin{aligned} \begin{aligned} E[a(t), b(t)] - E[a_{\infty },b_{\infty }]&\le (1+t)^{-K_2}(E[a_0,b_0] - E[a_{\infty },b_{\infty }])\\&\le e^{-\lambda C_1 t}e^{\lambda C_1T_0}(E[a_0,b_0] - E[a_{\infty },b_{\infty }]) \end{aligned} \end{aligned}$$
Due to the explicitness of \(T_0\) in (58), we eventually get the exponential convergence
$$\begin{aligned} E[a(t),b(t)] - E[a_{\infty },b_{\infty }] \le C_2e^{-{{\widehat{\lambda }}} t}(E[a_0,b_0] - E[a_{\infty },b_{\infty }]) \quad \text { for all } \quad t\ge 0, \end{aligned}$$
with the constant \(C_2=e^{\lambda C_1 T_0}\) and the rate \({{\widehat{\lambda }}} = \lambda C_1\). Note that \(C_2\) is explicit since \(T_0\) is explicit (see (58)). With another application of the Csiszár–Kullback–Pinsker inequality in Lemma 3.4, this yields
$$\begin{aligned}&\sum _{i=1}^{M}\Vert a_i(t) - a_{i,\infty }\Vert _1^2 + \sum _{j=1}^N\Vert b_j(t) - b_{j,\infty }\Vert _1^2 \\&\quad \le C_2C_\mathrm{CKP}^{-1}e^{-{\widehat{\lambda }}t}(E[a_0,b_0] - E[a_{\infty },b_{\infty }]) \le C_3e^{-{\widehat{\lambda }}t} \end{aligned}$$
with \(C_3 = C_2C_\mathrm{CKP}^{-1}(E[a_0,b_0] - E[a_{\infty },b_{\infty }])\). Finally, by combining the above exponential \(L^1\)-convergence with the at most polynomial grow \(L^{\infty }\) a priori estimates \(\Vert a_i\Vert _{L^{\infty }(Q_T)}, \Vert b_j\Vert _{L^{\infty }(Q_T)} \le C_T\), interpolation yields for any \(1<p<\infty \),
$$\begin{aligned} \Vert a_i(T) - a_{i,\infty }\Vert _p\le & {} \Vert a_i(T) - a_{i,\infty }\Vert _{\infty }^{\theta }\Vert a_i(T) - a_{i,\infty }\Vert _1^{1-\theta }\\\le & {} C_T^{\theta }C_3^{1-\theta }e^{-{\widehat{\lambda }}(1-\theta )T} \le C_4e^{-\lambda _p T} \end{aligned}$$
for some \(0< \lambda _p < {\widehat{\lambda }}(1-\theta )\) since \(C_T\) grows at most polynomially in T, and similarly
$$\begin{aligned} \Vert b_j(T) - b_{j,\infty }\Vert _p \le \Vert b_j(T) - b_{j,\infty }\Vert _{\infty }^{\theta }\Vert b_j(T) - b_{j,\infty }\Vert _1^{1-\theta }\le C_5e^{-\lambda _p T}. \end{aligned}$$
This concludes the proof of Theorem 1.3. \(\square \)

4 Entropy–entropy production inequality

Lemma 4.1

(Entropy–entropy production estimate) Let \(a_{\infty }\in (0,\infty )^{M}\) and \(b_{\infty }\in (0,\infty )^N\) satisfy
$$\begin{aligned} a_{\infty }^{\alpha } = b_{\infty }^{\beta } \end{aligned}$$
where \(\alpha \in [1,\infty )^M\) and \(\beta \in [1,\infty )^N\).
Then, there exists an explicit constant \(\lambda >0\) depending on \(a_{\infty }\), \(b_{\infty }\), \(\alpha \), \(\beta \) and the domain \(\Omega \), such that for any non-negative functions \(a = (a_i): \Omega \rightarrow {\mathbb {R}}_+^M\) and \(b = (b_j): \Omega \rightarrow {\mathbb {R}}_+^N\) satisfying
$$\begin{aligned} \beta _j\overline{a}_i + \alpha _i\overline{b}_j = \beta _ja_{i,\infty } + \alpha _ib_{j,\infty } \qquad \text { for all }\quad i=1,\ldots , M, \; j=1,\ldots , N, \end{aligned}$$
the following entropy–entropy production inequality holds
$$\begin{aligned} {\widetilde{D}}[a,b] \ge \lambda (E[a,b] - E[a_{\infty },b_{\infty }]) \end{aligned}$$
where
$$\begin{aligned} {\widetilde{D}}[a,b] = \sum _{i=1}^{M}\int _{\Omega }a_i\ln \frac{a_i}{\overline{a}_i}\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }b_j\ln \frac{b_j}{\overline{b}_j}\hbox {d}x + \int _{\Omega }(a^\alpha - b^\beta )\ln \frac{a^\alpha }{b^\beta }\hbox {d}x \end{aligned}$$
and
$$\begin{aligned} E[a,b] = \sum _{i=1}^{M}\int _{\Omega }(a_i\ln a_i - a_i + 1)\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }(b_j\ln b_j - b_j + 1)\hbox {d}x. \end{aligned}$$

Remark 4.1

The above entropy–entropy production inequality was first proved in [19] in a constructive way with explicit bounds on the constant \(\lambda \). The proof stated here follows the line of a significantly simplified version presented in [20].

Proof

First, by the additivity of the relative entropy, we have
$$\begin{aligned} \begin{aligned} E[a,b] - E[a_{\infty },b_{\infty }]&= (E[a,b] - E[\overline{a}, \overline{b}]) + (E[\overline{a}, \overline{b}] - E[a_{\infty },b_{\infty }])\\&=\left[ \sum _{i=1}^{M}\int _{\Omega }a_i\ln \frac{a_i}{\overline{a}_i}\hbox {d}x + \sum _{j=1}^{N}\int _{\Omega }b_j\ln \frac{b_j}{\overline{b}_j}\hbox {d}x\right] \\&\quad +\left[ \sum _{i=1}^{M}\left( \overline{a}_i\ln \frac{\overline{a}_i}{a_{i,\infty }} - \overline{a}_i + a_{i,\infty }\right) \right. \\&\quad \left. + \sum _{j=1}^{N}\left( \overline{b}_j\ln \frac{\overline{b}_j}{b_{j,\infty }} - \overline{b}_j + b_{j,\infty }\right) \right] \\&=: (I) + (II). \end{aligned} \end{aligned}$$
It is straightforward that (I) can be controlled by \({\widetilde{D}}[a,b]\), i.e.
$$\begin{aligned} \frac{1}{2} {\widetilde{D}}[a,b] \ge \frac{1}{2}\times (I). \end{aligned}$$
It remains to control (II). To do that, we first introduce the following useful notations and definitions
$$\begin{aligned} A_i= & {} \sqrt{a_i},\quad B_j = \sqrt{b_j}, \quad A_{i,\infty } = \sqrt{a_{i,\infty }}, \quad B_{j,\infty } = \sqrt{b_{j,\infty }}, \\ \delta _i(x)= & {} A_i(x) - \overline{A}_i, \qquad \eta _j(x) = B_j(x) - \overline{B}_j, \end{aligned}$$
and
$$\begin{aligned} A^{\alpha } = \prod _{i=1}^{M}A_i^{\alpha _i}, \quad B^{\beta } = \prod _{j=1}^{N}B_j^{\beta _j}. \end{aligned}$$
By the elementary inequality \((x-y)\ln (x/y) \ge 4(\sqrt{x} - \sqrt{y})^2\), we have
$$\begin{aligned} \int _{\Omega }a_i\ln \frac{a_i}{\overline{a}_i}\hbox {d}x = \int _{\Omega }\left( a_i\ln \frac{a_i}{\overline{a}_i} - a_i + \overline{a}_i\right) \hbox {d}x \ge 4\int _{\Omega }(\sqrt{a_i} - \sqrt{\overline{a}_i})^2\hbox {d}x \ge 4\Vert \delta _i\Vert ^2 \end{aligned}$$
and similarly \(\int _{\Omega }b_j\ln \frac{b_j}{\overline{b}_j}\hbox {d}x \ge 4\Vert \eta _j\Vert ^2\). Moreover, \(\int _{\Omega }(a^\alpha - b^\beta )\ln {\frac{a^\alpha }{b^\beta }}\hbox {d}x \ge 4\Vert A^\alpha - B^\beta \Vert ^2\). Therefore,
$$\begin{aligned} \frac{1}{2} {\widetilde{D}}[a,b] \ge 2\left[ \sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 + \Vert A^{\alpha } - B^{\beta }\Vert ^2\right] . \end{aligned}$$
(59)
In order to bound to estimate the right-hand side of (59) with an upper bound of (II), we first observe from the conservation laws
$$\begin{aligned} \beta _j\overline{a}_i + \alpha _i\overline{b}_j = \beta _ja_{i,\infty } + \alpha _ib_{j,\infty }, \quad \text { for all } i, j. \end{aligned}$$
that there exists a constant \(M_0>0\) such that
$$\begin{aligned} \overline{a}_i, \overline{b}_j \le M_0^2, \quad \text { for all } i, j. \end{aligned}$$
Next, we note that the two variables function
$$\begin{aligned} \Phi (x,y) = \frac{x\ln (x/y) - x + y}{(\sqrt{x} - \sqrt{y})^2} \end{aligned}$$
is continuous on \((0,\infty )^2\), and \(\Phi (\cdot , y)\) is increasing for each fixed y. Then, the term (II) is estimated as
$$\begin{aligned} (II)&= \sum _{i=1}^{M}\Phi (\overline{a}_i,a_{i,\infty })(\sqrt{\overline{a}_i} - \sqrt{a_{i,\infty }})^2+ \sum _{j=1}^{N}\Phi (\overline{b}_j,b_{j,\infty }) (\sqrt{\overline{b}_j} - \sqrt{b_{j,\infty }})^2\nonumber \\&\le \max _{i,j}\{\Phi (M_0^2,a_{i,\infty });\Phi (M_0^2, b_{j,\infty })\}\Biggl (\sum _{i=1}^M(\sqrt{\overline{A_i^2}} - {A_{i,\infty }})^2 \nonumber \\&\quad + \sum _{j=1}^N(\sqrt{\overline{B_j^2}} - {B_{j,\infty }})^2\Biggr ). \end{aligned}$$
(60)
From (59) and (60), it remains to show that
$$\begin{aligned}&\sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 + \Vert A^{\alpha } - B^{\beta }\Vert ^2 \nonumber \\&\quad \ge C_0\Biggl (\sum _{i=1}^M(\sqrt{\overline{A_i^2}} - {A_{i,\infty }})^2 + \sum _{j=1}^N(\sqrt{\overline{B_j^2}} - {B_{j,\infty }})^2\Biggr ) \end{aligned}$$
(61)
for some constant \(C_0>0\). By using Lemma 4.2, we have with \(\overline{A} = (\overline{A}_1, \ldots , \overline{A}_M)\) and \(\overline{B} = (\overline{B}_1, \ldots , \overline{B}_N)\)
$$\begin{aligned}&\sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 + \Vert A^{\alpha } - B^{\beta }\Vert ^2 \nonumber \\&\quad \ge C_1\Biggl (\sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 + \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2\Biggr ) \end{aligned}$$
(62)
for some constant \(C_1>0\). Using the ansatz
$$\begin{aligned} \overline{A_i^2} = A_{i,\infty }^2(1+\mu _i)^2 \quad \text { and } \quad \overline{B_j^2} = B_{j,\infty }^2(1+\zeta _j)^2, \qquad \text {where} \quad \mu _i, \zeta _j \in [-1,\infty ),\nonumber \\ \end{aligned}$$
(63)
the right-hand side of (61) writes as
$$\begin{aligned} \text {RHS of } (61) = C_0\Biggl (\sum _{i=1}^M\mu _i^2 + \sum _{j=1}^N\zeta _j^2\Biggr ). \end{aligned}$$
(64)
Moreover, the bounds \(\overline{a_i}=\overline{A_i^2} \le M_0^2\) and \(\overline{b_j}=\overline{B_j^2} \le M_0^2\) imply
$$\begin{aligned} -1 \le \mu _i \le M_1 \quad \text { and } -1\le \zeta _j \le M_1 \end{aligned}$$
(65)
for some constant \(M_1>0\). From the ansatz (63) (and similar to the proof of Lemma 3.3), we have
$$\begin{aligned} \overline{A}_i&= \sqrt{\overline{A_i^2}} - Q_i(A_i)\Vert \delta _i\Vert = A_{i,\infty }(1+\mu _i) - Q_i(A_i)\Vert \delta _i\Vert \\ \overline{B_j}&= \sqrt{\overline{B_j^2}} - R_j(B_j)\Vert \eta _j\Vert = B_{j,\infty }(1+\zeta _j) - R_j(B_j)\Vert \eta _j\Vert \end{aligned}$$
where
$$\begin{aligned} 0 \le Q_i(A_i) {:}{=} \frac{\Vert \delta _i\Vert }{\sqrt{\overline{A_i^2}} + \overline{A}_i} \le 1 \quad \text { and } \quad 0 \le R_j(B_j){:}{=} \frac{\Vert \eta _j\Vert }{\sqrt{\overline{B_j^2}} + \overline{B}_j} \le 1. \end{aligned}$$
Next, we use Taylor expansion to estimate
$$\begin{aligned} \overline{A_i}^{\alpha _i} = \left( A_{i,\infty }(1+\mu _i)- Q_i(A_i)\Vert \delta _i\Vert \right) ^{\alpha _i} = A_{i,\infty }^{\alpha _i}(1+\mu _i)^{\alpha _i} + {{\widehat{Q}}_i\Vert \delta _i\Vert } \end{aligned}$$
in which the Lagrange remainder term \({{\widehat{Q}}_i= {\widehat{Q}}(\mu _i, \Vert \delta _i\Vert )}\) is uniformly bounded above by a constant for all admissible values of \(\mu _i\) and \(\Vert \delta _i\Vert \) thanks to the boundedness of \(\mu _i\) and \(\Vert \delta _i\Vert \le \sqrt{\overline{A_i^2}} \le M_0\). Similarly,
$$\begin{aligned} \overline{B_j}^{\beta _j} = B_{j,\infty }^{\beta _j}(1+\zeta _j)^{\beta _j} + {{\widehat{R}}_j\Vert \eta _j\Vert } \end{aligned}$$
with uniformly bounded remainder \( {\widehat{R}}_j(\zeta _j,\Vert \eta _j\Vert )\). Thus,
$$\begin{aligned} \begin{aligned} \Bigl |\overline{A}^\alpha - \overline{B}^\beta \Bigr |^2&= \Biggl |\prod _{i=1}^M\overline{A}_i^{\alpha _i} - \prod _{j=1}^{N}\overline{B}_j^{\beta _j}\Biggr |^2\\&= \Biggl |\prod _{i=1}^M\left( A_{i,\infty }^{\alpha _i}(1+\mu _i)^{\alpha _i} + {\widehat{Q}}_i\Vert \delta _i\Vert \right) \\&\quad - \prod _{j=1}^{N} \left( B_{j,\infty }^{\beta _j}(1+\zeta _j)^{\beta _j} + {\widehat{R}}_j\Vert \eta _j\Vert \right) \Biggr |^2\\&= \Biggl |A_{\infty }^\alpha \prod _{i=1}^M(1+\mu _i)^{\alpha _i} - B_{\infty }^\beta \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j} \\&\quad + \Theta ({\widehat{Q}}_i, {\widehat{R}}_j)\Biggl (\sum _{i=1}^M\Vert \delta _i\Vert + \sum _{j=1}^N\Vert \eta _j\Vert \Biggr )\Biggr |^2 \end{aligned} \end{aligned}$$
with \(\Theta ({\widehat{Q}}_i, {\widehat{R}}_j)\) is also uniformly bounded. Thus, by using \((x+y)^2 \ge \frac{1}{2} x^2 - y^2\) and \(A_{\infty }^\alpha = \sqrt{a_{\infty }^\alpha } = \sqrt{b_{\infty }^\beta } = B_{\infty }^\beta \) and the Cauchy–Schwarz inequality,
$$\begin{aligned} \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2\ge & {} \frac{1}{2} A_{\infty }^\alpha \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |^2\nonumber \\&- |\Theta |^2(M+N)^2\Biggl (\sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\Biggr ). \end{aligned}$$
(66)
Hence, for any \(\delta \in (0,1)\) holds
$$\begin{aligned} \begin{aligned}&\sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2 + \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2\\&\quad \ge \sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\\&\qquad + \delta \Biggl (\frac{1}{2} A_{\infty }^\alpha \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j} \Biggr |^2 \\&\qquad - |\Theta |^2(M+N)^2\Biggl (\sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\Biggr )\Biggr )\\&\quad \ge \frac{\delta }{2}A_{\infty }^\alpha \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |^2 \end{aligned} \end{aligned}$$
by choosing \(\delta \) small enough such that \(1 \ge \delta |\Theta |^2(M+N)^2\) since \(\Theta \) is uniformly bounded above. This leads in combination with (62) to a lower bound of the left-hand side of (61)
$$\begin{aligned} \text {LHS of } (61) \ge C_1\frac{\delta }{2}A_{\infty }^\alpha \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |^2. \end{aligned}$$
(67)
From (64) and (67), it is sufficient to prove
$$\begin{aligned} \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |^2 \ge C_2\Biggl (\sum _{i=1}^M\mu _i^2 + \sum _{j=1}^N\zeta _j^2\Biggr ). \end{aligned}$$
(68)
In order to do so, we note that the conservation laws
$$\begin{aligned} \beta _j\overline{a}_i + \alpha _i\overline{b}_j = \beta _ja_{i,\infty } + \alpha _ib_{j,\infty } \end{aligned}$$
rewritten in terms of the ansatz (63), i.e.
$$\begin{aligned} \beta _jA_{i,\infty }^2(\mu _i^2 + 2\mu _i) + \alpha _iB_{j,\infty }^2(\zeta _j^2 + 2\zeta _j) = 0. \end{aligned}$$
imply \(\mu _i\zeta _j \le 0\) thanks to \(\mu _i, \zeta _j \ge -1\) for all ij. Without loss of generality, we assume \(\mu _i \ge 0\) and \(\zeta _j \le 0\) for all ij. Then, for any \(1\le i_0 \le M\) and \(1\le j_0 \le N\),
$$\begin{aligned} \begin{aligned} \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |&\ge \prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\\&\ge (1+\mu _{i_0})^{\alpha _{i_0}} - (1+\zeta _{j_0})^{\beta _{j_0}}\\&\ge (1+\mu _{i_0}) - (1+\zeta _{j_0}) \ge \mu _{i_0} - \zeta _{j_0} \ge 0. \end{aligned} \end{aligned}$$
Thus,
$$\begin{aligned} \Biggl |\prod _{i=1}^M(1+\mu _i)^{\alpha _i} - \prod _{j=1}^{N}(1+\zeta _j)^{\beta _j}\Biggr |^2 \ge (\mu _{i_0} - \zeta _{j_0})^2 = \mu _{i_0}^2 - 2\mu _{i_0}\zeta _{j_0} + \zeta _{j_0}^2 \ge \mu _{i_0}^2 + \zeta _{j_0}^2. \end{aligned}$$
Since \(1\le i_0\le M\) and \(1 \le j_0\le N\) are arbitrary, we finally obtain (68) with \(C_2 = 1/\max \{M; N\}\). \(\square \)

Lemma 4.2

Let \(a_i, b_j\) be functions defined in Lemma 4.1. Then, there exists a constant C such that
$$\begin{aligned} \sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 + \Vert A^{\alpha } - B^{\beta }\Vert ^2 \ge C\left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2. \end{aligned}$$

Proof

Fix a constant \(L>0\). Denote by
$$\begin{aligned} S= & {} \{x\in \Omega : |\delta _i(x)| \le L, |\eta _j(x)|\le L \text { for all } i=1,\ldots , M, \; j=1,\ldots , N\} \quad \text { and } \\ S^\perp= & {} \Omega \backslash S. \end{aligned}$$
Recalling \(\overline{A_i} \le \sqrt{\overline{A_i^2}} \le M_0\) and \(\overline{B_j} \le \sqrt{\overline{B_j^2}}\le M_0\), we use Taylor expansion to estimate
$$\begin{aligned} \Vert A^\alpha - B^\beta \Vert ^2&\ge \int _{S}\biggl |\prod _{i=1}^{M}(\overline{A}_i + \delta _i(x))^{\alpha _i} - \prod _{j=1}^N(\overline{B}_j + \eta _j(x))^{\beta _j} \biggr |^2\hbox {d}x\nonumber \\&\ge \frac{1}{2} \Bigl |\overline{A}^\alpha - \overline{B}^\beta \Bigr |^2|S| - {\widetilde{R}}(\overline{A}_i, \overline{B}_j, |\delta _i|, |\eta _j|)\Biggl (\sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\Biggr ) \end{aligned}$$
(69)
where \(|{\widetilde{R}}| \le C(M_0, L)\) due to the boundedness of \(\delta _i\) and \(\eta _j\) in S. In \(S^\perp \), we have
$$\begin{aligned} \sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 \ge \int _{S^\perp }\Biggl (\sum _{i=1}^M|\delta _i(x)|^2 + \sum _{j=1}^N|\eta _j(x)|^2\Biggr )\hbox {d}x \ge L^2|S^\perp |. \end{aligned}$$
Next, there clearly exists a constant \(\Lambda >0\) such that \(\left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2 \le \Lambda \) since \(\overline{A}_i, \overline{B}_j \le M_0\). Therefore,
$$\begin{aligned} \sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^{N}\Vert \eta _j\Vert ^2 \ge L^2|S^\perp | \ge \frac{L^2}{\Lambda }\left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2|S^\perp |. \end{aligned}$$
(70)
Combining (69) and (70), we find for any \(\theta _1, \theta _2 \in (0,1)\)
$$\begin{aligned} \begin{aligned}&\sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2 + \Vert A^\alpha - B^\beta \Vert ^2\\&\quad \ge \theta _1\frac{L^2}{\Lambda }\left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2|S^\perp | + (1-\theta _1) \Biggl (\sum _{i=1}^M\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\Biggr )\\&\qquad + \theta _2\frac{1}{2} \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2|S| - \theta _2|{\widetilde{R}}|\Biggl (\sum _{i=1}^{M}\Vert \delta _i\Vert ^2 + \sum _{j=1}^N\Vert \eta _j\Vert ^2\Biggr )\\&\quad \ge \min \left\{ \theta _1\frac{L^2}{\Lambda }; \theta _2\frac{1}{2} \right\} \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2(|S| + |S^\perp |)\\&\quad = \min \left\{ \theta _1\frac{L^2}{\Lambda }; \theta _2\frac{1}{2} \right\} \left| \overline{A}^\alpha - \overline{B}^\beta \right| ^2 \end{aligned} \end{aligned}$$
by choosing \(\theta _1, \theta _2\) small enough such that \(1 - \theta _1 - \theta _2|{\widetilde{R}}| \ge 0\) and using \(|S| + |S^\perp | = |\Omega | = 1\). The proof of Lemma 4.2 is hence complete. \(\square \)

5 Proof Theorem 1.1: existence of global weak solution to (S)

In this section, we give a proof Theorem 1.1 about the global existence of weak solutions to (S) under the conditions (G)–(M)–(P). Consider the approximating system
$$\begin{aligned}&\partial _t u_{i,\varepsilon } - d_i\Delta (u_{i,\varepsilon }^{m_i}) = f_{i,\varepsilon }(u_{\varepsilon }){:}{=} \frac{f_i(u_{\varepsilon })}{1+\varepsilon \sum _{i=1}^{S}|f_i(u_{\varepsilon })|},\nonumber \\&\quad \nabla (u_{i,\varepsilon }^{m_i})\cdot \overrightarrow{n} = 0, \quad u_{i,\varepsilon }(x,0) = u_{i,0,\varepsilon }(x) \end{aligned}$$
(71)
where \(u_{\varepsilon } = (u_{1,\varepsilon }, \ldots , u_{S,\varepsilon })\) and the sequence of approximating non-negative initial data \(u_{i,0,\varepsilon }\in L^{\infty }(\Omega )\) converges to \(u_{i,0}\) in \(L^2(\Omega )\). By the construction of the approximative system, it directly follows that the nonlinearities \(f_{i,\varepsilon }\) still satisfy the conditions (M) and (P). Moreover, for \(\varepsilon >0\)
$$\begin{aligned} |f_{i,\varepsilon }(u_{\varepsilon })| \le \frac{|f_i(u_{\varepsilon })|}{1+\varepsilon \sum _{i=1}^{S}|f_i(u_{\varepsilon })|} \le \frac{1}{\varepsilon } \quad \text { for all } u_{\varepsilon }\in {\mathbb {R}}^S. \end{aligned}$$
Hence, by a classical result for the porous medium equation with \(L^{\infty }\) data, there exists a strong non-negative solution \(u_{\varepsilon } = (u_{i,\varepsilon })_{i=1\ldots S}\) (see e.g. [43, Section 8]) in the sense that
$$\begin{aligned}&u_{i,\varepsilon }^{m_i}\in L^2_{loc}(0,+\infty ;H^1(\Omega )), \quad \!\! \partial _tu_{i,\varepsilon } = d_i\Delta (u_{i,\varepsilon }^{m_i}) + f_{i,\varepsilon }(u_{\varepsilon }) \in L^1_{loc}(0,+\infty ;L^1(\Omega )), \\&\quad u_{i,\varepsilon }\in C([0,T); L^1(\Omega )) \text { and } u_{i,\varepsilon }(0) = u_{i,0,\varepsilon }, \end{aligned}$$
and the equation for \(u_{i,\varepsilon }\) holds a.e. in \(Q_T\) for any \(T>0\). Therefore, it follows immediately that
$$\begin{aligned}&-\int _{\Omega }u_{i,0,\varepsilon }\psi (0)\hbox {d}x - \int _0^T\int _{\Omega }(\partial _t \psi u_{i,\varepsilon } + u_{i,\varepsilon }^{m_i}\Delta \psi )\hbox {d}x\hbox {d}t \nonumber \\&\quad = \int _0^T\int _{\Omega }f_{i,\varepsilon }(u_{\varepsilon })\psi \hbox {d}x\hbox {d}t \end{aligned}$$
(72)
for any test function \(\psi \in C^{2,1}({\overline{\Omega }}\times [0,T])\) with \(\psi (T) = 0\) and \(\nabla \psi \cdot \overrightarrow{n} = 0\) on \(\partial \Omega \times (0,T)\). As for the existence of weak solutions, it can be obtained by classical methods, for instance following the ideas in [1] and more precisely, derive a Lyapunov functional similar to the one on p. 39. One can also use similar arguments in [26, Proof of Lemma 2.3] with a few modifications to adapt to Neumann boundary conditions.

In order to pass to the limit as \(\varepsilon \rightarrow 0\) in the weak formula (72), we use the following uniform a priori estimates, which are a consequence of a duality argument in the spirit of e.g. [36] and references therein.

Lemma 5.1

(Duality estimates and uniform a priori estimates for the approximating solutions, cf. [26]) Let \(u_{\varepsilon } = (u_{1,\varepsilon },\ldots , u_{S,\varepsilon })\) be the non-negative solutions to the approximating system (71). Then,
$$\begin{aligned} \Vert u_{i,\varepsilon }\Vert _{L^{m_i+1}(Q_T)} \le C_T \quad \text { for all } \quad T>0 \quad \text { and } \quad i = 1,\ldots , S, \end{aligned}$$
(73)
where the \(\varepsilon \)-independent constant \(C_T\) depends only polynomially in T. Moreover, we have
$$\begin{aligned} \Vert f_{i,\varepsilon }(u_{\varepsilon })\Vert _{L^{1+\delta }(Q_T)} \le C_T \end{aligned}$$
for some \(\delta > 0\), where the constant \(C_T\) depends at most polynomially in \(T>0\)

Proof

The proof follows [26] with straightforward changes due to the considered Neumann (instead of Dirichlet) boundary conditions. By setting
$$\begin{aligned} Z = \sum _{i=1}^{S}\lambda _iu_{i,\varepsilon }\quad \text { and } \quad W = \sum _{i=1}^{S}d_i\lambda _iu_{i,\varepsilon }^{m_i} \end{aligned}$$
and by summing up the equations of systems (S), the mass dissipation property (M) implies
$$\begin{aligned} \partial _t Z - \Delta W \le 0 \quad \text { and } \quad \nabla W \cdot \overrightarrow{n} = 0. \end{aligned}$$
Then, integration over (0, t) and multiplication with W(t) in \(L^2(\Omega )\) (due to the regularity of the approximative solutions) lead after integration over \(\Omega \) to
$$\begin{aligned} \int _{\Omega }\left( Z(t)-Z(0)\right) W(t)\hbox {d}x - \int _{\Omega } W(t)\Delta \int _0^tW(s)\hbox {d}s \hbox {d}x \le 0. \end{aligned}$$
(74)
Next, we integrate by parts with homogeneous Neumann boundary conditions the second term on the left-hand side and calculate
$$\begin{aligned} - \int _{\Omega } W(t) \Delta \int _0^tW(s)\hbox {d}s\; \hbox {d}x= & {} \int _{\Omega }\nabla W(t)\cdot \nabla \int _0^tW(s)\hbox {d}s\;\hbox {d}x \\= & {} \frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}\int _{\Omega }|\nabla \int _0^tW(s)\hbox {d}s|^2\hbox {d}x. \end{aligned}$$
Therefore, by integrating (74) with respect to t on (0, T), we obtain
$$\begin{aligned} \int _{0}^{T}\int _{\Omega }Z(t)W(t)\hbox {d}x\hbox {d}t + \frac{1}{2}\int _{\Omega }|\nabla \int _0^TW(s)\hbox {d}s|^2\hbox {d}x \le \int _{0}^{T}\int _{\Omega }Z(0)W(t)\hbox {d}x\hbox {d}t. \end{aligned}$$
(75)
Moreover, we note that
$$\begin{aligned} \int _0^T\int _{\Omega }Z(t)W(t)\hbox {d}x\hbox {d}t= & {} \int _0^T\int _{\Omega }\left( \sum _{i=1}^{S}\lambda _iu_{i,\varepsilon }\right) \left( \sum _{i=1}^{S}d_i\lambda _iu_{i,\varepsilon }^{m_i}\right) \hbox {d}x\hbox {d}t\nonumber \\\ge & {} \sum _{i=1}^{S}d_i\lambda ^2_i\Vert u_{i,\varepsilon }\Vert _{L^{m_i+1}(Q_T)}^{m_i+1} \end{aligned}$$
(76)
due to the non-negativity of functions \(u_{i,\varepsilon }\) and the constant \(\lambda _i\). To estimate the right-hand side of (75) in terms of the \(L^2\)-norm of Z(0), we first notice from \(\partial _t Z - \Delta W \le 0\) that
$$\begin{aligned} Z(T) - \Delta \int _0^TW\hbox {d}t \le Z(0). \end{aligned}$$
Define \(\varphi (x) = \int _0^TW(x,t)\hbox {d}t\), we have, thanks to \(Z(T) \ge 0\),
$$\begin{aligned} -\Delta \varphi \le Z(0) \; \text { in }\; \Omega \quad \text { and } \quad \nabla \varphi \cdot \overrightarrow{n} = 0 \; \text { on } \; \partial \Omega . \end{aligned}$$
Multiplying this inequality by \(\varphi \ge 0\) and using the Poincaré–Wirtinger inequality \(\Vert \nabla \varphi \Vert ^2 \ge C_P\Vert \varphi - {\overline{\varphi }}\Vert ^2\) yield
$$\begin{aligned} \begin{aligned}&C_P\Vert \varphi - {\overline{\varphi }}\Vert ^2 \le \Vert \nabla \varphi \Vert ^2 \le \int _{\Omega }Z(0)\varphi \hbox {d}x \\&\quad = \int _{\Omega }Z(0)(\varphi - {\overline{\varphi }})\hbox {d}x + {\overline{\varphi }}\int _{\Omega }Z(0)\hbox {d}x\\&\quad \le \frac{C_P}{2}\Vert \varphi - {\overline{\varphi }}\Vert ^2 + \frac{1}{2C_P}\Vert Z(0)\Vert ^2 + {\overline{\varphi }}\int _{\Omega }Z(0)\hbox {d}x. \end{aligned} \end{aligned}$$
where \({\overline{\varphi }} = \frac{1}{|\Omega |}\int _{\Omega }\varphi \hbox {d}x\). Thus,
$$\begin{aligned} \Vert \varphi - {\overline{\varphi }}\Vert ^2 \le C\Vert Z(0)\Vert ^2 + {\overline{\varphi }}\Vert Z(0)\Vert _{L^1(\Omega )}. \end{aligned}$$
We can now estimate
$$\begin{aligned} \int _0^T\int _{\Omega }Z(0)W(t)\hbox {d}x\hbox {d}t&= \int _{\Omega }\varphi Z(0)\hbox {d}x = \int _{\Omega }(\varphi - {\overline{\varphi }})Z(0)\hbox {d}x + {\overline{\varphi }}\int _{\Omega }Z(0)\hbox {d}x\\&\le 2\Vert \varphi - {\overline{\varphi }}\Vert ^2 + 2\Vert Z(0)\Vert ^2 + {\overline{\varphi }}\int _{\Omega }Z(0)\hbox {d}x\\&\le C\Vert Z(0)\Vert ^2 + C\Vert Z(0)\Vert {\overline{\varphi }}. \end{aligned}$$
By inserting this into (75) and (76), we obtain
$$\begin{aligned}&\sum \limits _{i=1}^{S}d_i\lambda ^2_i\Vert u_{i,\varepsilon }\Vert _{L^{m_i+1}(Q_T)}^{m_i+1} \le C\Vert Z(0)\Vert ^{2} + C\Vert Z(0)\Vert {\overline{\varphi }} \\&= C\Vert Z(0)\Vert ^2 + C\Vert Z(0)\Vert \sum _{i=1}^Sd_i\lambda _i \Vert u_{i,\varepsilon }\Vert _{L^{m_i}(Q_T)}^{m_i}. \end{aligned}$$
An application of Young’s inequality gives us the first a priori estimate (73) of Lemma 5.1.
Concerning the second uniform a priori estimate for the nonlinearities, we have
$$\begin{aligned} |f_{i,\varepsilon }(u_{\varepsilon })| \le |f_{i}(u_{\varepsilon })| \le C(1+|u_{\varepsilon }|^{\nu }), \end{aligned}$$
where C does not depend on \(\varepsilon \). By the assumption \(m_{i} > \nu - 1\) and the estimate of \(\Vert u_{i,\varepsilon }\Vert _{L^{m_i+1}(Q_T)}\), we obtain \(\Vert f_{i,\varepsilon }(u_{\varepsilon })\Vert _{L^{1+\delta }(Q_T)} \le C_T\). \(\square \)

The following compactness lemma allows to extract a converging subsequence from the approximating system.

Lemma 5.2

[3] Let \(m> (d-2)_+/d\) with \((d-2)_+ = \max \{0,d-2\}\). The mapping \(L^1(\Omega )\times L^1(Q_T)\ni (u_0, f) \mapsto u \in L^1(Q_T)\) where \(u\in C([0,T];L^1(\Omega ))\) is the weak solution to
$$\begin{aligned} \partial _t u - \delta \Delta (u^{m}) = f, \quad \nabla (u^m)\cdot \overrightarrow{n} = 0, \quad u(0) = u_0, \end{aligned}$$
with \(\delta >0\), is compact.

Proof of Theorem 1.1

Thanks to the uniform bounds of the nonlinearities in Lemma 5.1 and the compactness Lemma 5.2, there exists a subsequence (not relabelled) \(\{u_{i,\varepsilon }\}_{\varepsilon }\) which converges in \(L^1(Q_T)\) to limit functions \(u_{i}\in L^1(Q_T)\). From the \(L^{m_i+1}\)-bound in Lemma 5.1, it holds in fact that \(u_{i,\varepsilon }\) (up to another subsequence) converges strongly to \(u_{i}\) in \(L^{m_i}(Q_T)\). For the nonlinearities, we first notice from Lemma 5.1 that the sequence \(\{f_{i,\varepsilon }(u_{\varepsilon })\}\) is uniformly integrable. Moreover, for another subsequence \(u_{i,\varepsilon } \rightarrow u_i\) a.e. in \(Q_T\), it follows that
$$\begin{aligned} f_{i,\varepsilon }(u_{\varepsilon }) \rightarrow f_i(u_i) \quad \text { a.e. in } \quad Q_T. \end{aligned}$$
Therefore, we can apply Vitali’s Lemma, see e.g. [41, Chapter 16], to obtain \(f_{i,\varepsilon }(u_{\varepsilon }) \rightarrow f_{i}(u_i)\) strongly in \(L^1(Q_T)\). All this allows to pass to the limit in the weak formulation (72) for any test function \(\psi \in C^{2,1}({\overline{\Omega }}\times [0,T])\) with \(\psi (T) = 0\) and \(\nabla \psi \cdot \overrightarrow{n} = 0\) on \(\partial \Omega \times (0,T)\). Hence, we get
$$\begin{aligned} -\int _{\Omega }\psi (0)u_{i,0}\hbox {d}x - \int _{Q_T}(\partial _t \psi u_{i} + u_{i}^{m_i}\Delta \psi )\hbox {d}x\hbox {d}t = \int _{Q_T}f_{i}(u)\psi \hbox {d}x\hbox {d}t. \end{aligned}$$
The additional regularity \(u^{m_i}_i \in L^1(0,T;W^{1,1}(\Omega ))\) follows immediately from [29, Lemma 4.7],1 where
$$\begin{aligned} \int _0^T\int _{\Omega }|\nabla u_i^{m_i}|^{\beta }\hbox {d}x\hbox {d}t \le C(T, \Vert u_{i,0}\Vert _1, \Vert f_i(u)\Vert _{L^1(Q_T)}) \quad \text { for all } 1\le \beta < 1 + \frac{1}{1+m_id}. \end{aligned}$$
From the above estimate and \(f_i(u)\in L^1(Q_T)\), we also have \(\partial _tu_i \in L^1(0,T;(W^{1,1}(\Omega ))^*)\) which implies in particular \(u_i \in C([0,T];L^1(\Omega ))\). This completes the proof of existence of global weak solutions. \(\square \)

Footnotes

  1. 1.

    The results presented in [29] are for homogeneous Dirichlet boundary conditions. However, similar results for homogeneous Neumann boundary conditions can be obtained with slight modifications.

Notes

Acknowledgements

Open access funding provided by University of Graz. We would like to thank the referee for the comments and suggestions which help to significantly improve the paper. The second author was supported by the DFG Project CH 955/3-1. This work is partially supported by International Research Training Group IGDK 1754 and NAWI Graz.

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Authors and Affiliations

  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.University of MannheimMannheimGermany

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