Analysis of mass controlled reaction–diffusion systems with nonlinearities having critical growth rates

Abstract

We analyze semilinear reaction–diffusion systems that are mass controlled, and have nonlinearities that satisfy critical growth rates. The systems under consideration are only assumed to satisfy natural assumptions, namely the preservation of non-negativity and a control of the total mass. It is proved in dimension one that if nonlinearities have (slightly super-) cubic growth rates then the system has a unique global classical solutions. Moreover, in the case of mass dissipation, the solution is bounded uniformly in time in sup-norm. One key idea in the proof is the Hölder continuity of gradient of solutions to parabolic equation with possibly discontinuous diffusion coefficients and low regular forcing terms. When the system possesses additionally an entropy inequality, the global existence and boundedness of a unique classical solution is shown for nonlinearities satisfying a cubic intermediate sum condition, which is a significant generalization of cubic growth rates. The main idea in this case is to combine a modified Gagliardo-Nirenberg inequality and the newly developed $L^p$-energy method in Fitzgibbon et al. (SIAM J Math Anal 53(6):6771–6803, 2021) and Morgan and Tang (Commun Contemp Math, 2022). This idea also allows us to deal with the case of discontinuous diffusion coefficients in higher dimensions, which has only recently been touched in the context of mass controlled reaction–diffusion systems.

1 Introduction and main results

Let $\Omega =(0,L)$ for $L>0$. We study the reaction–diffusion system for vector of concentrations $u=(u_1,\ldots ,u_m): \Omega \times \mathbb {R}_+ \rightarrow \mathbb {R}^m$,$m\ge 1$, given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u_i-d_i\partial _{xx}u_i=f_i(x,t,u), &{}\quad x\in \Omega ,~ t>0,\\ \partial _xu_i(0,t)=\partial _xu_i(L,t)=0, &{}\quad t>0,\\ u_i(x,0)=u_{i,0}(x), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $d_i>0$ are diffusion coefficients, the initial data are non-negative and bounded, i.e. $0\le u_{i,0}\in L^{\infty }(\Omega ), \forall i=1,\ldots , m$. The nonlinearities satisfy the following assumptions:

• (A1) (Local Lipschitz) For all $i=1,\ldots , m$, $f_i:\Omega \times \mathbb {R}_+\times \mathbb {R}_+^m\rightarrow \mathbb {R}$ is locally Lipschitz continuous in the third argument, continuously differentiable in the second argument, and two times continuously differentiable in the first argument.

  (Quasi-positivity) Moreover, they are quasi-positive, that is for any $i\in {1,\ldots ,m}$ and any $(x,t)\in \Omega \times \mathbb {R}_+$, it holds

  $$\begin{aligned} f_i(x,t,u)\ge 0 \text { provided } u\in \mathbb {R}^m_+ \text { and } u_i=0. \end{aligned}$$

• (A2) (Mass control)

  $$\begin{aligned} \sum ^m_{i=1}f_i(x,t,u)\le k_0+k_1\sum ^m_{j=1}u_j,~~\forall u\in \mathbb {R}_+^m, \; \forall (x,t)\in \Omega \times \mathbb {R}_+, \end{aligned}$$

  for some $k_0\ge 0, k_1\in \mathbb {R}.$

The local Lipschitz continuity of nonlinearities in (A1) ensures the existence of a local classical solution in a maximal time interval. The quasi-positivity is a preservation of non-negativity. That is, if the initial data are non-negative, then the solution is non-negative as long as it exists. This property has a simple physical interpretation. If a concentration is zero at a time then it cannot be consumed in a reaction. Reaction–diffusion systems satisfying (A1) and (A2) appear naturally in modeling many real life phenomena, ranging from chemistry, biology, ecology, or social sciences. Remarkably, these two natural assumptions are not enough to ensure global existence of bounded solutions as it was pointed out by counterexamples in [22, 23]. In fact, without further assumptions on the nonlinearities, it is not known if (1.1) possesses a global solution in any sense. One main reason is that from (A1) and (A2) only limited a priori estimates have been derived. More precisely, by summing the equations of (1.1), integrating in $\Omega$ and using (A2), one (formally) obtains

$$\begin{aligned} \frac{d}{dt}\sum _{i=1}^{m}\int _{\Omega }u_i(x,t)dx \le k_0|\Omega | + k_1\sum _{i=1}^{m}\int _{\Omega }u_i(x,t)dx. \end{aligned}$$

The standard Gronwall inequality implies for any $T>0$,

$$\begin{aligned} \sum _{i=1}^{m}\int _{\Omega }u_i(x,t)dx \le e^{k_1T}\sum _{i=1}^{m}\int _{\Omega }u_i(x,0)dx + \frac{k_0}{k_1}|\Omega |e^{k_1T} \quad \forall t\in (0,T). \end{aligned}$$

Thanks to the non-negativity of solutions implied by (A1), we get

$$\begin{aligned} u_i\in L^{\infty }(0,T;L^{1}(\Omega )) \end{aligned}$$

(1.2)

$\text { for all } i=1,\ldots , m, \; \text { and all } \; T>0.$ With this estimate, it was shown early in [17] that (1.1) has a global classical solution if the nonlinearities are bounded by

$$\begin{aligned} |f_i(u)| \le C\left( 1+\sum _{j=1}^m u_j\right) ^r \quad \forall i=1,\ldots , m, \; \forall u\in \mathbb {R}_+^m, \end{aligned}$$

(1.3)

where the growth rate r is sub-critical¹

$$\begin{aligned} 1\le r < r_{\text {critical}}:= 1 + \frac{2}{n}, \end{aligned}$$

(1.4)

where $n\in {\mathbb {N}}$ is the spatial dimension, i.e. $\Omega \subset \mathbb {R}^n$. It is observed that in one dimension, the global existence is obtained for the nonlinearities that have sub-cubic growth rates. For two and higher dimensions, the results in [17] require the nonlinearities to be sub-quadratic. By using a duality method, it can be shown, see e.g. [17, 18, 22], that

$$\begin{aligned} u_i \in L^{2}(0,T;L^2(\Omega )) \end{aligned}$$

(1.5)

for all $i=1,\ldots , m$, and all $T>0$. When the nonlinearities have at most quadratic growth rates, this estimate implies global weak solutions in all dimensions [5]. An improved duality technique in [1] showed that (1.5) can be slightly improved, i.e. there exists $\varepsilon >0$ depending only on the domain and diffusion coefficients such that

$$\begin{aligned} u_i\in L^{2+\varepsilon }(0,T;L^{2+\varepsilon }(\Omega )) \end{aligned}$$

(1.6)

for all $i=1,\ldots , m$, and all $T>0$. This allows the authors in [1] to obtain global classical solutions for systems of type (1.1) with quadratic nonlinearities in dimension two. In fact, global existence for quadratic nonlinearities with space dimension $n\le 2$ was in fact first proved in [21] even with quasilinear diffusion. Systems of form (1.1) with quadratic nonlinearities are in fact of high interest and importance due to their relevance in bimolecular reactions and population dynamics such as Lotka-Volterra or SIR systems. The global existence of classical solutions for such systems in three and higher dimensions was open until recently when it was proved affirmatively in four works [2, 8, 9, 25].

The global existence of bounded solution to (1.1) with super-quadratic nonlinearities is widely open. Up to our knowledge, all existing results are conditional, in the sense that additional assumptions are imposed: for instance, when the diffusion coefficients are close to each other [1, 19], when diffusion coefficients are large enough [3], or when the initial data are small enough [3]. This is the main motivation of the this paper where we investigate the global existence of classical solution in one dimensional for (1.1) with nonlinearities having cubic or (slightly) higher growth rates.

Our first main result is the following theorem.

Theorem 1.1

(Global classical solutions with (slightly super-)cubic nonlinearities) Let $\Omega = (0,L)$ for some $L>0$. Assume (A1) and (A2). Then there exists $\varepsilon >0$ depending on $\Omega , m, d_i, k_0$ such that if the nonlinearities satisfy

• (A3)
  $$\begin{aligned} |f_i(x,t,u)| \le C\left( 1+|u|^{3+\varepsilon }\right) \quad \forall i=1,\ldots , m,\; \forall u\in \mathbb {R}_+^m,\; \forall (x,t)\in \Omega \times \mathbb {R}_+ \end{aligned}$$

for some constant $C>0$, the system (1.1) possesses a unique global classical solution. Moreover, if $k_1 = k_0 = 0$, then

$$\begin{aligned} \sup _{i}\limsup _{t\rightarrow \infty }\Vert u_i(t)\Vert _{L^{\infty }(\Omega )} < +\infty . \end{aligned}$$

Remark 1

• It is clear that the results of Theorem 1.1 still hold if (A2) is replaced by

  $$\begin{aligned} \sum ^m_{i=1}\alpha _if_i(x,t,u)\le k_0 + k_1 \sum _{i=1}^{m}u_i \qquad \text {for all}\quad u\in \mathbb {R}_+^m, (x,t)\in \Omega \times \mathbb {R}_+, \end{aligned}$$

  for some $(\alpha _i)_{i=1,\ldots ,m}\in (0,\infty )^m.$

• It is interesting that if a convective term is added to any of the equations, then the arguments fall apart in Theorem 1.1. This is due to the fact that a new Hölder continuity results does not seem to hold.

To prove Theorem 1.1, we first consider the case where (A2) is replaced by a (seemingly) stronger assumption

• (A2’) there exists $g\in L^{\infty }(Q_T)$ with $\partial _xg\in L^{\infty }(Q_T)$ such that

  $$\begin{aligned} \sum _{i=1}^{m}f_i(x,t,u) = g(x,t), \quad \forall u\in \mathbb {R}_+^m, \; \forall (x,t)\in \Omega \times \mathbb {R}_+. \end{aligned}$$

We will prove the following theorem.

Theorem 1.2

Let $\Omega = (0,L)$ for some $L>0$. Assume (A1), (A2’) and (A3). Then for any non-negative, bounded initial data, (1.1) has a unique global classical solutions. Moreover, if $g \equiv 0$, the solution is uniformly sup-norm bounded, i.e.

$$\begin{aligned} \sup _{i}\limsup _{t\rightarrow \infty }\Vert u_i(t)\Vert _{L^{\infty }(\Omega )}<+\infty . \end{aligned}$$

(1.7)

At the first glance, (A2’) is stronger that (A2), and consequently Theorem 1.2 is weaker than Theorem 1.1. However, with a change of variable and introducing a new equation to the system (see Sect. 3 or [8]), it can be shown that Theorem 1.1 can be implied from Theorem 1.2.

Let us now describe the ideas to prove Theorem 1.2. We follow the approach of Kanel [13], which was used there for the case of mass conservation, i.e. when (A2) is fulfilled with an equality sign and $k_0 =k_1 = 0$. By summing the equations of (1.1), it follows from (A2’) that

$$\begin{aligned} \partial _t\left( \sum _{i=1}^{m}u_i\right) - \partial _{xx}\left( \sum _{i=1}^{m}d_iu_i\right) = g(x,t). \end{aligned}$$

Integrating this relation with respect to time on (0, t) gives

$$\begin{aligned}{} & {} \sum _{i=1}^{m}u_i(x,t) - \partial _{xx}\left( \int _0^t\sum _{i=1}^{m}d_iu_i(x,s)ds\right) \nonumber \\{} & {} \quad = \sum _{i=1}^{m}u_{i,0}(x) + \int _0^tg(x,s)ds=:G(x,t). \end{aligned}$$

(1.8)

By defining $v(x,t) = \int _0^t\sum _{i=1}^{m}d_iu_i(x,s)ds$, it follows that

$$\begin{aligned} \sup _{i=1,\ldots , m}\sup _{\Omega \times (0,T)}|u_i| \le m\left[ \sup _{\Omega \times (0,T)}|\partial _{xx}v| + \Vert G\Vert _{L^{\infty }(Q_T)}\right] . \end{aligned}$$

(1.9)

The equation (1.8) can also be written as

$$\begin{aligned} b(x,t)\partial _tv(x,t) - \partial _{xx}v(x,t) = G(x,t), \end{aligned}$$

(1.10)

where b(x, t) is bounded from above and below by positive constants. One cornerstone of [8] is that (1.10) implies v is Hölder continuous with an exponent $\gamma \in (0,1)$. In case when the nonlinearities are bounded by (slightly super-)quadratic polynomials, i.e. $|f_i(u)| \lesssim 1 + |u|^{2+\varepsilon }$, this Hölder continuity of v allows us to ultimately estimate

$$\begin{aligned} \sup _{\Omega \times (0,T)}|\partial _{xx}v| \lesssim 1 + \left[ \sup _{i=1,\ldots , m}\sup _{\Omega \times (0,T)}|u_i|\right] ^{\frac{3+\varepsilon }{4}+ \frac{1-\gamma }{2(2-\gamma )}}. \end{aligned}$$

Inserting this into (1.9) gives, thanks to $\frac{3+\varepsilon }{4}+ \frac{1-\gamma }{2(2-\gamma )}<1$ for small enough $\varepsilon >0$,

$$\begin{aligned} {\sup _{i=1,\ldots , m}\sup _{\Omega \times (0,T)}|u_i| \lesssim C\left( T,\gamma , \Vert G\Vert _{L^{\infty }(Q_T)}, \text {initial data}\right) ,} \end{aligned}$$

(1.11)

which implies the global existence and uniform sup-norm boundedness of (1.1) in the case of (slightly super-)quadratic nonlinearities in all dimensions (see [8, 9]). In dimension one, similar to [13, Theorems 2 & 3] in the mass conservation case, we will prove that the spatial derivative $\partial _xv$ is also Hölder continuous with an exponent $\alpha \in (0,1)$. This improvement is the key element to deal with (slightly super-) cubic nonlinearities as in (A3). Indeed, by using the Hölder continuity of $\partial _xv$ and (A3), we can estimate

$$\begin{aligned} \sup _{\Omega \times (0,T)}|\partial _{xx}v| \lesssim 1 + \left[ 1+\sup _{i=1,\ldots ,m}\sup _{\Omega \times (0,T)}|u_i|\right] ^{\left( 2+\frac{\varepsilon }{2}\right) \frac{1-\alpha }{2-\alpha -\delta }} \end{aligned}$$

for some $0<\delta <\alpha$. Now, by using $\left( 2+\frac{\varepsilon }{2}\right) \frac{1-\alpha }{2-\alpha -\delta } < 1$ for small enough $\varepsilon >0$, the estimate (1.11) follows, hence the global existence of classical solutions. Finally, to show the uniform-in-time boundedness of solution (1.7), we study (1.1) on time intervals of fixed length with the help of a cut-off function and show that the solution is bounded independent of the intervals.

The assumption (A3) requires that all nonlinearities are bounded by (slightly super-)cubic polynomials. Our second main result shows global existence of bounded solutions to (1.1) with nonlinearities having critical growth rates $r_{\text {critical}}$ and satisfying an entropy inequality and a cubic intermediate sum condition. This condition means that only one nonlinearity is assumed to be bounded by a cubic polynomial, while the others just need to satisfy an intermediate sum condition of order three. Intermediate sum conditions of various orders have been studied in [17, 18] and revisited in [10, 19]. In general, an intermediate sum condition of order r means that

• (A4) there exists a lower triangular matrix $A = (a_{ij}) \in \mathbb {R}^{m\times m}$ with non-negative elements and positive diagonal elements such that for any $i=1,\ldots , m$,

  $$\begin{aligned} \sum _{j=1}^{i}a_{ij}f_j(x,t,u) \le C\left( 1+\sum _{j=1}^mu_j\right) ^r \quad \text { for all } \;u\in \mathbb {R}_+^m, \; (x,t)\in \Omega \times \mathbb {R}_+ \end{aligned}$$

  where $C>0$ is a fixed constant.

It is remarked that (A4) is significantly more general than (1.3) (see Example 1.5). Using the estimate (1.2), it was shown in [10, 17] that under the additional assumption (A4), system (1.1) has a unique global classical solution if the growth rate r is sub-critical as in (1.4). Note that these results do not allow to have quadratic growth in dimensions higher than three. The case of quadratic nonlinearities in two dimension has been recently revisited in [19] by utilizing the improved duality method first proved in [1]. The case of cubic intermediate sums has not been treated in the literature. In this work, we show global classical solutions to (1.1) in one dimension with nonlinearities satisfying a cubic intermediate sum condition and the so-called entropy inequality:

• (E) there exists $\mu _1, \ldots , \mu _m \in \mathbb {R}$ and $k_2, k_3\ge 0$ such that

  $$\begin{aligned} \sum _{i=1}^mf_i(x,t,u)(\log u_i + \mu _i)\le k_2\sum _{i=1}^{m}u_i(\log u_i + \mu _i-1) + k_3, \end{aligned}$$

  for all $u\in (0,\infty )^m$ and all $(x,t)\in \Omega \times \mathbb {R}_+.$

This entropy inequality appears frequently in (bio-)chemical reactions and therefore has been studied extensively in the literature. By assuming (E), it was proved in [12] that (1.1) is globally well-posed in one and two dimensions, respectively, with cubic nonlinearities, i.e. $r=3$ in (1.3), and quadratic nonlinearities, i.e. $r=2$ in (1.3). When the nonlinearities are strictly sub-quadratic, i.e. $r<2$ in (1.3), the global existence of classical solution was shown in all dimensions, see [17] for a separable Lyupanov approach and [4] for a De Giorgi approach. A breakthrough was shown in [6] where the author showed global renormalized solutions to (1.1) under (E) without any growth assumptions on the nonlinearities. Our result in the following theorem shows global classical solutions to (1.1) under a critical intermediate sum condition, i.e. $r=r_{\text {critical}} = 3$ in (A4) in one dimension, and the entropy inequality (E), which significantly generalizes previous results.

Theorem 1.3

(Global classical solutions with cubic intermediate sum) Let $\Omega = (0,L)$ for some $L>0$. Assume (A1), (E) and (A4) with $r=3$. Moreover, assume that there exist $\ell >0$ and $C>0$ satisfying for all $i=1,\ldots , m$,

$$\begin{aligned} f_i(x,t,u) \le C\left( 1+\sum _{j=1}^{m}u_j^\ell \right) \end{aligned}$$

(1.12)

for all $u\in \mathbb {R}_+^m$ and all $(x,t)\in \Omega \times \mathbb {R}_+$. Then for any non-negative, bounded initial data, (1.1) has a unique global classical solution. Moreover, if $k_2 = k_3 = 0$ or $k_2<0$ in (E), the solution is bounded uniformly in time in sup-norm, i.e.

$$\begin{aligned} \sup _{i}\limsup _{t\rightarrow \infty }\Vert u_i(t)\Vert _{L^{\infty }(\Omega )}<+\infty . \end{aligned}$$

We emphasize that by assuming (E), the mass control assumption (A2) can be relaxed. Moreover, the assumption (1.12) indicates that the nonlinearities are bounded above by polynomials, but we do not impose any restriction on the growth rate $\ell$.

Our key idea in proving Theorem 1.3 is to combine a new $L^p$-energy method, see [10, 20] and a modified Gagliardo-Nirenberg inequality. The $L^p$-energy method deduces that, under the intermediate sum condition (A4), one can choose for any $2\le p \in {\mathbb {N}}$ coefficients $\theta _\beta$ such that (see Lemma 4.4) the energy function

$$\begin{aligned} \mathscr {E}_p[u]:= \sum _{|\beta | = \beta _1+\cdots + \beta _m = p}\int _{\Omega }\left( \theta _\beta \prod _{i=1}^{m}u_i^{\beta _i}\right) dx \end{aligned}$$

satisfies

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha _p\sum _{i=1}^{m}\int _{\Omega }\left| \partial _x(u_i^{p/2}) \right| ^2dx \le C\left( 1+\sum _{i=1}^{m}\int _{\Omega }u_i^{p-1+r}dx\right) . \end{aligned}$$

For $r = 3$, choosing $p = 2$ yields

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \alpha _2\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \le C\left( 1+\sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^4\right) . \end{aligned}$$

To deal with the terms involving the $L^{4}(\Omega )$-norm on the right-hand side, we first utilize the entropy condition (E) to obtain a bound of $\Vert u_i\log |u_i|\Vert _{L^{1}(\Omega )}$, then apply a modified Gagliardo-Nirenberg inequality in one dimension to show that the $L^{4}(\Omega )$-norm can be controlled by the $H^1(\Omega )$-norm and $\Vert u_i\log |u_i|\Vert _{L^{1}(\Omega )}$. This leads to bounds of $\mathscr {E}_2[u]$ and consequently $L^\infty (0,T;L^2(\Omega ))$ bounds, which in turn gives $L^\infty (0,T;L^\infty (\Omega ))$ bounds by considering general $2\le p \in {\mathbb {N}}$.

There is another advantage of the $L^p$-energy method: it allows us to deal with the case of discontinuous diffusion coefficients. More precisely, consider a variant of (1.1) in arbitrary dimension, i.e. $\Omega \subset \mathbb {R}^n$, $n\ge 1$, with possibly discontinuous diffusion coefficients

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_i - \nabla _x\cdot (D_i(x,t)\nabla _x u_i) = f_i(x,t,u), &{} \quad x\in \Omega , \; t>0,\\ u_i(x,t) = 0, &{} \quad x\in \partial \Omega , t>0,\\ u_{i,0}(x) = u_{i}(x,0), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(1.13)

where the diffusion matrix $D_i:\Omega \times [0,\infty )\rightarrow \mathbb {R}^{n\times n}$ satisfies

$$\begin{aligned} \lambda |\xi |^2\le \xi ^\top D_i(x,t)\xi ,~\forall (x,t)\in \Omega \times [0,\infty ),\forall \xi \in \mathbb {R}^n,\forall i=1,\ldots ,m, \end{aligned}$$

(1.14)

for some $\lambda >0$ and

$$\begin{aligned} D_i\in L^\infty _{\text {loc}}(\mathbb {R}_+;L^{\infty }(\Omega )),\forall i=1,\ldots ,m. \end{aligned}$$

(1.15)

Due to the low regularity of diffusion coefficients, one cannot expect classical solutions to (1.13). The suitable framework is weak solutions. By using similar methods to the proof of Theorem 1.3, we show that under (A1), (E) and (A4), the system (1.13) has a unique global bounded weak solution. While all previous results focused on dimension one, the following theorem is proved in all dimensions.

Theorem 1.4

Let $\Omega \subset \mathbb {R}^n$, $n\ge 1$. Assume (A1), (E), (1.14), (1.15) and (A4) with r in (A4) fulfills

$$\begin{aligned} 1 \le r \le r_{\text {critical}} = 1 + \frac{2}{n}. \end{aligned}$$

Moreover, assume that there exist $\ell >0$ and $C>0$ satisfying for all $i=1,\ldots , m$,

$$\begin{aligned} f_i(x,t,u) \le C\left( 1+\sum _{j=1}^{m}u_j^\ell \right) \end{aligned}$$

(1.16)

for all $u\in \mathbb {R}_+^m$ and all $(x,t)\in \Omega \times \mathbb {R}_+$. Then, for any non-negative, bounded initial data $u_0 \in L^{\infty }(\Omega )^{m}$, there exists a unique global bounded, non-negative solution to (1.13). Moreover, if $k_2 = k_3 = 0$ in (E), the solution is bounded uniformly in time, i.e.

$$\begin{aligned} \sup _{i}\limsup _{t\rightarrow \infty }\Vert u_i(t)\Vert _{L^{\infty }(\Omega )} < +\infty . \end{aligned}$$

Remark 2

• Obviously, Theorem 1.3 is a special case of Theorem 1.4 in dimension $n = 1$ when all diffusion coefficients are constants.

• Theorem 1.4 improves results of [10] by allowing the critical value $r = 1 + \frac{2}{n}$.

• In case of constant diffusion, Theorem 1.4 recovers results in [19] in two dimensions concerning quadratic intermediate sum condition at the cost of (E). In higher dimensions, still with constant diffusion coefficients, results in [19] allow intermediate sum conditions of order $r<r_* \le 1 + \frac{4}{n+2}$, which is obviously better than $1+\frac{2}{n}$ as soon as $n\le 3$. However one novelty of Theorem 1.4 is that it deals with discontinuous diffusion coefficients, which are out of reach for the duality method used in [19].

Example 1.5

Consider the reversible reaction for three chemical species $\mathcal {U}, \mathcal {V}, \mathcal {W}$ as given by

$$\begin{aligned} \alpha {\mathcal {U}} + \beta {\mathcal {V}} \leftrightharpoons \gamma W \end{aligned}$$

with stoichiometric coefficients $\alpha , \beta , \gamma \in {\mathbb {N}}$, and, for the sake of simplicity, assume the reaction rate constants are one. By applying the mass action law, one obtains the following one dimensional reaction–diffusion system with $\Omega = (0,L)$

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \partial _{x}(d_u(x,t)\partial _xu) = f_1(u,v,w):= -\alpha \left( u^\alpha v^\beta - w^{\gamma }\right) , &{}\quad x\in \Omega ,\\ \partial _t v- \partial _{x}(d_v(x,t)\partial _xv) = f_2(u,v,w):= -\beta \left( u^\alpha v^\beta - w^\gamma \right) , &{}\quad x\in \Omega ,\\ \partial _t w - \partial _{x}(d_w(x,t)\partial _xw) = f_3(u,v,w):= \gamma \left( u^\alpha v^\beta - w^\gamma \right) , &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(1.17)

subject to homogeneous Neumann boundary conditions and non-negative, bounded initial data. It is easy to see that if $\gamma = 3$ and $\alpha , \beta \in {\mathbb {N}}$ arbitrary, the cubic intermediate sum condition is satisfied for (1.17). Moreover, thanks to the reversibility, the entropy inequality condition (E) is also fulfilled since

$$\begin{aligned} f_1\times \log u + f_2 \times \log v + f_3\times \log w = -\left( u^\alpha v^\beta - w^\gamma \right) \left( \log \left( u^\alpha v^\beta \right) - \log \left( w^\gamma \right) \right) \le 0. \end{aligned}$$

Therefore, one can apply Theorem 1.4 to obtain global existence and uniform-in-time bounds of a unique weak solution to (1.17). When the diffusion coefficients $d_u, d_v, d_w$ are smooth or constants, the solution is classical. We emphasize that global classical solutions to systems of type (1.17) have been studied many times before, see e.g. [7, 14, 19, 24], but none of this results are applicable to the case $\gamma =3$ and $\alpha , \beta \in {\mathbb {N}}$ arbitrary, even with constant diffusion coefficients, unless $\alpha =\beta =1$ and $\gamma$ is arbitrary.

Organization of the paper. In the next section, we show the global existence and boundedness of solutions for systems with cubic nonlinearities in one dimension. We prove Theorem 1.2 in Sect. 2. The proof of Theorem 1.1, as a consequence of Theorem 1.2, is presented in Sect. 3. Section 4 is devoted to case of nonlinearities satisfying a cubic intermediate sum condition, where systems with constant diffusion and discontinuous coefficients are considered in Sects. 4.1 and 4.2 respectively.

Notation. In this paper we will use the following notation, some of which will be recalled from time to time:

• For $T>0$ and $p\in [1,\infty ]$, $Q_T:= \Omega \times (0,T)$ and

  $$\begin{aligned} L^p(Q_T):= L^p(0,T;L^p(\Omega )) \end{aligned}$$

  equipped with the usual norm

  $$\begin{aligned} \Vert f\Vert _{L^{p}(Q_T)}:= \left( \int _0^T\int _{\Omega }|f|^pdxdt\right) ^{1/p} \end{aligned}$$

  for $1\le p < \infty$ and

  $$\begin{aligned} \Vert f\Vert _{L^{\infty }(Q_T)}:= \underset{(x,t)\in Q_T}{\text {ess sup}}|f(x,t)|. \end{aligned}$$

• For $p\in [1,\infty ]$, $\tau \ge 0$ and $\delta >0$, we denote by

  $$\begin{aligned} Q_{\tau ,\tau +\delta }:= \Omega \times (\tau ,\tau +\delta ), \end{aligned}$$

  and

  $$\begin{aligned} L^p(Q_{\tau ,\tau +\delta }):= L^p(\tau ,\tau +\delta ; L^p(\Omega )). \end{aligned}$$

2 Proof of Theorem 1.2

2.1 Preliminaries

We start with the definition of classical solutions.

Considering (1.1) in arbitrary dimension, i.e. $\Omega \subset \mathbb {R}^n$, $n\ge 1$, be a bounded domain. The reaction–diffusion system for vector of concentrations $u=(u_1,\ldots ,u_m): \Omega \times (0,T) \rightarrow \mathbb {R}^m$, $m\ge 1$, given by

$$\begin{aligned} {\left\{ \begin{array}{ll} {\partial _{t}u_i-d_i\Delta u_i=f_i(x,t,u)}, &{}\quad (x,t)\in \Omega \times (0,T),\\ \nabla _xu_i\cdot \nu =0, &{}\quad (x,t)\in \partial \Omega \times (0,T),\\ u_i(x,0)=u_{i,0}(x), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(2.1)

where $d_i>0$ are diffusion coefficients.

Definition 2.1

(Classical solutions) Let $0<T\le \infty$. A classical solution to (2.1) on (0,T) is a vector of concentrations $u=(u_1,\ldots ,u_m),m\ge 1$, satisfying for all $i=1,\ldots ,m$, $u_i\in C([0,T];L^p(\Omega ))\cap L^\infty ((0,T)\times \Omega )\cap C^{1,2}((\tau ,T)\times {\bar{\Omega }})$ for all $p>1$ and all $0<\tau <T$, and u satisfies each equation in (2.1) pointwise in $Q_T$.

Theorem 2.2

(Local existence, [8], Proposition 3.1) Assume (A1). Then, for any bounded, nonnegative initial data, (2.1) possesses a local nonnegative classical solution on a maximal interval $[0,T_{\max })$. Moreover, if

$$\begin{aligned} \begin{aligned} \limsup _{t\rightarrow T^-_{\max }}\Vert u_i(t)\Vert _{L^\infty (\Omega )}<\infty \quad \text {for all}~i=1,2,\ldots ,m, \end{aligned} \end{aligned}$$

(2.2)

then $T_{\max }=+\infty .$

Thanks to Theorem 2.2, the global existence of strong solutions to (1.1) follows if we can show that

$$\begin{aligned} \sup _{i=1,\ldots , m}\sup _{T\in (0,T_{\max })}\Vert u_i\Vert _{L^{\infty }(Q_T)} < +\infty . \end{aligned}$$

(2.3)

Moreover, due to the smoothing effect, we can shift the initial time to $0<\tau <T_{\max }$ to assume w.l.o.g. that the initial data $u_{i,0}\in C^2({\bar{\Omega }})$, $i=1,\ldots , m$, and satisfy the compatibility condition $\nabla u_{i,0}\cdot \nu = 0$ on $\partial \Omega$. We will use these regular initial data for the rest of this paper.

The following interpolation lemma was proved in [8] and it holds in all dimensions.

Lemma 2.3

(Regularity Interpolation, Neumann boundary conditions) [8, 13] Let $\Omega \subset \mathbb {R}^n$, $n\ge 1$, be a bounded domain with smooth boundary $\partial \Omega$. For some constant $d>0$, let u be the solution to the inhomogeneous linear heat equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u-d\Delta u=\phi (x,t), &{}\quad (x,t)\in \Omega \times (0,T),\\ \nabla u\cdot \nu =0, &{}\quad (x,t)\in \partial \Omega \times (0,T),\\ u(x,0)=u_0(x), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

Assume that there exists $\gamma \in [0, 1)$ such that for all $x, x'\in \Omega$, and all $t\in (0, T)$,

$$\begin{aligned} |u(x,t)-u(x',t)|\le H|x-x'|^\gamma . \end{aligned}$$

Then, the following gradient estimate follows:

$$\begin{aligned} \sup _{Q_T}|\nabla u(x,t)|\le C\Vert u_0\Vert _{C^1(\Omega )}+BH^{\frac{1}{2-\gamma }}\Vert \phi \Vert _{L^{\infty }(Q_T)}^{\frac{1-\gamma }{2-\gamma }}, \end{aligned}$$

where $B>0$ and $C>0$ are constants depending only on $\Omega , d$ and $\gamma$.

We have a similar lemma for the case of homogeneous Dirichlet boundary conditions.

Lemma 2.4

(Regularity Interpolation, Dirichlet boundary conditions) Let $\Omega \subset \mathbb {R}^n$, $n\ge 1$, be a bounded domain with smooth boundary $\partial \Omega$. Let $0\le \tau < T$. For some constant $d>0$, let u be the solution to the inhomogeneous linear heat equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u-d\Delta u=\phi (x,t), &{}\quad (x,t)\in Q_{\tau ,T},\\ u(x,t)=0, &{}\quad (x,t)\in \partial \Omega \times (\tau ,T),\\ u(x,\tau )=u_0(x), &{} \quad x\in \Omega . \end{array}\right. } \end{aligned}$$

Assume there exists $\gamma \in [0, 1)$ such that for all $x, x'\in \Omega$, and all $t\in (\tau , T)$,

$$\begin{aligned} |u(x,t)-u(x',t)|\le H|x-x'|^\gamma . \end{aligned}$$

Then, the following uniform gradient estimate follows:

$$\begin{aligned} \sup _{Q_{\tau ,T}}|\nabla u(x,t)|\le C\Vert u_0\Vert _{C^1(\Omega )}+C_{T-\tau }H^{\frac{1-\delta }{2-\gamma -\delta }}\Vert \phi \Vert _{L^{\infty }(Q_{\tau ,T})}^{\frac{1-\gamma }{2-\gamma -\delta }}, \end{aligned}$$

(2.4)

where C and $C_{T-\tau }$ are constants depending only on $T-\tau >0$, $\Omega , d$, $\gamma$ and $\delta \in (0,2-\gamma )$.

Remark 3

Using similar method as in [8] it is possible to show (2.4) for $\delta = 0$ and the constant $C_T$ on the right hand is independent of T. The only modification is that one needs gradient estimates of the Green function with homogeneous Dirichlet, instead of Neumann, boundary conditions. Here we present a simpler proof for (2.4) using maximal-regularity of the parabolic equation. Note that (2.4) is enough for our later purpose by choosing $\delta \le \gamma /3$.

Proof

Let $u=v+w$, where

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}v-d\Delta v=0, &{}\quad (x,t)\in Q_{\tau ,T},\\ v(x,t)=0, &{}\quad (x,t)\in \partial \Omega \times (\tau ,T),\\ v(x,\tau )=u_0(x), &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(2.5)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}w-d\Delta w=\phi (x,t), &{}\quad (x,t)\in Q_{\tau ,T},\\ w(x,t)=0, &{}\quad (x,t)\in \partial \Omega \times (\tau ,T),\\ w(x,\tau )=0, &{} \quad x\in \Omega . \end{array}\right. } \end{aligned}$$

(2.6)

From equation (2.5), we can obtain

$$\begin{aligned} \Vert v(t)\Vert _{C^1(\Omega )}\le C\Vert u_0\Vert _{C^1(\Omega )}. \end{aligned}$$

(2.7)

From equation (2.63.3), using $L^p$-max-regularity [15], we have

$$\begin{aligned} \Vert w\Vert _{W^{1,2}_p(Q_{\tau ,T})}\le C_p\Vert \phi \Vert _{L^p(Q_{\tau ,T})},\quad 1<p<\infty \end{aligned}$$

(2.8)

where the space $W^{1,2}_p(Q_{\tau ,T})$ is defined as

$$\begin{aligned} W^{1,2}_p(Q_{\tau ,T}):= \left\{ f\in L^{p}(Q_{\tau ,T}):\, \Vert f\Vert _{W^{1,2}_p(Q_{\tau ,T})}:= \sum _{2r+s\le 2}\Vert \partial _t^r\partial _x^sf\Vert _{L^{p}(Q_{\tau ,T})} < +\infty \right\} . \end{aligned}$$

We have the embedding (see [16, Lemma II.3.4]), for all $t\in (\tau ,T)$,

$$\begin{aligned} \Vert w(t)\Vert _{W^{2(1-\frac{1}{p}),p}(\Omega )}\le C_{p,T}\Vert w\Vert _{W^{1,2}_p(Q_{\tau ,T})},\quad 1<p<\infty . \end{aligned}$$

(2.9)

Moreover, by choosing $p> \frac{n+2}{2-\gamma } > \frac{n+2}{2}$ we have

$$\begin{aligned} W^{2(1-\frac{1}{p}),p}(\Omega )\hookrightarrow W^{2-\delta ,\infty }(\Omega ), \end{aligned}$$

(2.10)

where

$$\begin{aligned} \frac{1}{\infty }<\frac{1}{p}-\frac{2(1-\frac{1}{p})-2+\delta }{n} \quad \Leftrightarrow \quad \delta< \frac{n+2}{p} < 2 - \gamma . \end{aligned}$$

(2.11)

Using interpolation, we have

$$\begin{aligned} \Vert w\Vert _{C^1(\Omega )}\le & {} C\Vert w\Vert _{C^{\gamma }(\Omega )}^\theta \cdot \Vert w\Vert _{W^{2-\delta ,\infty }(\Omega )}^{1-\theta }\nonumber \\\le & {} C_p\Vert w\Vert _{C^{\gamma }(\Omega )}^\theta \cdot \Vert \phi \Vert _{L^p(Q_{\tau ,T})}^{1-\theta }\nonumber \\\le & {} C_{p,T-\tau }\Vert w\Vert _{C^{\gamma }(\Omega )}^\theta \cdot \Vert \phi \Vert _{L^\infty (Q_{\tau ,T})}^{1-\theta }, \end{aligned}$$

(2.12)

where $1=\gamma \theta +(2-\delta )(1-\theta )$, $\theta =\frac{1-\delta }{2-\delta -\gamma }$ and $0<\delta <2-\gamma .$ Thus

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{C^1(\Omega )}&\le C\Vert u_0\Vert _{C^1(\Omega )}+C_{T-\tau }H^{\frac{1-\delta }{2-\gamma -\delta }} \Vert \phi \Vert _{L^{\infty }(Q_{\tau ,T})}^{\frac{1-\gamma }{2-\gamma -\delta }}. \end{aligned} \end{aligned}$$

(2.13)

$\square$

The following results are specifically designed for the case of one dimension.

Lemma 2.5

Let $\Omega = (0,L)$ for $L>0$, $0\le \tau < T$. Let $f\in L^{\infty }(Q_{\tau ,T})$ and $a:Q_{\tau ,T}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} 0<\alpha _1 \le a(x,t) \le \alpha _1^{-1} \quad \forall (x,t)\in Q_{\tau ,T}, \end{aligned}$$

(2.14)

for some $\alpha _1>0$. Let u be the solution of the parabolic equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{1}{a(x,t)}\partial _t u - \partial _{xx} u = f, &{}\quad (x,t)\in Q_{\tau ,T},\\ \partial _xu(0,t) = \partial _xu(L,t) = 0, &{}\quad t\in (\tau ,T),\\ u(x,\tau ) = 0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

(2.15)

If $u\in L^{\infty }(Q_{\tau ,T})$, $u\ge 0$, and $u_t\ge 0$, then we have

$$\begin{aligned} \Vert \partial _x u\Vert _{L^{\infty }(Q_{\tau ,T})}\le C, \end{aligned}$$

where C depends on $T-\tau$, $\alpha _1$, $\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}$ and $\Vert u\Vert _{L^{\infty }(Q_{\tau ,T})}$.

Proof

We define ${\bar{Q}}_{\tau ,T}=(-L,2L)\times (\tau ,T)$, and ${{\bar{u}}}$, ${{\bar{a}}}$, ${{\bar{f}}}$: ${\bar{Q}}_{\tau ,T} \rightarrow \mathbb {R}$ as follows

$$\begin{aligned} {\bar{u}}(x,t)= & {} {\left\{ \begin{array}{ll} u(x,t),&{}\quad x\in [0,L],\\ u(2L-x,t),&{}\quad x\in (L,2L],\\ u(-x,t),&{}\quad x\in [-L,0), \end{array}\right. } \end{aligned}$$

(2.16)

$$\begin{aligned} {\bar{a}}= & {} {\left\{ \begin{array}{ll} a(x,t),&{}\quad x\in [0,L],\\ a(2L-x,t),&{} \quad x\in (L,2L],\\ a(-x,t),&{}\quad x\in [-L,0) \end{array}\right. } \end{aligned}$$

(2.17)

and

$$\begin{aligned} {\bar{f}}={\left\{ \begin{array}{ll} f(x,t),&{}\quad x\in [0,L],\\ f(2L-x,t),&{}\quad x\in (L,2L],\\ f(-x,t),&{} \quad x\in [-L,0). \end{array}\right. } \end{aligned}$$

(2.18)

Since u solves (2.15), it follows that ${\bar{u}}$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{1}{{{\bar{a}}}}\partial _t {\bar{u}} - \partial _{xx} {\bar{u}} = {\bar{f}}, &{} \quad (x,t)\in {\bar{Q}}_{\tau ,T},\\ \partial _x{\bar{u}}(-L,t) = \partial _x{\bar{u}}(2L,t) = 0, &{}\quad t\in (\tau ,T),\\ {\bar{u}}(x,\tau ) = 0, &{}\quad x\in (-L,2L). \end{array}\right. } \end{aligned}$$

(2.19)

Moreover,

$$\begin{aligned} \Vert {\bar{f}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}=\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}\quad \text {and }~ \Vert {\bar{u}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}=\Vert u\Vert _{L^{\infty }(Q_{\tau ,T})}. \end{aligned}$$

(2.20)

Fix $\rho \in (0,L)$ and $x_0\in (0,L)$, and let $\zeta =\zeta (|x-x_0|)$ with $\zeta \in C^{\infty }_0(\mathbb {R})$ satisfying

$$\begin{aligned}\zeta (s)= {\left\{ \begin{array}{ll} 1,~&{}\quad 0\le s\le \frac{\rho }{2},\\ 0,~&{}\quad s\ge \rho >0, \end{array}\right. } \end{aligned}$$

$0\le \zeta \le 1$ and $|\zeta '|\le \frac{1}{\rho }$. By multiplying the pde in (2.19) with ${\bar{u}}\zeta ^2$ and integrating over $[x_0-\rho ,x_0+\rho ]\times (\tau ,T)$, we obtain that

$$\begin{aligned} \begin{aligned}&\int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }\frac{1}{{{\bar{a}}}}{\bar{u}}_t{\bar{u}}\zeta ^2 dxdt+\int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }(\partial _{x}{\bar{u}})^2\zeta ^2 dxdt\\&\quad =\int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }-2 {\bar{u}}\partial _{x} {\bar{u}}\zeta \partial _x\zeta dxdt+\int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }{\bar{f}}{\bar{u}}\zeta ^2 dxdt. \end{aligned} \end{aligned}$$

The first term on the left hand side is non-negative due to the assumptions on a and $u, u_t \ge 0$. By applying Cauchy’s inequality $-2{\bar{u}}\partial _{x}{\bar{u}}\zeta \partial _x\zeta \le \frac{1}{2}(\partial _{x}{\bar{u}})^2\zeta ^2+2{\bar{u}}^2(\partial _x\zeta )^2$, we have

$$\begin{aligned}{} & {} \int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }(\partial _x{\bar{u}})^2\zeta ^2 dxdt\nonumber \\{} & {} \quad \le 4\int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }{\bar{u}}^2(\partial _x\zeta )^2 dxdt+2\Vert {\bar{f}}\Vert _{L^{\infty }({\bar{Q}}_T)}\Vert {\bar{u}}\Vert _{L^{\infty }({\bar{Q}}_T)} \int ^T_\tau \int ^{x_0+\rho }_{x_0-\rho }\zeta ^2dxdt\nonumber \\{} & {} \quad \le \frac{8 (T-\tau )}{\rho }\Vert {\bar{u}}\Vert ^2_{L^{\infty }({\bar{Q}}_{\tau ,T})}+4\rho T\Vert {\bar{f}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}\Vert {\bar{u}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}:=C_0, \end{aligned}$$

(2.21)

note that, where (2.20) implies $C_0$ depends only on $\rho ,~T-\tau ,~\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}$ and $\Vert u\Vert _{L^{\infty }(Q_{\tau ,T})}$. Therefore

$$\begin{aligned} \begin{aligned} \int ^T_\tau \int ^{x_0+\frac{\rho }{2}}_{x_0-\frac{\rho }{2}}(\partial _x{\bar{u}})^2dxdt\le C_0. \end{aligned} \end{aligned}$$

(2.22)

Differenting equation (2.19) with respect to x gives

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t (\partial _x{\bar{u}}) - \partial _x({\bar{a}}\partial _{xx} {\bar{u}}) =\partial _x({\bar{a}}{\bar{f}}), &{}\quad (x,t)\in {\bar{Q}}_{\tau ,T},\\ \partial _x{\bar{u}}(-L,t) = \partial _x{\bar{u}}(2L,t) = 0, &{}\quad t\in (\tau ,T),\\ \partial _x{\bar{u}}(x,\tau ) = 0, &{}\quad x\in (-L,2L). \end{array}\right. } \end{aligned}$$

Let ${\widehat{Q}}_{\tau ,T}=[x_0-\frac{\rho }{3},x_0+\frac{\rho }{3}]\times (\tau ,T)$. From (2.22), (2.14) and the fact ${\bar{a}}{\bar{f}}\in L^{\infty }({\bar{Q}}_{\tau ,T})$, by [16, Chapter 3, Section 8, Theorem 8.1], we have

$$\begin{aligned} \begin{aligned} \Vert \partial _x {\bar{u}}\Vert _{L^{\infty }({\widehat{Q}}_{\tau ,T})}\le C_1, \end{aligned} \end{aligned}$$

where $C_1$ depends on $C_0$ and the constant $\alpha _1$, but is independent of $x_0$. Thus, since $x_0 \in (0,L)$ arbitrary and $\rho >0$, we can obtain

$$\begin{aligned} \Vert \partial _x u\Vert _{L^{\infty }(Q_{\tau ,T})}\le C_1. \end{aligned}$$

$\square$

Lemma 2.6

(Hölder continuity in one dimension) Let $\Omega = (0,L)$ for some $L>0$, and $0\le \tau < T$. Assume that $f\in L^{\infty }(Q_{\tau ,T})$ and $a:Q_{\tau ,T}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} 0<\alpha _2 \le a(x,t) \le \alpha _2^{-1} \quad \forall (x,t)\in Q_{\tau ,T}, \end{aligned}$$

for some $\alpha _2>0$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t w - \partial _x(a\partial _x w ) = \partial _xf, &{}\quad (x,t)\in Q_{\tau ,T},\\ w(0,t) = w(L,t) = 0, &{}\quad t\in (\tau ,T),\\ w(x,\tau ) = 0, &{}\quad x\in \Omega \end{array}\right. } \end{aligned}$$

(2.23)

has a unique weak solution. Moreover, if $w\in L^{\infty }(Q_{\tau ,T})$, then w is Hölder continuous with an exponent $\delta \in (0,1)$, i.e.

$$\begin{aligned} |w(x,t) - w(x',t')| \le K_0\left( |x - x'|^{\delta } + |t-t'|^{\delta /2}\right) , \quad \forall (x,t), (x',t') \in Q_{\tau ,T}, \end{aligned}$$

where the constant $K_0$ depends on $\Vert w\Vert _{L^{\infty }(Q_{\tau ,T})},~\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}$ and $\alpha _2$.

Proof

Similar to the proof of Lemma 2.5, define ${\bar{Q}}_{\tau ,T}=(-L,2L)\times (\tau ,T)$, and set

$$\begin{aligned} {\bar{w}}(x,t)={\left\{ \begin{array}{ll} w(x,t),&{} \quad x\in [0,L],\\ -w(2L-x,t),&{}\quad x\in (L,2L],\\ -w(-x,t),&{}\quad x\in [-L,0). \end{array}\right. } \end{aligned}$$

(2.24)

Then ${\bar{w}}$ is the weak solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t {\bar{w}} - \partial _x({\bar{a}}\partial _{x} {\bar{w}}) =\partial _x{\bar{f}}, &{}\quad (x,t)\in {\bar{Q}}_{\tau ,T},\\ {\bar{w}}(-L,t) = {\bar{w}}(2L,t) = 0, &{}\quad t\in (\tau ,T),\\ {\bar{w}}(x,\tau ) = 0, &{}\quad x\in (-L,2L), \end{array}\right. } \end{aligned}$$

(2.25)

where ${\bar{a}}$ and ${\bar{f}}$ are given similarly to (2.24). Obviously,

$$\begin{aligned} \Vert {\bar{f}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}=\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}\quad \text {and }~ \Vert {\bar{w}}\Vert _{L^{\infty }({\bar{Q}}_{\tau ,T})}=\Vert w\Vert _{L^{\infty }(Q_{\tau ,T})}. \end{aligned}$$

We fix $(x_0, t_0)\in Q_{\tau ,T}$ and $0<\varrho <L$, denote

$$\begin{aligned} B_\varrho = \{x\in \Omega : x\in (x_0-\varrho ,x_0+\varrho )\} \end{aligned}$$

and, for $0<r\le t_0$,

$$\begin{aligned} Q(\varrho ,r) = B_\varrho \times (t_0-\tau ,t_0) = \{(x,t): x\in (x_0-\varrho ,x_0+\varrho ), t_0 - r< t < t_0\}. \end{aligned}$$

For any $0< \tau _2< \tau _1 < \min \{1,r\}$ and $0< \varrho _2< \varrho _1 < \varrho$ such that $Q(\varrho _1, \tau _1)\subset {\bar{Q}}_T$, let $\xi : {\bar{Q}}_T \rightarrow [0,1]$ be a smooth cut-off function such that

$$\begin{aligned} \xi (x,t) = {\left\{ \begin{array}{ll} 1 &{}\quad \text { if } \quad (x,t) \in Q(\varrho _2, \tau _2),\\ 0 &{}\quad \text { if } \quad (x,t) \not \in Q(\varrho _1, \tau _1). \end{array}\right. } \end{aligned}$$

For any $k>0$, we denote by ${\bar{w}}^{(k)} = ({\bar{w}}-k)_+$. By multiplying the pde in (2.25) with ${\bar{w}}^{(k)}\xi ^2$. Integrating over $B_{\varrho _1}\times (t_0-\tau _1, t)$, $t_0 - \tau _2< t < t_0$, and by integrating by parts, we calculate

$$\begin{aligned}{} & {} \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}\partial _t{\bar{w}} {\bar{w}}^{(k)}\xi ^2dx ds + \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds\nonumber \\{} & {} \quad = -2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}} {\bar{a}}(\partial _x {\bar{w}}^{(k)}\cdot \partial _x \xi ) {\bar{w}}^{(k)}\xi dxds -\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{f}}\partial _x{\bar{w}}^{(k)}\xi ^2dxds\nonumber \\{} & {} \qquad -2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{f}}{\bar{w}}^{(k)}\xi \partial _x\xi dxds. \end{aligned}$$

(2.26)

Since

$$\begin{aligned}{} & {} \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}\partial _t{\bar{w}} {\bar{w}}^{(k)}\xi ^2dx ds \nonumber \\{} & {} \quad = \frac{1}{2}\int _{B_{\varrho _1}}({\bar{w}}^{(k)}(t))^2\xi (t)^2dx - \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}({\bar{w}}^{(k)})^2\xi \partial _t\xi dxds\nonumber \\{} & {} \quad \ge \frac{1}{2}\int _{B_{\varrho _2}}|{\bar{w}}^{(k)}(t)|^2dx - \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}({\bar{w}}^{(k)})^2\xi \partial _t\xi dxds\nonumber \\{} & {} \quad \ge \frac{1}{2}\int _{B_{\varrho _2}}|{\bar{w}}^{(k)}(t)|^2dx - \int _{Q(\varrho _1,\tau _1)}|{\bar{w}}^{(k)}|^2|\xi _t|dxds, \end{aligned}$$

(2.27)

where we used $\xi (\cdot , t_0-\tau _1) = 0$ at the first step and $\xi |_{Q(\rho _2,\tau _2)} \equiv 1$ at the second step. By applying the Cauchy-Schwarz inequality, we get

$$\begin{aligned}{} & {} -2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}(\partial _x {\bar{w}}^{(k)}\cdot \partial _x\xi ) {\bar{w}}^{(k)}\xi dxds \nonumber \\{} & {} \le \frac{1}{4} \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds+ 4\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}} {\bar{a}} |{\bar{w}}^{(k)}|^2|\partial _x \xi |^2dxds\nonumber \\{} & {} \quad \le \frac{1}{4} \int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds+ 4\int _{Q(\varrho _1,\tau _1)}{\bar{a}}|{\bar{w}}^{(k)}|^2|\partial _x \xi |^2dxds.\nonumber \\ \end{aligned}$$

(2.28)

Denoting by $\chi _{\{{\bar{w}} > k \}}$ the characteristic function of the set $\{(x,t)\in {\bar{Q}}_{\tau ,T}: {\bar{w}}(x,t) > k\}$,

we can estimate

$$\begin{aligned}{} & {} -\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{f}}\partial _x {\bar{w}}^{(k)}\xi ^2dxds-2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{f}}{\bar{w}}^{(k)}\xi \partial _x\xi dxds\nonumber \\{} & {} \quad \le \frac{1}{4}\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds + \Vert {\bar{f}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}\frac{1}{{\bar{a}}}\xi ^2\chi _{\{{\bar{w}}> k\}}dxds\nonumber \\{} & {} \qquad +2\Vert {\bar{f}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}|\xi |^2\chi _{\{{\bar{w}}> k\}}dxds+\frac{1}{2}\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}|{\bar{w}}^{(k)}|^2|\partial _x\xi |^2dxds\nonumber \\{} & {} \quad \le \frac{1}{4}\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds + \frac{\Vert {\bar{f}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2}{{\alpha _2}}\int _{Q(\varrho _1,\tau _1)}\chi _{\{{\bar{w}}> k\}}dxds\nonumber \\{} & {} \qquad +2\Vert {\bar{f}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2\int _{Q(\varrho _1,\tau _1)}\chi _{\{{{\bar{w}}}> k\}}dxds+\frac{1}{2}\int _{Q(\varrho _1,\tau _1)}|{\bar{w}}^{(k)}|^2|\partial _x\xi |^2dxds\nonumber \\{} & {} \quad \le \frac{1}{4}\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}{\bar{a}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds + C_{T,\tau }\int _{Q(\varrho _1,\tau _1)}\left( |w^{(k)}|^2|\partial _x\xi |^2 + \chi _{\{{{\bar{w}}} > k \}} \right) dxds\nonumber \\ \end{aligned}$$

(2.29)

where

$$\begin{aligned} C_{T,\tau }:= \max \left\{ \frac{\Vert {{\bar{f}}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2}{{\alpha _2}}; \frac{1}{2}; 2\Vert {{\bar{f}}}\Vert _{L^{\infty }(Q_{\tau ,T})}^2 \right\} . \end{aligned}$$

We now insert the estimates (2.27), (2.28) and (2.29) into (2.26) to obtain that, for all $t\in (t_0-\tau _2, t_0)$,

$$\begin{aligned}{} & {} \frac{1}{2}\Vert {\bar{w}}^{(k)}(t)\Vert _{L^2(B_{\varrho _2})}^2 + \frac{{\alpha _2}}{2}\int _{t_0-\tau _1}^t\int _{B_{\varrho _1}}|\partial _x {\bar{w}}^{(k)}|^2\xi ^2dxds\nonumber \\{} & {} \quad \le \int _{Q(\varrho _1,\tau _1)}|{\bar{w}}^{(k)}|^2|\xi _t|dxds+\frac{4}{{\alpha _2}}\int _{Q(\varrho _1,\tau _1)}|{\bar{w}}^{(k)}|^2|\partial _x \xi |^2dxds\nonumber \\{} & {} \qquad + C_{T,\tau } \int _{Q(\varrho _1,\tau _1)}\Big (|{\bar{w}}^{(k)}|^2|\partial _x\xi |^2+\chi _{\{{\bar{w}}> k\}}\Big )dxds\nonumber \\{} & {} \quad \le C_{T,\tau } \int _{Q(\varrho _1,\tau _1)}\Big (|{\bar{w}}^{(k)}|^2(|\xi _t|+|\partial _x \xi |^2) + \chi _{\{{\bar{w}}> k\}}\Big )dxds. \end{aligned}$$

(2.30)

By adding the inequality $\frac{{\alpha _2}}{2} \int _{t_0-\tau _1}^{t}\int _{B_{\varrho _1}}|{\bar{w}}^{(k)}|^2\xi ^2dxds \le \frac{{\alpha _2}}{2} \int _{Q(\varrho _1,\tau _1)}|{\bar{w}}^{(k)}|^2dxds$ on both sides of (2.30), and taking the supremum over $t\in (t_0 - \tau _2, t_0)$, we have that

$$\begin{aligned}{} & {} \frac{1}{2}\left( \sup _{t_0 - \tau _2< t < t_0}\Vert {\bar{w}}^{(k)}(t)\Vert _{L^2(B_{\varrho _2})}^2 + \int _{t_0 - \tau _2}^{t_0}\Vert {\bar{w}}^{(k)}\Vert _{H^1(B_{\varrho _2})}^2ds\right) \nonumber \\{} & {} \quad \le C_{T,\tau } \int _{Q(\varrho _1,\tau _1)}\Big (|{\bar{w}}^{(k)}|^2(|\xi _t|+|\partial _x \xi |^2) +\chi _{\{{\bar{w}}> k\}}\Big )dxds. \end{aligned}$$

(2.31)

Finally, due to the definition of the cut-off function $\xi$, there exists a constant $C\ge 1$ independent of $\varrho _i$ and $\tau _i$ such that $|\partial _x \xi | \le C(\varrho _1-\varrho _2)^{-1}$ and $|\partial _t \xi | \le C(\tau _1 - \tau _2)^{-1}$. Noting also $1 \le (\tau _1 - \tau _2)^{-1}$ since $\tau _1, \tau _2 \in (0,1)$, we get from (2.31) the energy estimate

$$\begin{aligned}{} & {} \sup _{t_0-\tau _2< t < t_0}\Vert ({\bar{w}}-k)_+\Vert _{L^2(B_{\varrho _2})}^2+ \int _{t_0-\tau _2}^{t_0}\Vert ({\bar{w}}-k)_+\Vert _{H^1(B_{\varrho _2})}^2ds\nonumber \\{} & {} \quad \le C_{T,\tau }\left[ ((\varrho _1-\rho _2)^{-2} + (\tau _1 - \tau _2)^{-1})\Vert ({\bar{w}} -k)_+\Vert _{L^2(Q(\varrho _1,\tau _1))}^2 + \int _{Q(\varrho _1,\tau _1)}\chi _{\{{\bar{w}}>k\}}dxds\right] ,\nonumber \\ \end{aligned}$$

(2.32)

where the constant C depends only on $\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}$ and $\alpha$, but is independent of $x_0$ and $t_0$.

By the arbitrariness of $x_0\in (0,L)$ and $t_0\in (\tau ,T)$, combining with the estimates (2.32) and the fact $\Vert w\Vert _{L^\infty (Q_{\tau ,T})} \le C_{T,\tau }$, thanks to [16, Theorem II.7.1, Lemmas 7.2 and 7.3], we can obtain the local Hölder continuity of ${\bar{w}}$ on $(-\varrho _2,L+\varrho _2)\times (\tau ,T)$, which implies the global Hölder continuity of w on $Q_{\tau ,T}$ immediately, that is, there exist constant $\delta \in (0,1)$ such that

$$\begin{aligned} |w(x,t) -w(x', t')| \le C(|x - x'|^{\delta } + |t - t'|^{\delta /2}) \quad \text {for all} \quad (x,t), (x', t') \in Q_{\tau ,T}, \end{aligned}$$

where the constant C depends only on $\Vert f\Vert _{L^{\infty }(Q_{\tau ,T})}$, $\Vert w\Vert _{L^\infty (Q_{\tau ,T})}$ and ${\alpha _2}$. $\square$

2.2 Global existence

The following bound in $L^\infty (0,T;L^1(\Omega ))$ is immediate.

Lemma 2.7

Assume (A1) and (A2). For any $0<T<T_{\max }$, it holds

$$\begin{aligned} \sup _{i=1,\ldots , m}\Vert u_i(t)\Vert _{L^{1}(\Omega )} \le C_T \quad \text { for all } \quad t\in (0,T). \end{aligned}$$

Proof

By summing the equations in (1.1), integrating on $\Omega$ and using (A2), we have

$$\begin{aligned} \frac{d}{dt}\sum _{i=1}^{m}\int _{\Omega }u_i(x,t)dx \le k_0 + k_1\sum _{i=1}^{m}\int _{\Omega }u_i(x,t)dx. \end{aligned}$$

The classical Gronwall inequality gives the desired estimate. $\square$

Summing the equations of (1.1), it follows from (A2’) that

$$\begin{aligned} \sum _{i=1}^{m}u_i(x,t)\! -\! \partial _{xx} \int _0^t\sum _{i=1}^{m}d_iu_i(x,s)ds = \sum _{i=1}^{m}u_{i,0}(x) \!+\! \int ^t_0g(x,s)ds =: G(x,t).\nonumber \\ \end{aligned}$$

(2.33)

Denote by

$$\begin{aligned} v(x,t):= \int _0^t\sum _{i=1}^{m}d_iu_i(x,s)ds. \end{aligned}$$

(2.34)

Then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} b(x,t)\partial _tv(x,t) - \partial _{xx} v(x,t) = G(x,t), &{}\quad x\in \Omega , \; t>0,\\ \partial _xv(0,t) = \partial _xv(L,t) = 0, &{}\quad t>0,\\ v(x,0) = 0, &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(2.35)

where

$$\begin{aligned} \min _{i=1,\ldots , m}\left\{ \frac{1}{d_i} \right\} \le b(x,t):= \frac{\sum _{i=1}^{m}\nolimits u_i(x,t)}{\sum _{i=1}^{m}\nolimits d_iu_i(x,t)} \le \max _{i=1,\ldots , m}\left\{ \frac{1}{d_i} \right\} . \end{aligned}$$

(2.36)

It follows from (2.33) that v is a solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv(x,t) - \partial _{xx} v(x,t) = U_d(x,t) + G(x,t), &{}\quad x\in \Omega , \; t>0,\\ \partial _x v(0,t) = \partial _xv(L,t) = 0, &{}\quad t>0,\\ v(x,0) = 0, &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(2.37)

where

$$\begin{aligned} U_d(x,t):= \sum _{i=1}^{m}(d_i - 1)u_i(x,t). \end{aligned}$$

(2.38)

The following lemma is crucial as it shows that, in one dimension, also the spatial derivative of v is Hölder continuous.

Lemma 2.8

The function v defined in (2.34) satisfies

$$\begin{aligned} |\partial _xv(x,t) - \partial _xv(x',t)| \le {\mathcal {C}}_0|x'-x|^{\alpha } \quad \forall (x',t),(x,t)\in Q_T, \end{aligned}$$

where ${\mathcal {C}}_0$ and $\alpha \in (0,1)$ depend only on T, $\Vert G\Vert _{L^{\infty }(Q_T)}$, L, and $d_i$, $0<T<T_{\max }$.

Proof

We first claim that

$$\begin{aligned} \Vert \partial _xv(x,t)\Vert _{L^{\infty }(Q_T)} \le C_T. \end{aligned}$$

(2.39)

To prove this we first show

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(Q_T)} \le C_T. \end{aligned}$$

(2.40)

This was in fact showed in [8], but in one dimension, it can be shown by elementary arguments. Note also that $C_T$ in (2.39) and (2.40) depend only on T, $\Vert G\Vert _{L^{\infty }(Q_T)}$, L, and $d_i$. Indeed, thanks to Lemma 2.7 we have

$$\begin{aligned} \Vert v(t)\Vert _{L^{1}(\Omega )} \le \sum _{i=1}^{m}d_i\int _0^t\Vert u_i(s)\Vert _{L^{1}(\Omega )}ds \le C_T \quad \text { for all } t\in (0,T). \end{aligned}$$

(2.41)

Multiplying both sides of (2.33) by v, integrating over $\Omega$, and using $u_i \ge 0$ and $v\ge 0$, we have

$$\begin{aligned} \int _{\Omega }|\partial _x v(x,t)|^2dx \le \int _{\Omega }G(x,t)v(x,t)dx \le \frac{1}{4}\Vert G(t)\Vert _{L^{2}(\Omega )}^2 + \Vert v(t)\Vert _{L^{2}(\Omega )}^2. \end{aligned}$$

Adding $\Vert v(t)\Vert _{L^{2}(\Omega )}^2$ to both sides gives

$$\begin{aligned} \Vert v\Vert _{H^1(\Omega )}^2 \le \frac{|\Omega |}{4}\Vert G\Vert _{L^{\infty }(Q_T)}^2 + 2\Vert v(t)\Vert _{L^{2}(\Omega )}^2. \end{aligned}$$

The interpolation inequality $\Vert v(t)\Vert _{L^2(\Omega )} \le C\Vert v(t)\Vert _{H^1(\Omega )}^{1/3}\Vert v(t)\Vert _{L^1(\Omega )}^{2/3}$ and Cauchy-Schwarz’s inequality lead to

$$\begin{aligned} {\Vert v\Vert _{H^1(\Omega )} \le C\Vert G\Vert _{L^{\infty }(Q_T)} \le C_T.} \end{aligned}$$

Now we use the one dimensional embedding $H^1(\Omega )\hookrightarrow L^\infty (\Omega )$ to eventually obtain the estimate (2.40).

From (2.40) and the fact that $v\ge 0$, $\partial _t v\ge 0$, we can apply Lemma 2.5 to obtain the boundedness of the gradient (2.39).

Define $a(x,t):= (b(x,t))^{-1}$ we get from (2.35) that

$$\begin{aligned} \partial _tv(x,t) - a(x,t)\partial _{xx}v(x,t) = a(x,t)G(x,t). \end{aligned}$$

By differentiating this equation with respect to x, and denoting $w(x,t):= \partial _x v(x,t)$, we obtain

$$\begin{aligned} \partial _t w(x,t) - \partial _x(a(x,t)\partial _x w(x,t)) = \partial _x(a(x,t)G(x,t)). \end{aligned}$$

Note that w satisfies homogeneous Dirichlet boundary condition since $w(0,t) = \partial _xv(0,t) = 0$, and $w(L,t) = \partial _xv(L,t) = 0$.

Now we can apply Lemma 2.6 to obtain the Hölder continuity of w, which finishes the proof of Lemma 2.8. $\square$

Lemma 2.9

Assume (A1) and (A2). Let v be the solution to (2.37). It holds, for any $0<\delta < 2- \alpha$,

$$\begin{aligned} \sup _{Q_T}|\partial _{xx}v| \le \mathcal {C}_1\mathcal {C}_0^{\frac{1-\delta }{2-\alpha -\delta }}\left[ \sup _{Q_T}|\partial _x U_d + \partial _x G| \right] ^{\frac{1-\alpha }{2-\alpha -\delta }} \end{aligned}$$

where the constants $\mathcal {C}_0$ and $\alpha$ are given in Lemma 2.8, and $\mathcal {C}_1=\mathcal {C}_1(T, \mathcal {C}_0, \alpha , \delta$).

Proof

By differentiating (2.37) with respect to x, we have

$$\begin{aligned} \partial _t(\partial _x v) - \partial _{xx}(\partial _xv(x,t)) = \partial _x\left( U_d(x,t) + G(x,t)\right) , \end{aligned}$$

and $\partial _xv$ satisfies homogeneous Dirichlet boundary condition $\partial _xv(0,t) = \partial _xv(L,t) = 0$. Thanks to the Hölder continuity of $\partial _xv$ in Lemma 2.8, we can apply Lemma 2.4 to conclude the proof of Lemma 2.9. $\square$

We are now ready to show the global existence part of Theorem 1.2.

Proof of Theorem 1.2 - Global existence

We denote by

$$\begin{aligned} U:= \sup _{Q_T}\sup _{i}|u_i(x,t)|. \end{aligned}$$

From the equation of $u_i$

$$\begin{aligned} \partial _t u_i - d_i\partial _{xx}u_i = f_i(x,t,u), \end{aligned}$$

assumption (A3), and Lemma 2.3, we have

$$\begin{aligned} \sup _{Q_T}|\partial _x u_i| \le \sup _{\Omega }|\partial _xu_{i,0}| + CU^{\frac{1}{2}}\left[ 1+U^{3+\varepsilon }\right] ^{\frac{1}{2}} \le C\left( 1+U^{2+\frac{\varepsilon }{2}}\right) . \end{aligned}$$

From (2.38), we know

$$\begin{aligned} \sup _{Q_T}|\partial _x U_d|\le \sum _{i=1}^{m}(d_i - 1)\sup _{Q_T}|\partial _x u_i|. \end{aligned}$$

It follows that

$$\begin{aligned} \sup _{Q_T}|\partial _x U_d| \le C\left( 1+U^{2+\frac{\varepsilon }{2}}\right) . \end{aligned}$$

Therefore, Lemma 2.9 implies

$$\begin{aligned} \sup _{Q_T}|\partial _{xx}v| \le C_T\left[ 1+U^{2+\frac{\varepsilon }{2}}\right] ^{\frac{1-\alpha }{2-\alpha -\delta }} \le C_T\left[ 1+U^{(2+\frac{\varepsilon }{2})\frac{1-\alpha }{2-\alpha -\delta }}\right] . \end{aligned}$$

It then follows from (2.33) that

$$\begin{aligned} U \le \sup _{Q_T}|\partial _{xx}v| + \Vert G\Vert _{L^{\infty }(Q_T)} \le C_T\left[ 1 + U^{(2+\frac{\varepsilon }{2})\frac{1-\alpha }{2-\alpha -\delta }}\right] . \end{aligned}$$

We can choose $\varepsilon$ and $\delta$ small enough such that

$$\begin{aligned} \left( 2+\frac{\varepsilon }{2}\right) \frac{1-\alpha }{2-\alpha -\delta }<1 \quad \text {or equivalently}\quad \varepsilon <\frac{2(\alpha -\delta )}{1-\alpha }. \end{aligned}$$

By apply Young’s inequality to finally obtain

$$\begin{aligned} U \le C_T, \end{aligned}$$

which confirms the global existence of (1.1). $\square$

2.3 Uniform-in-time bounds

Now we consider the case where $g\equiv 0$ in the assumption (A2’). To obtain the uniform-in-time bound for this solution, we just need to show that

$$\begin{aligned} \sup _{\tau \in {\mathbb {N}}}\Vert u_i\Vert _{L^\infty (Q_{\tau ,\tau +1})}\le C. \end{aligned}$$

where we recall $Q_{\tau , \tau +1} = \Omega \times (\tau ,\tau +1)$.

Let $\tau \in {\mathbb {N}}$ and $\varphi _\tau : \mathbb {R}_+ \rightarrow [0,1]$ a smooth function such that $\varphi _\tau (s) = 0$ for $s\le \tau$, $\varphi _\tau (s) = 1$ for $s\ge \tau +1$, and $\varphi _\tau '(s) \ge 0$ for all $s\in \mathbb {R}_+$. By multiplying the equation (1.1) by $\varphi _\tau$ and denoting

$$\begin{aligned} w_i(x,t) = \varphi _\tau (t)u_i(x,t), \end{aligned}$$

(2.42)

we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t w_i - d_i\partial _{xx}w_i = \varphi _\tau ' u_i + \varphi _\tau f_i(x,t,u), &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xw_i(0,t) = \partial _xw_i(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ w_i(x,\tau ) = 0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

(2.43)

We sum the equations of (2.43), use $g\equiv 0$, and integrate the result on $(\tau ,t)$, $t\in (\tau ,\tau +2)$ to obtain

$$\begin{aligned} \sum _{i=1}^{m}w_i(x,t) - \partial _{xx}\int _{\tau }^{t}\sum _{i=1}^{m}d_iw_i(x,s)ds = z(x,t):= \sum _{i=1}^{m}\int _{\tau }^{t}\varphi _\tau ' u_i(x,s)ds.\nonumber \\ \end{aligned}$$

(2.44)

With

$$\begin{aligned} v(x,t):= \int _{\tau }^{t}\sum _{i=1}^{m}d_iw_i(x,s)ds \end{aligned}$$

(2.45)

and the function b(x, t) defined as in (2.36) we also have

$$\begin{aligned} {\left\{ \begin{array}{ll} b(x,t)\partial _t v - \partial _{xx}v = z(x,t), &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xv(0,t) = \partial _xv(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ v(x,\tau ) = 0, &{}\quad x\in \Omega \end{array}\right. } \end{aligned}$$

(2.46)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v - \partial _{xx}v = \Phi (x,t):= \sum _{i=1}^{m}(d_i-1)w_i(x,t) + z(x,t), &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xv(0,t) = \partial _xv(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ v(x,\tau ) = 0, &{}\quad x\in \Omega . \end{array}\right. }\nonumber \\ \end{aligned}$$

(2.47)

Lemma 2.10

(Lemma 2.1, [9]) Let $\Omega \subset \mathbb {R}^n$ be bounded with smooth boundary, let $d>0$ and $0\le \tau < T$. Let u be the solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - d\Delta u = \psi (x,t), &{}\quad (x,t)\in Q_{\tau ,T},\\ \nabla u\cdot \nu = 0, &{}\quad (x,t)\in \partial \Omega \times (\tau ,T),\\ u(x,\tau ) = u_0(x), &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

Assume that there exists $\gamma \in [0,1)$ such that

$$\begin{aligned} {|u(x,t) - u(x',t)| \le H|x - x'|^{\gamma } \quad \forall (x,t), (x',t)\in Q_{\tau ,T}.} \end{aligned}$$

Then

$$\begin{aligned} \sup _{Q_{\tau ,T}}|\nabla u(x,t)| \le C\Vert u_0\Vert _{C^1(\Omega )} + BH^{\frac{1}{2-\gamma }}\Vert \psi \Vert _{L^{\infty }(Q_{\tau ,T})}^{\frac{1-\gamma }{2-\gamma }} \end{aligned}$$

where $B>0$ and $C>0$ are constants depending only on $\Omega$, n, d and $\gamma$.

Remark 4

It is remarked that Lemma 2.10 looks similar to Lemma 2.3, except for the fact that it is considered in the cylinder $\Omega \times (\tau ,T)$. Due to t-dependence of $\psi$, the problem is non-autonomous. Nevertheless, the proof is in fact similar to that of Lemma 2.3.

Lemma 2.11

Assume (A1) and (A2). There exists a constant $C>0$ such that

$$\begin{aligned} \sup _{\tau \in {\mathbb {N}}} \left( \Vert v\Vert _{L^{\infty }(Q_{\tau ,\tau +2})} + \Vert z\Vert _{L^\infty (Q_{\tau ,\tau +2})}\right) \le C. \end{aligned}$$

Proof

The bound of v was proved in [9, Lemma 3.3]. The bound of z follows immediately due to its definition in (2.44) and the bound of v. $\square$

Lemma 2.12

Assume (A1) and (A2). There exists a constant $C>0$ such that

$$\begin{aligned} \sup _{\tau \in {\mathbb {N}}}\Vert \partial _x v\Vert _{L^{\infty }(Q_{\tau ,\tau +2})} \le C. \end{aligned}$$

(2.48)

Proof

To prove this lemma, we apply Lemma 2.5 to the equation (2.46) on the interval $(\tau ,\tau +2)$. Then, it holds

$$\begin{aligned} \Vert \partial _x v\Vert _{L^{\infty }(Q_{\tau ,\tau +2})} \le \mathscr {C} \end{aligned}$$

where the constant $\mathscr {C}$ depends only on $d_i$ and the bounds $\Vert z\Vert _{L^{\infty }(Q_{\tau ,\tau +2})}$ and $\Vert v\Vert _{L^{\infty }(Q_{\tau ,\tau +2})}$. Since these last two quantities are independent of $\tau \in {\mathbb {N}}$, the desired estimate (2.48) follows. $\square$

Lemma 2.13

Assume (A1) and (A2). There exist $\beta \in (0,1)$ and $H>0$, which is independent of $\tau \in {\mathbb {N}}$, such that

$$\begin{aligned} |\partial _x v(x,t) - \partial _xv(x',t)| \le H|x-x'|^{\beta }, \quad \forall (x,t), (x',t)\in Q_{\tau ,\tau +2}. \end{aligned}$$

Proof

By dividing both sides of (2.46) by b(x, t) then differentiating with respect to x, we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t (\partial _x v) - \partial _x(b(x,t)^{-1}\partial _x(\partial _xv)) = \partial _x(b(x,t)^{-1}z(x,t)), &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xv(0,t) = \partial _xv(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ \partial _xv(x,0) = 0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

Now we can apply Lemma 2.6 to this equation (with w is equal to $\partial _x v$) to obtain the Hölder continuity

$$\begin{aligned} |\partial _x v(x,t) - \partial _xv(x',t)| \le H|x-x'|^{\beta }, \quad \forall (x,t), (x',t)\in Q_{\tau ,\tau +2} \end{aligned}$$

where we notice that the constant H depends on $\Vert \partial _xv\Vert _{L^{\infty }(Q_{\tau ,\tau +2})}$ and $\Vert b^{-1}z\Vert _{L^{\infty }(Q_{\tau ,\tau +2})}$, which are in turn independent of $\tau \in {\mathbb {N}}$, thanks to Lemmas 2.11 and 2.12. $\square$

We also need the following elementary lemma, whose proof is straightforward.

Lemma 2.14

Let $\{y_n\}_{n\ge 0}$ be a nonnegative sequence. Define $\mathcal {N}=\{n\in \mathbb {N}:y_{n-1}\le y_n\}.$ If there exists $c_0>0$ such that

$$\begin{aligned} y_n\le c_0 \quad \text {for all}\quad n\in \mathcal { N}, \end{aligned}$$

then

$$\begin{aligned} y_n\le c=\max \{y_0,c_0\},\quad \text {for all} \quad n\in \mathbb {N}, \end{aligned}$$

where a constant c independent of n.

Proof of Theorem 1.2 - Uniform-in-time bounds

By differentiating (2.47) with respect to x we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t(\partial _x v) - \partial _{xx}(\partial _xv) = \partial _x \Phi , &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xv(0,t) = \partial _x(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ \partial _xv(x,\tau ) = 0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

(2.49)

By applying Lemmas 2.13 and 2.4 to (2.49) we have, for any $0<\delta <2-\beta$,

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}|\partial _{xx}v|\le & {} CH^{\frac{1-\delta }{2-\beta -\delta }}\left[ \sup _{Q_{\tau ,\tau +2}}|\partial _x \Phi |\right] ^{\frac{1-\beta }{2-\beta -\delta }}\nonumber \\\le & {} C\left[ \sup _{Q_{\tau ,\tau +2}}\sum _{i=1}^{m}|\partial _x w_i| + |\partial _x z| \right] ^{\frac{1-\beta }{2-\beta -\delta }}. \end{aligned}$$

(2.50)

Define

$$\begin{aligned} U:= \sup _{Q_{\tau ,\tau +2}}\sup _{i=1,\ldots , m}|u_i(x,t)|. \end{aligned}$$

By applying Lemma 2.10 to the equation (2.43) and using the growth (A3), we have

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}|\partial _x w_i|\le & {} C\sup _{Q_{\tau ,\tau +2}}|w_i|^{1/2}\left[ \sup _{Q_{\tau ,\tau +2}}\left( |\varphi _\tau ' u_i| + |\varphi _\tau f_i(x,t,u)|\right) \right] ^{1/2}\nonumber \\\le & {} CU^{1/2}\left[ U + 1 + U^{3+\varepsilon }\right] ^{1/2}\nonumber \\\le & {} C\left[ 1 + U^{2+\frac{\varepsilon }{2}}\right] . \end{aligned}$$

(2.51)

By integrating (1.1) on $(\tau ,t)$ and denoting $y_i(x,t) = \int _{\tau }^tu_i(x,s)$ we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t y_i - d_i\partial _{xx}y_i = \int _{\tau }^{t}f_i(x,s,u)ds, &{}\quad (x,t)\in Q_{\tau ,\tau +2},\\ \partial _xy_i(0,t) = \partial _xy_i(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ y_i(x,\tau ) = 0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

(2.52)

Using (A3) and Lemma 2.11 we can estimate

$$\begin{aligned}&\sup _{(x,t)\in Q_{\tau ,\tau +2}}\left| \int _{\tau }^{t}f_i(x,s,u)ds \right| \\&\le C\sup _{Q_{\tau ,\tau +2}}\int _{\tau }^{\tau }\left[ 1+\sum _{i=1}^{m}u_i(x,s)\right] \left[ 1 + \sum _{i=1}^{m}u_{i}(x,s)\right] ^{2+\varepsilon }ds\\&\le C\left( 1 + U\right) ^{2+\varepsilon }\sup _{Q_{\tau ,\tau +2}}\int _{\tau }^{\tau +2}\left[ 1+\sum _{i=1}^{m}u_i(x,s)\right] ds\\&\le C\left( 1+U\right) ^{2+\varepsilon }. \end{aligned}$$

Now, we can apply Lemma 2.10 to (2.52) to have

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}|\partial _x y_i| \le C\left[ \sup _{Q_{\tau ,\tau +2}}|y_i|\right] ^{1/2}\left[ \sup _{Q_{\tau ,\tau +2}}\left| \int _{\tau }^t f_i(x,s,u)ds\right| \right] ^{1/2} \le C\left( 1+U\right) ^{1+\frac{\varepsilon }{2}} \end{aligned}$$

(2.53)

where we used $\sup _{Q_{\tau ,\tau +2}}|y_i| \le C$ at the last step thanks to Lemma 2.11. From the definition of z in (2.44) and (2.53) it follows that

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}|\partial _x z| \le C\left[ 1+U^{1+\frac{\varepsilon }{2}}\right] . \end{aligned}$$

Inserting this and (2.51) into (2.50) yields

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}|\partial _{xx}v| \le C\left[ 1 + U^{2+\frac{\varepsilon }{2}}\right] ^{\frac{1-\beta }{2-\beta -\delta }} \le C\left[ 1 + U^{\left( 2+\frac{\varepsilon }{2}\right) \frac{1-\beta }{2-\beta -\delta }}\right] . \end{aligned}$$

From this and (2.44), it implies

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}\sup _{i=1,\ldots , m}|w_i(x,t)| \le \sup _{Q_{\tau ,\tau +2}}|\partial _{xx}v| + C\sup _{Q_{\tau ,\tau +2}}|y_i| \le C\left[ 1 + U^{\left( 2+\frac{\varepsilon }{2}\right) \frac{1-\beta }{2-\beta -\delta }}\right] . \end{aligned}$$

We consider the set

$$\begin{aligned} \mathscr {N}:= \left\{ \tau \in {\mathbb {N}}:\, \sup _{Q_{\tau ,\tau +1}}\sup _{i=1,\ldots , m}|u_i(x,t)| \le \sup _{Q_{\tau +1,\tau +2}}\sup _{i=1,\ldots , m}|u_i(x,t)|\right\} . \end{aligned}$$

Now for all $\tau \in \mathscr {N}$, we use $\varphi _\tau \ge 0$ and $w_i(x,t) = u_i(x,t)$ for all $(x,t)\in Q_{\tau +1,\tau +2}$ to have

$$\begin{aligned} U \le 2\sup _{Q_{\tau +1,\tau +2}}\sup _{i=1,\ldots , m}|u_i(x,t)| \le C\left[ 1 + U^{\left( 2+\frac{\varepsilon }{2}\right) \frac{1-\beta }{2-\beta -\delta }}\right] . \end{aligned}$$

(2.54)

Therefore, for $\varepsilon >0$ and $\delta >0$ small enough,

$$\begin{aligned} \left( 2+\frac{\varepsilon }{2}\right) \frac{1-\beta }{2-\beta -\delta } < 1, \end{aligned}$$

and consequently, (2.54) and Young’s inequality imply

$$\begin{aligned} \sup _{Q_{\tau ,\tau +2}}\sup _{i=1,\ldots , m}|u_i(x,t)| = U \le C \quad \forall \tau \in \mathscr {N}, \end{aligned}$$

(2.55)

where C is independent of $\tau$. Finally, by applying Lemma 2.14, the estimate (2.55) is true for all $\tau \in {\mathbb {N}}$, and the proof of uniform-in-time bounds of Theorem 1.2 is finished. $\square$

3 Proof of Theorem 1.1

Proof of Theorem 1.1

Similar to Theorem 1.2, the global existence follows if one can show that

$$\begin{aligned} \sup _{i=1,\ldots , m}\sup _{T\in (0,T_{\max })}\Vert u_i\Vert _{L^{\infty }(Q_T)} < +\infty , \end{aligned}$$

where $(0,T_{\max })$ is the maximal interval of the local strong solution.

Assume for contradiction that $T_{\max }<\infty$. We follow the idea from [8] which states that with a suitable change of unknowns, and especially adding one more appropriate equation, we can transform a system with the mass control condition (A2) into a system with condition (A2’), that keeps the essential features (A1) and (A3).

In order to do that, we define

$$\begin{aligned} \begin{aligned} w_i(x,t)=e^{-k_1t}u_i(x,t) \quad \text {or equivalently} \quad u_i(x,t)=e^{k_1t}w_i(x,t). \end{aligned} \end{aligned}$$

Direct computations give

$$\begin{aligned} \begin{aligned} \partial _tw_i&=e^{-k_1t}(\partial _tu_i-k_1u_i)\\&=e^{-k_1t}(d_i\partial _{xx} u_i+f_i(x,t,u)-k_1u_i)\\&=d_i\partial _{xx} w_i+g_i(x,t,w), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} g_i(x,t,w)=e^{-k_1t}(f_i(x,t,u)-k_1e^{k_1t}w_i). \end{aligned}$$

(3.1)

Note that

$$\begin{aligned} \begin{aligned} \sum ^m_{i=1}g_i(x,t,w)=e^{-k_1t}\sum ^m_{i=1}(f_i(x,t,u)-k_1u_i)\le e^{-k_1t}k_0 \end{aligned} \end{aligned}$$

due to the assumption (A2). Introduce a new unknown $w_{m+1}:\Omega \times (0, T_{\max })\rightarrow \mathbb {R}_{+}$ which solves

$$\begin{aligned} \begin{aligned} \partial _tw_{m+1}-\partial _{xx} w_{m+1}=k_0e^{-k_1t}-\sum ^m_{i=1}g_i(x,t,w)=:g_{m+1}(x,t,w)\ge 0 \end{aligned} \end{aligned}$$

(3.2)

with homogeneous Neumann boundary condition $\partial _x w_{m+1}(0,t) = \partial _x w_{m+1}(L,t)=0$ and zero initial data $w_{m+1}(x,0)=w_{m+1,0}(x)=0$ for $x\in \Omega$. With a slight abuse of notation we write the new vector of concentrations ${\widetilde{w}}=(w_1, \ldots , w_m, w_{m+1})$ and the nonlinearities $g_i(x,t,{\widetilde{w}}):=g_i(x,t,w_1, \ldots , w_m)$ for all $i=1,\ldots , m$ while $g_{m+1}(x,t,{\widetilde{w}})=k_0e^{-k_1t}-\sum ^m_{i=1}g_i(x,t,w)$. We have arrived at the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}w_i-d_i\partial _{xx}w_i=g_i(x,t,{\widetilde{w}}), &{}\quad (x,t)\in \Omega \times (0,T_{\max }),~ i=1,2,\ldots ,m+1,\\ \partial _xw_i(0,t)=\partial _xw_i(L,t)=0, &{}\quad t>0,\;i=1,2,\ldots ,m+1,\\ w_i(x,0)=u_{i,0}(x),&{}\quad x\in \Omega ,~i=1,2,\ldots ,m,\\ w_{m+1}(x,0)=w_{m+1,0}(x)=0, &{}\quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(3.3)

where $d_{m+1}=1$. It’s obvious to check that the nonlinearities $g_i, i=1,\ldots ,m+1$ satisfy the assumption (A1). Moreover, due to the definition $w_i(x, t)=e^{-k_1t}u_i(x, t)$, it follows from (A3) and (3.1) the growth control

$$\begin{aligned} |g_i(x,t,{\widetilde{w}})|\le & {} e^{-k_1t}(|f_i(x,t,u)|+k_1e^{k_1t}|w_i|)\nonumber \\\le & {} e^{-k_1t}(C(1+|u|^{3+\varepsilon })+k_1e^{k_1t}|w_i|)\nonumber \\\le & {} Ce^{(2+\varepsilon )k_1T_{\max }}(1+|{\widetilde{w}}|^{3+\varepsilon }). \end{aligned}$$

(3.4)

Moreover, the nonlinearities of (2.63.3) satisfies the condition (A2’), i.e.

$$\begin{aligned} \sum ^{m+1}_{i=1}g_i(x,t,{\widetilde{w}})=k_0e^{-k_1t} \end{aligned}$$

(3.5)

thanks to (3.2). Now we can apply the results of Theorem 1.2 to get that (2.63.3) has a global classical solution ${\widetilde{w}}$. Changing back to the original unknowns $u_i(x, t) =e^{k_1t}w_i(x, t)$ for $i=1,\ldots , N$, we obtain finally the global existence of classical solution to (1.1).

If, additionally, $k_0 = k_1 = 0$, the condition (3.5) becomes

$$\begin{aligned} \sum _{i=1}^{m+1}g_i(x,t,{\widetilde{w}}) = 0. \end{aligned}$$

Therefore, Theorem 1.2 implies that

$$\begin{aligned} \sup _{i=1,\ldots , m}\sup _{t\ge 0}\Vert w_i(t)\Vert _{L^{\infty }(\Omega )} < +\infty , \end{aligned}$$

which in turn gives, since with $k_1 = 0$, $w_i(x,t) = u_i(x,t)$, the uniform-in-time bound of solutions to (1.1). The proof of Theorem 1.1 is, therefore, finished. $\square$

4 Cubic intermediate sum condition

4.1 Constant diffusion coefficients: Proof of Theorem 1.3

We first show that the entropy dissipation condition (E) implies the boundedness of solution in $L\log L(\Omega ):=\{u: \Omega \rightarrow \mathbb {R}, ~\int _{\Omega }|u|\log |u|\,dx <+\infty \}$ and consequently in $L^1(\Omega )$.

Lemma 4.1

Assume (E). Then, there exists constant $C>0$ depending only on $\Vert u_0\Vert _{L^{\infty }(\Omega )}$ and $|\Omega |$, such that for any $T\in (0, T_{\max })$,

$$\begin{aligned} \Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )} \le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T). \end{aligned}$$

As a consequence,

$$\begin{aligned} \Vert u_i(t)\Vert _{L^{1}(\Omega )} \le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T). \end{aligned}$$

Proof

From the entropy inequality (E) we have

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\int _{\Omega }\sum ^m_{i=1}(u_i(\log u_i+\mu _i)-u_i)dx=\int _{\Omega }\sum ^m_{i=1}(\log u_i+\mu _i)\partial _tu_idx\\&\quad =-\int _{\Omega }\sum ^m_{i=1}d_i\frac{|\nabla u_i|^2}{u_i}dx+\int _{\Omega }\sum ^m_{i=1}(\log u_i+\mu _i)f_i(x,t,u)dx\\&\quad \le -4\int _{\Omega }\sum ^m_{i=1}d_i|\nabla \sqrt{u_i}|^2+ k_2\sum _{i=1}^{m}\int _{\Omega }\left( u_i\log u_i + (\mu _i-1)u_i\right) dx + k_3|\Omega |\\&\quad \le k_2\sum _{i=1}^{m}\int _{\Omega }\left( u_i\log u_i + (\mu _i-1)u_i\right) dx + k_3|\Omega |. \end{aligned} \end{aligned}$$

By standard Gronwall’s inequality we have

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^{m}\int _{\Omega }\left( u_i(t)\log u_i(t)+ (\mu _i-1)u_i(t)\right) dx\\&\quad \le \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0} + (\mu _i-1)u_{i0}\right) dx + k_3|\Omega |e^{k_2T} \quad \forall t\in [0,T). \end{aligned} \end{aligned}$$

We rewrite this inequality as

$$\begin{aligned}{} & {} \sum _{i=1}^{m}\int _{\Omega }(u_i(t)\log u_i(t)-u_i(t)+1) dx\le \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0}+ (\mu _i-1)u_{i0}\right) dx \nonumber \\{} & {} \quad + k_3|\Omega |e^{k_2T}+m|\Omega |-\sum _{i=1}^{m}\int _{\Omega }\mu _i u_i(t)dx \quad \forall t\in [0,T). \end{aligned}$$

(4.1)

Using the inequality $x\log x-x+1\ge Lx-e^L+1$ for all $L>0$ we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\int _{\Omega }( u_i(t)\log u_i(t)-u_i(t)+1) \,dx\ge k\sum _{i=1}^{m}\int _{\Omega }u_i(t)dx-e^km|\Omega |+m|\Omega |\,dx \end{aligned} \end{aligned}$$

with $k=2\max _{i=1,\ldots ,m}|\mu _i|$. Therefore we obtain the estimate

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\int _{\Omega }u_i(t)dx&\le \frac{2}{3k}\left( \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0} + (\mu _i-1)u_{i0}\right) dx + k_3|\Omega |e^{k_2T}+e^km|\Omega |\right) \\&\le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T). \end{aligned} \end{aligned}$$

Which, together with the positivity of the solution, leads to the uniform in time $L^1-$bound. The bound of $\Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )}$ immediately follows from (4.1) and $x\log x\ge x-1$ for all $x\ge 0$, i.e.

$$\begin{aligned} \begin{aligned} \Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )}\le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T). \end{aligned} \end{aligned}$$

$\square$

The following modified interpolation inequality is important for the sequel.

Lemma 4.2

(A modified Gagliardo-Nirenberg inequality) For any $\varepsilon >0$ and $\Omega \subset \mathbb {R}^n$, there exists $c_\varepsilon >0$ such that for all $f\in H^1(\Omega )$,

$$\begin{aligned} \begin{aligned} \Vert f\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}\le \varepsilon \Vert f\Vert ^2_{H^1(\Omega )}\Vert f\log |f|\Vert ^{\frac{2}{n}}_{L^{1}(\Omega )}+c_\varepsilon \Vert f\Vert _{L^1(\Omega )}. \end{aligned} \end{aligned}$$

Remark 5

This inequality was proved in [26] for the case $n=1$ and $n=2$.

Proof

Fix a constant $N>1$. Define a function $\chi :\mathbb {R}\rightarrow \mathbb {R}$ as $\chi (s)=0$ if $|s|\le N,~\chi (s)=2(|s|-N)$ when $N<|s|<2N$ and $\chi (s)=|s|$ when $|s|>2N.$ In this proof we use

$$\begin{aligned} \Omega \{|f|\ge N\}:= & {} \{x\in \Omega :|f(x)|\ge N\},\\ \Omega \{|f|\le N\}:= & {} \{x\in \Omega :|f(x)|\le N\} \end{aligned}$$

and

$$\begin{aligned} \Omega \{|f|\le 2N\}:=\{x\in \Omega :|f(x)|\le 2N\}. \end{aligned}$$

First we write

$$\begin{aligned} \Vert f\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}\le & {} \left( \Vert \chi (f)\Vert _{L^{2+\frac{2}{n}}(\Omega )}+\Vert |f|-\chi (f)\Vert _{L^{2+\frac{2}{n}}(\Omega )}\right) ^{2+\frac{2}{n}}\nonumber \\\le & {} 2^{1+\frac{2}{n}}\left( \Vert \chi (f)\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}+\Vert |f|-\chi (f)\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}\right) \end{aligned}$$

(4.2)

and then estimate each term separately. It is easy to see that

$$\begin{aligned} \Vert |f|-\chi (f)\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}= & {} \int _{\Omega }||f|-\chi (f)|^{2+\frac{2}{n}}dx\nonumber \\\le & {} (2N)^{1+\frac{2}{n}}\int _{\Omega \{|f|\le 2N\}}|f|dx\le (2N)^{1+\frac{2}{n}}\Vert f\Vert _{L^{1}(\Omega )}.\nonumber \\ \end{aligned}$$

(4.3)

Concerning the other term, we use the usual Gagliardo-Nirenberg inequality

$$\begin{aligned} \begin{aligned} \Vert \chi (f)\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}\le C\Vert \chi (f)\Vert ^2_{H^{1}(\Omega )}\Vert \chi (f)\Vert ^{\frac{2}{n}}_{L^{1}(\Omega )}, \end{aligned} \end{aligned}$$

(4.4)

for some $C>0$ depending only on n and $\Omega$. On the one hand

$$\begin{aligned} \begin{aligned} \Vert \chi (f)\Vert ^2_{H^{1}(\Omega )}=\Vert \chi '(f)\nabla f\Vert ^2_{L^{2}(\Omega )}+\Vert \chi (f)\Vert ^2_{L^{2}(\Omega )}\le 4\Vert f\Vert ^2_{H^1(\Omega )} \end{aligned} \end{aligned}$$

(4.5)

and on the other hand

$$\begin{aligned} \Vert \chi (f)\Vert ^{\frac{2}{n}}_{L^{1}(\Omega )}\le & {} \left( \int _{\Omega \{|f|\ge N\}}|f|dx\right) ^{\frac{2}{n}}\nonumber \\\le & {} \frac{1}{(\log N)^{\frac{2}{n}}}\Vert f\log |f|\Vert _{L^{1}(\Omega )}^{\frac{2}{n}}. \end{aligned}$$

(4.6)

By combining (4.2)-(4.6) we obtain

$$\begin{aligned} \begin{aligned} \Vert f\Vert ^{2+\frac{2}{n}}_{L^{2+\frac{2}{n}}(\Omega )}\le 2^{3+\frac{2}{n}}\frac{C}{(\log N)^{\frac{2}{n}}} \Vert f\Vert ^2_{H^1(\Omega )}+2^{1+\frac{2}{n}}(2N)^{1+\frac{2}{n}}\Vert f\Vert _{L^{1}(\Omega )}. \end{aligned} \end{aligned}$$

At this point we can choose N to be large enough to obtain the desired inequality. $\square$

Remark 6

A similar argument can be used if $f\log |f|$ in Lemma 4.2 is replaced by $f\Phi (f)$ for some strictly increasing function $\Phi : (0,\infty )\rightarrow \mathbb {R}$ satisfying $\lim _{z\rightarrow 0}(z\Phi (z)) = 0$ and $\lim _{z\rightarrow \infty }\Phi (z) = +\infty$. We leave the details for the interested reader.

We need another result from [20], which provides an important consequence of the intermediate sum condition (A4), which in turn becomes essential in building appropriate $L^p$-energy functions.

Lemma 4.3

[20, Lemma 2.2] Assume (A1) and (A4). Then there exist componentwise increasing functions $g_i: \mathbb {R}^{m-i} \rightarrow \mathbb {R}_+$ for $i=1,\ldots , m-1$ such that: if $\theta = (\theta _1,\ldots , \theta _m)\in (0,\infty )^m$ satisfies $\theta _i \ge g_i(\theta _{i+1},\ldots , \theta _m)$ for all $i=1,\ldots , m-1$, then

$$\begin{aligned} \sum _{i=1}^{m}\theta _i f_i(x,t,u) \le K_\theta \left( 1+\sum _{i=1}^{m}u_i\right) ^r \end{aligned}$$

for some constant $K_\theta$ depending on $\theta$, $g_i$.

The following $L^p$-energy function has been developed in [10, 20]. For any $2\le p \in {\mathbb {N}}$, we define

$$\begin{aligned} E_p[u]:= \sum _{\beta \in \mathbb {Z}_+^m, |\beta | = p}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u^{\beta } \end{aligned}$$

(4.7)

and

$$\begin{aligned} \mathscr {E}_p[u]:= \int _{\Omega }E_p[u(x)]dx, \end{aligned}$$

(4.8)

where

$$\begin{aligned} \begin{pmatrix}p\\ \beta \end{pmatrix} = \frac{p!}{\beta _1!\ldots \beta _m!}, \quad \theta ^{\beta ^2} = \prod _{j=1}^m \theta _j^{\beta _j^2}, \quad u^{\beta } = \prod _{j=1}^{m}u_j^{\beta _j}. \end{aligned}$$

Lemma 4.4

Assume (A1) and (A4). Then for any $2\le p \in {\mathbb {N}}$, there exists $\theta = (\theta _1,\ldots , \theta _m)\in (0,\infty )^m$ such that the energy function $\mathscr {E}_p[u]$ defined in (4.8) satisfies

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha _p\sum _{i=1}^{m}\int _{\Omega }|\partial _{x}(u_i^{p/2})|^2dx \le C\left( 1+\sum _{i=1}^{m}\int _{\Omega }u_i^{p-1+r}dx\right) \end{aligned}$$

along the trajectory of solution to (1.1), for some constant $\alpha _p>0$ depending on $\theta$ and p, $C=C(p,\theta )$.

Proof

Let u solve (1.1) and $\mathscr {E}_p(t):=\mathscr {E}_p[u](t)$ be defined in (4.8). Then by [20, Lemma A.1], we have

$$\begin{aligned} \begin{aligned} \mathscr {E}'_p(t)&=\int _{\Omega }\sum _{|\beta |=p-1}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }\sum ^m_{j=1}\theta _j^{2\beta _j+1}\frac{\partial }{\partial t}u_j(x,t)dx\\&= \int _{\Omega }\sum _{|\beta |=p-1}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }\sum ^m_{j=1}\theta _j^{2\beta _j+1}\left( d_j\partial _{xx}u_j+f_j(x,t,u)\right) dx. \end{aligned} \end{aligned}$$

If we apply [20, Lemma A.2] and integration by parts, we have

$$\begin{aligned}{} & {} \int _{\Omega }\sum _{|\beta |=p-1}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }\sum ^m_{j=1}\theta _j^{2\beta _j+1}d_j\partial _{xx}u_jdx\nonumber \\{} & {} \quad =-\int _{\Omega }\sum _{|\beta |=p-2}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }\sum _{i=1}^{m}\sum ^m_{j=1}a_{ij}\partial _x u_i\partial _x u_jdx\nonumber \\{} & {} \quad =:(I) \end{aligned}$$

(4.9)

with

$$\begin{aligned} a_{ij}={\left\{ \begin{array}{ll} \frac{d_i+d_j}{2}\theta _i^{2\beta _i+1}\theta _j^{2\beta _j+1} &{}\quad \text {if}\quad i\ne j,\\ d_i\theta _i^{4\beta _i+4} &{}\quad \text {if}\quad i= j. \end{array}\right. } \end{aligned}$$

Denote by ${\mathscr {A}} = (a_{i,j})_{i,j=1,\ldots , m}$, $\mathscr {C} = \text {diag}(\theta _i^{-2\beta _i - 1})$, and

$$\begin{aligned} \mathscr {M} = (m_{ij}):= {\left\{ \begin{array}{ll} d_i\theta _i^2, &{}\quad \text { when } i=j,\\ \frac{d_i+d_j}{2}, &{}\quad \text { when } i\ne j. \end{array}\right. } \end{aligned}$$

It’s easy to check that

$$\begin{aligned} {\mathscr {A}} = \mathscr {C}^{-1}\mathscr {M}\mathscr {C}^{-1}. \end{aligned}$$

By choosing $\theta _i, i=1,\ldots , m$ large enough, the matrix $\mathscr {M}$ is diagonally dominant, and therefore positive definite. This implies that ${\mathscr {A}}$ is also positive definite. Therefore, there exists $\omega >0$ such that

$$\begin{aligned} (\partial _x u)^{\top }\mathscr {A}(\partial _x u) \ge \omega |\partial _x u|^2 \end{aligned}$$

where $\partial _x u = (\partial _x u_i)_{i=1,\ldots ,m}$ is a column vector in $\mathbb {R}^m$. It then follows from (4.9) that

$$\begin{aligned} \begin{aligned} (I)&=-\int _{\Omega }\sum _{|\beta |=p-2}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }(\partial _x u)^{\top } \mathscr {A}(\partial _x u)dx\\&\le \alpha _p\sum _{i=1}^m\int _{\Omega }\left| \partial _x(u_i^{\frac{p}{2}})\right| ^2dx. \end{aligned} \end{aligned}$$

Consequently, returning to above, we obtain

$$\begin{aligned}{} & {} \mathscr {E}'_p(t)+\alpha _p\sum _{i=1}^{m}\int _{\Omega }|\partial _x(u_i^{\frac{p}{2}})|^2dx\nonumber \\{} & {} \quad \le \int _{\Omega }\sum _{|\beta |=p-1}\begin{pmatrix}p\\ \beta \end{pmatrix}\theta ^{\beta ^2}u(x,t)^{\beta }\sum _{i=1}^{m}\theta _i^{2\beta _i+1}f_i(x,t,u)dx. \end{aligned}$$

(4.10)

So, we choose the components of $\theta = (\theta _i)$ inductively so that $\theta _i$ are sufficiently large that the previous positive definiteness condition of $\mathscr {M}$ is satisfied, and

$$\begin{aligned} \begin{aligned} \theta _i\ge g_i(\theta _{i+1}^{2p-1},\ldots ,\theta _{m}^{2p-1}) \quad \text {for}\quad i=1,\ldots ,m-1, \end{aligned} \end{aligned}$$

(4.11)

where $g_i$ are functions constructed in Lemma 4.3. Note that $\theta _i \le \theta _i^{2\beta _i + 1} \le \theta _i^{2p - 1}$. Since $g_i$ is componentwise increasing, the relation (4.11) implies

$$\begin{aligned} \theta _i^{2\beta _i + 1} \ge g_i\left( \theta _{i+1}^{2\beta _{i+1}+1}, \ldots , \theta _m^{2\beta _m + 1}\right) , \quad \forall i=1,\ldots , m-1. \end{aligned}$$

Now we can apply Lemma 4.3, to obtain some $K_{{\widetilde{\theta }}}$ so that for all $\beta \in \mathbb {Z}_+$ with $|\beta |=p-1$, we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\theta _{i}^{2\beta _i+1}f_i(x,t,u)\le K_{{\widetilde{\theta }}}\left( 1+\sum _{i=1}^{m}u_i^r\right) \quad \text {for all}\quad (x,t,u)\in \Omega \times \mathbb {R}_+\times \mathbb {R}^m_+. \end{aligned} \end{aligned}$$

It follows that there exists $C_p>0$, such that (4.10) implies

$$\begin{aligned} \begin{aligned}&\mathscr {E}'_p(t)+\alpha _p\sum _{i=1}^{m}\int _{\Omega }|\partial _x(u_i^{\frac{p}{2}})|^2dx\le C_p\left( \sum _{i=1}^{m}\int _{\Omega }u_i^{p-1+r}dx+1\right) . \end{aligned} \end{aligned}$$

$\square$

Lemma 4.5

Assume (E), (A1) and (A4) with $r=3$. Then for any $T\in (0,T_{\max })$,

$$\begin{aligned} \sup _{i=1,\ldots , m}\Vert u_i(t)\Vert _{L^{2}(\Omega )} \le C(T) \quad \text { for all } \quad t\in (0,T), \end{aligned}$$

with C(T) depends continuously on $T\in (0,\infty )$.

Proof

From Lemma 4.4, it follows by choosing $p=2$ that

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \alpha _2\sum _{i=1}^{m}\int _{\Omega }|\partial _x u_i|^2dx \le C\left( 1+\sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^{4}\right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \alpha _2\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \le C\left( 1+\sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^{4}\right) . \end{aligned}$$

Fix $\varepsilon >0$. Thanks to Lemmas 4.1 and 4.2,

$$\begin{aligned} \sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^4 \le \varepsilon \sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 + C_\varepsilon . \end{aligned}$$

By choosing $\varepsilon >0$ small enough, it follows that

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \frac{\alpha _2}{2}\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \le C, \end{aligned}$$

which finishes the proof of Lemma 4.5. $\square$

Lemma 4.6

Assume (E), (A1) and (A4) with $r=3$. Then for any $2\le p \in {\mathbb {N}}$ and any $T\in (0,T_{\max }]$,

$$\begin{aligned} \sup _{i=1,\ldots , m}\Vert u_i(t)\Vert _{L^{p}(\Omega )} \le C(T,p) \quad \text { for all } \quad t\in (0,T) \end{aligned}$$

(4.12)

with C(T, p) depends continuously on $T>0$.

Proof

By using the $E_p[u]$ defined in (4.7) and Lemma 4.4, we obtain

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha _p\sum _{i=1}^{m}\int _{\Omega }|\partial _{x}(u_i^{p/2})|^2dx \le C\left( 1+\sum _{i=1}^{m}\int _{\Omega }u_i^{p+2}dx\right) . \end{aligned}$$

(4.13)

Since we are in one dimension and have $\Vert u_i(t)\Vert _{L^{2}(\Omega )} \le C(T)$, we can apply [20, Lemma 2.3] to obtain

$$\begin{aligned} \int _{\Omega }u_i^{p+2}dx \le \varepsilon \Vert u_i^{p/2}\Vert _{H^1(\Omega )}^2 + C_\varepsilon (T). \end{aligned}$$

(4.14)

By adding $\alpha _p\sum _{i=1}^{m}\int _{\Omega }u_i^pdx$ to both sides of (4.13) and using (4.14) with $\varepsilon = \alpha _p/2$, it yields

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \frac{\alpha _p}{2}\sum _{i=1}^{m}\int _{\Omega }u_i^{p}dx \le C(T). \end{aligned}$$

This leads to

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha \mathscr {E}_p[u] \le C(T) \end{aligned}$$

for some constant $\alpha >0$. Gronwall’s lemma yields

$$\begin{aligned} \sup _{t\in (0,T)}\mathscr {E}_p[u(t)]\le C_1(T), \end{aligned}$$

which implies (4.12). $\square$

We now can prove the global existence part of Theorem 1.3

Proof of Theorem 1.3 - Global existence

The existence of local in time, classical, nonnegative solutions $u=(u_1,u_2,\ldots ,u_m)$ on some maximal time interval $[0,T_{\max })$ follows from classical results.

Our aim in the following is to prove for all finite $T\in (0,T_{\max }]$,

$$\begin{aligned} \limsup _{t\rightarrow T_{}}\Vert u_i(t)\Vert _{L^\infty (\Omega )}<\infty , \quad \text {for all}~i=1,2,\ldots ,m. \quad \Rightarrow T_{\max }=+\infty . \end{aligned}$$

From the polynomial bound (1.12), we have by the comparison principle that $u_i \le v_i$ where $v_i$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v_i - d_i\partial _{xx}v_i = h_i:= C\left( 1+\sum _{j=1}^m u_j^{\ell }\right) , &{}\quad (x,t)\in Q_T,\\ \partial _xv_i(0,t) = \partial _xv_i(L,t) = 0, &{}\quad t\in (0,T),\\ v_i(x,t) = u_{i,0}(x), &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

Now thanks to Lemma 4.6, $h_i \in L^p(Q_T)$ for any $1\le p <\infty$. By the smoothing effect of the heat operator $\partial _t - d_i\partial _{xx}$, it follows that for some $p>\frac{3}{2}$,

$$\begin{aligned} \Vert v_i\Vert _{L^{\infty }(Q_T)} \le C\left( \Vert u_{i,0}\Vert _{L^{\infty }(\Omega )}, \Vert h_i\Vert _{L^{p}(Q_T)}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \Vert u_i\Vert _{L^\infty (Q_T)}\le C_T, \end{aligned}$$

and consequently, $T_{\max } = +\infty$. $\square$

We now turn to the uniform-in-time bounds for the case when $k_2 = k_3 = 0$ or $k_2<0$ in (E), i.e. the case of entropy dissipation. In order to do that, it’s sufficient to prove there exists $C>0$ independent of $\tau >0$ such that

$$\begin{aligned} \begin{aligned} \sup _{\tau \in \mathbb {N}}\Vert u_i\Vert _{L^\infty (\Omega \times (\tau ,\tau +1))}\le C. \end{aligned} \end{aligned}$$

(4.15)

Since $k_2=k_3=0$, we have from (E)

$$\begin{aligned} \sum _{i=1}^{m}f_i(x,t,u)(\log u_i+\mu _i)\le 0. \end{aligned}$$

This implies the following uniform bounds.

Lemma 4.7

Assume (E) with $k_2 = k_3 = 0$ or $k_2<0$. Then, there exists constant $C>0$ depending only on $\Vert u_0\Vert _{L^{\infty }(\Omega )}$ and $|\Omega |$, such that

$$\begin{aligned} \sup _{t>0}\Vert u_i(t)\Vert _{L^{1}(\Omega )} \le C \quad \text {and} \quad \sup _{t>0}\Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )} \le C. \end{aligned}$$

Proof

From the entropy inequality (E) we have

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\int _{\Omega }\sum ^m_{i=1}(u_i(\log u_i+\mu _i)-u_i)dx=\int _{\Omega }\sum ^m_{i=1}(\log u_i+\mu _i)\partial _tu_idx\\&\quad =-\int _{\Omega }\sum ^m_{i=1}d_i\frac{|\nabla u_i|^2}{u_i}dx+\int _{\Omega }\sum ^m_{i=1}(\log u_i+\mu _i)f_i(x,t,u)dx\le 0. \end{aligned} \end{aligned}$$

Integration on (0, t) provides

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\int _{\Omega }&\left( u_i(t)\log u_i(t)+ (\mu _i-1)u_i(t)\right) dx\le \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0} + (\mu _i-1)u_{i0}\right) dx . \end{aligned} \end{aligned}$$

We rewrite this inequality as

$$\begin{aligned}{} & {} \sum _{i=1}^{m}\int _{\Omega }( u_i(t)\log u_i(t)-u_i(t)+1)dx\le \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0}+ (\mu _i-1)u_{i0}\right) dx \nonumber \\{} & {} \quad +m|\Omega |-\sum _{i=1}^{m}\int _{\Omega }\mu _iu_i(t)dx, \quad \forall t\in [0,T). \end{aligned}$$

(4.16)

Using the inequality $x\log x-x+1\ge Kx-e^K+1$ for all $K>0$ we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\int _{\Omega } (u_i(t)\log u_i(t)-u_i(t)+1)\,dx\ge k\sum _{i=1}^{m}\int _{\Omega }u_i(t)dx-e^km|\Omega |+m|\Omega | \end{aligned} \end{aligned}$$

with $k=2\max _{i=1,\ldots ,m}|\mu _i|$. Therefore we obtain the estimate

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{m}\int _{\Omega }u_i(t)dx&\le \frac{2}{3k}\left( \sum _{i=1}^{m}\int _{\Omega }\left( u_{i0}\log u_{i0} + (\mu _i-1)u_{i0}\right) dx +e^km|\Omega |\right) \\&\le C. \end{aligned} \end{aligned}$$

Which, together with the positivity of the solution, leads to the uniform in time $L^1$-bound. The bound of $\Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )}$ follows immediately from (4.16) and $x\log x\ge x-1$ for all $x\ge 0$, i.e.

$$\begin{aligned} \begin{aligned} \Vert (u_i\log u_i)(t)\Vert _{L^{1}(\Omega )}\le C. \end{aligned} \end{aligned}$$

$\square$

Lemma 4.8

Assume (E) with $k_2 = k_3 = 0$ or $k_2<0$, (A1) and (A4) with $r=3$. Then

$$\begin{aligned} \sup _{t>0}\Vert u_i(t)\Vert _{L^{2}(\Omega )} \le C. \end{aligned}$$

Proof

From Lemma 4.4, it follows by choosing $p=2$ that

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \alpha _2\sum _{i=1}^{m}\int _{\Omega }|\partial _x u_i|^2dx \le C\left( 1+\sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^{4}\right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \alpha _2\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \le C\left( 1+\sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^{4}\right) . \end{aligned}$$

Fix $\varepsilon >0$. Thanks to Lemmas 4.7 and 4.2,

$$\begin{aligned} \sum _{i=1}^{m}\Vert u_i\Vert _{L^{4}(\Omega )}^4 \le \varepsilon \sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 + C_\varepsilon . \end{aligned}$$

By choosing $\varepsilon >0$ small enough, it follows that

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \frac{\alpha _2}{2}\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \le C, \end{aligned}$$

where the constant C is independent of time $t>0$. By using $\sum _{i=1}^{m}\Vert u_i\Vert _{H^1(\Omega )}^2 \ge \beta _0\mathscr {E}_2[u]$ for some $\beta _0>0$, we obtain

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_2[u] + \frac{\alpha _2\beta _0}{2}\mathscr {E}_2[u] \le C. \end{aligned}$$

With the classical Gronwall lemma, we can finish the proof of Lemma 4.8. $\square$

Lemma 4.9

Assume (E) with $k_2 = k_3 = 0$ or $k_2<0$, (A1) and (A4) with $r=3$. Then for any $2\le p \in {\mathbb {N}}$

$$\begin{aligned} \sup _{t>0}\Vert u_i(t)\Vert _{L^{p}(\Omega )} \le C(p). \end{aligned}$$

(4.17)

Proof

By using the $E_p[u]$ defined in (4.7) and Lemma 4.4, we obtain

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha _p\sum _{i=1}^{m}\int _{\Omega }|\partial _{x}(u_i^{p/2})|^2dx \le C\left( 1+\sum _{i=1}^{m}\int _{\Omega }u_i^{p+2}dx\right) . \end{aligned}$$

Adding $\alpha _p\sum _{i=1}^{m}\int _{\Omega }u_i^p$ to both sides and using Young’s inequality yield

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \alpha _p\sum _{i=1}^{m}\Vert u_i^{p/2}\Vert _{H^1(\Omega )}^2 \le C\left( 1+\sum _{i=1}^{m}\int _{\Omega }u_i^{p+2}dx\right) . \end{aligned}$$

(4.18)

Thanks to Lemma 4.8, we can apply [20, Lemma 2.3] to estimate for any $\delta >0$

$$\begin{aligned} \int _{\Omega }u_i^{p+2} \le \delta \Vert u_i^{p/2}\Vert _{H^1(\Omega )}^2 + C_\delta \quad \forall i=1,\ldots , m. \end{aligned}$$

By choosing $\delta = \alpha _p/2$, inserting this into (4.18), and using $\sum _{i=1}^{m}\Vert u_i^{p/2}\Vert _{H^1(\Omega )}^2 \ge \beta _1\mathscr {E}_p[u]$ for some $\beta _1>0$, we get

$$\begin{aligned} \frac{d}{dt}\mathscr {E}_p[u] + \frac{\alpha _p \beta _1}{2}\mathscr {E}_p[u] \le C, \end{aligned}$$

where the constant C is independent of the time $t>0$. The classical Gronwall lemma yields the desired bound (4.17). $\square$

Proof

In order to prove (4.15), we define an cut-off function $\varphi _\tau$ as

$$\begin{aligned} \varphi _\tau (t)= {\left\{ \begin{array}{ll} 0&{} \quad \text {for on} \quad [0,\tau ],\\ 1 &{}\quad \text {for on} \quad [\tau +1,\infty ) \end{array}\right. } \end{aligned}$$

and $0\le \varphi _\tau '\le 2$. Direct computations leads to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t(\varphi _\tau u_i)-d_i\partial _{xx}(\varphi _\tau u_i)=\varphi '_\tau u_i+\varphi _\tau f_i(x,t,u),&{}\quad (x,\tau )\in \Omega \times (\tau ,\tau +2),\\ \partial _x(\varphi _\tau u_i)(0,t) = \partial _x(\varphi _\tau u_i)(L,t) = 0, &{}\quad t\in (\tau ,\tau +2),\\ (\varphi _\tau u_i)(x,\tau )=0, &{}\quad x\in \Omega . \end{array}\right. } \end{aligned}$$

Thanks to Lemma 4.9 and polynomial upper bound (1.12), we have

$$\begin{aligned} \varphi '_\tau u_i+\varphi _\tau f_i(x,t,u)\le G_i(x,t,u):=C\left( 1+\sum ^m_{k=1}u^\ell _k\right) . \end{aligned}$$

Thanks to Lemma 4.9 for any $2\le p<\infty$ a constant $C_p>0$ such that

$$\begin{aligned} \Vert G_i(x,t,u)\Vert _{L^p(\Omega \times (\tau ,\tau +2))}\le C_p,\quad \forall i=1,\ldots ,m. \end{aligned}$$

where $C_p$ is a constant independent of $\tau \in {\mathbb {N}}$. Therefore, by the smoothing effect of the heat operator $\partial _t - d_i\partial _{xx}$ and comparison principle we get

$$\begin{aligned} \Vert \varphi _\tau u_i\Vert _{L^\infty (\Omega \times (\tau ,\tau +2))}\le C,\quad \forall i=1,\ldots ,m, \end{aligned}$$

where C is a constant independent of $\tau \in \mathbb {N}$. Thanks to $\varphi _\tau \ge 0$ and $\varphi |_{(\tau +1,\tau +2)}\equiv 1$, we obtain finally the uniform-in-time bound

$$\begin{aligned} \sup _{\tau \in \mathbb {N}}\Vert u_i\Vert _{L^\infty (\Omega \times (\tau ,\tau +1))}\le C,\quad \forall i=1,\ldots ,m, \end{aligned}$$

and the proof of Theorem 1.3 is complete. $\square$

4.2 Discontinuous diffusion coefficients: Proof of Theorem 1.4

Due to the discontinuity of the diffusion coefficients, we do not expect to obtain strong solutions to (1.13). Therefore, we will work with weak solutions whose concept is defined in the following

Definition 4.10

A vector of non-negative state variables $u=(u_1,\ldots ,u_m)$ is called a weak solution to (1.13) on (0, T) if

$$\begin{aligned} u_i\in C([0,T];L^2(\Omega ))\cap L^2(0,T;H^1_0(\Omega )),~f_i(x,t,u)\in L^2(0,T;L^2(\Omega )), \end{aligned}$$

with $u_i(\cdot ,0)=u_{i,0}(\cdot )$ for all $i=1,\ldots ,m$, and for any test function $\varphi \in L^2(0,T;H^1_0(\Omega ))$ with $\partial _t\varphi \in L^2(0,T;H^{-1}(\Omega ))$, one has

$$\begin{aligned} \begin{aligned}&\int _{\Omega }u_i(x,T)\varphi (x,T)dx-\int ^T_0\int _{\Omega }u_i\partial _t\varphi dxdt +\int ^T_0\int _{\Omega }D_i(x,t)\nabla u_i\cdot \nabla \varphi dxdt\\&\quad =\int _{\Omega }u_{i,0}\varphi (x,0)dx + \int ^T_0\int _{\Omega }f_i(x,t,u)\varphi dxdt. \end{aligned} \end{aligned}$$

Proof of Theorem 1.4

We start with a truncated system where the nonlinearities are regularized to be bounded. Then crucial estimates which are uniform with respect to the regularization parameters are derived using the $L^p$-energy estimates as in the case of constant diffusion coefficients. Of importance are the bounds in $L^{\infty }(Q_T)$ of the approximate solutions. The global existence of solutions follows straightforwardly thanks to these bounds and passing to the limit of the regularization. Finally, the uniform-in-time bounds are obtained similarly to the proof of Theorem 1.3.

For any $\varepsilon >0$, we consider the truncated system: for all $i=1,\ldots ,m$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_i^{\varepsilon } - \nabla _x\cdot (D_i(x,t)\nabla _x u_i^{\varepsilon }) = f_i^{\varepsilon }(x,t,u^{\varepsilon }), &{}\quad x\in \Omega , \; t>0,\\ D_i(x,t)\nabla _x u_i^{\varepsilon }(x,t)\cdot \nu (x) = 0, &{}\quad x\in \partial \Omega , t>0,\\ u_{i}^{\varepsilon }(x,0) = u_{i,0}^{\varepsilon }(x), &{} \quad x\in \Omega , \end{array}\right. } \end{aligned}$$

(4.19)

where

$$\begin{aligned} \begin{aligned} f_i^{\varepsilon }(x,t,u^{\varepsilon }):=\frac{f_i(x,t,u^{\varepsilon })}{[1+\varepsilon \sum ^m_{j=1}|f_j(x,t,u^{\varepsilon })|]^{-1}}. \end{aligned} \end{aligned}$$

and $u_{i,0}^{\varepsilon }\in L^{\infty }(\Omega )$ such that $\Vert u_{i,0}^{\varepsilon }-u_{i,0}\Vert _{L^{\infty }(\Omega )}{\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}}0$. Note that since $u_{i,0}\in L^{\infty }(\Omega )$ one can take $u_{i,0}^{\varepsilon }=u_{i,0}$ to obtain the global existence. Following [10, Lemma 2.1], for any fixed $\varepsilon >0$, there exists a global bounded, non-negative weak solution to (4.19).

We would like to emphasize that all constants in this subsection are independent of $\varepsilon$. By using the observation

$$\begin{aligned} \sum _{i=1}^{m}f_i^{\varepsilon }(u^\varepsilon )(\log u_i^\varepsilon + \mu _i)&= \frac{1}{1+\varepsilon \sum _{i=1}^{m}|f_i(u^\varepsilon )|}\sum _{i=1}^{m}f_i(u^\varepsilon )(\log u_i^\varepsilon + \mu _i)\\&\le \frac{1}{1+\varepsilon \sum _{i=1}^{m}|f_i(u^\varepsilon )|}\left( k_2\sum _{i=1}^{m}u_i(\log u_i + \mu _i - 1) + k_3\right) \\&\le C\sum _{i=1}^{m}u_i(\log u_i + \mu _i - 1) + C. \end{aligned}$$

We obtain, similarly to Lemma 4.1

$$\begin{aligned} \Vert (u^{\varepsilon }_i\log u^{\varepsilon }_i)(t)\Vert _{L^{1}(\Omega )} \le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T), \end{aligned}$$

and as a consequence,

$$\begin{aligned} \Vert u^{\varepsilon }_i(t)\Vert _{L^{1}(\Omega )} \le C\left( 1+e^{CT}\right) \quad \text { for all } \quad t\in [0,T). \end{aligned}$$

We establish next the estimates in $L^p$-norm by using the $L^p$-energy estimates similarly to the case of constant diffusion coefficients. Thanks to [10, Lemma 2.3], we get similar results to Lemma 4.3 where $f_i$ are replaced by $f_i^{\varepsilon }$, i.e.

$$\begin{aligned} \sum _{i=1}^{m}\theta _i f_i^{\varepsilon }(x,t,u^\varepsilon ) \le K_\theta \left( 1+\sum _{i=1}^{m}u_i^\varepsilon \right) ^r. \end{aligned}$$

This allows us to repeat the arguments in Lemmas 4.4 and 4.5 to obtain

$$\begin{aligned} \sup _{i=1,\ldots ,m}\Vert u_i^\varepsilon (t)\Vert _{L^{2}(\Omega )} \le C(T), \quad \forall t\in (0,T). \end{aligned}$$

(4.20)

We remark that the discontinuous diffusion coefficients complicate the integration by parts when differentiating $\mathscr {E}_p(t)$, but this issue is overcome by using the results in [10, Lemma 4.2]. From (4.20) and the modified Gagliardo-Nirenberg inequality in Lemma 4.2, we can repeat the arguments in Lemma 4.6 to get for any $2\le p \in {\mathbb {N}}$,

$$\begin{aligned} \sup _{i=1,\ldots , m}\Vert u_i^{\varepsilon }(t)\Vert _{L^{p}(\Omega )} \le C(T,p), \quad \forall t\in (0,T). \end{aligned}$$

This is enough to obtain bounds in $L^{\infty }(Q_T)$ of the approximate solutions $u^\varepsilon$ to system (4.19) which are uniform in $\varepsilon >0$. It follows that

$$\begin{aligned} \Vert f_i(x,t,u^{\varepsilon })\Vert _{L^{\infty }(Q_T)}\le C_T, \quad \forall i=1,\ldots ,m. \end{aligned}$$

By multiplying (4.19) by $u^{\varepsilon }_i$ then integrating on $Q_T$ gives

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\Vert u^{\varepsilon }_i(T)\Vert ^2_{L^{2}(\Omega )} +\int ^T_0\int _{\Omega }D_i(x,t)\nabla u^{\varepsilon }_i\cdot \nabla u^{\varepsilon }_idxdt\\&\quad =\frac{1}{2}\Vert u^{\varepsilon }_{i,0}\Vert ^2_{L^{2}(\Omega )}+\int ^T_0\int _{\Omega }f^{\varepsilon }_i(u^{\varepsilon })u^{\varepsilon }_idxdt. \end{aligned} \end{aligned}$$

Using (1.14) we obtain

$$\begin{aligned} \begin{aligned} \Vert u^{\varepsilon }_i(T)\Vert ^2_{L^{2}(\Omega )}+2\lambda \Vert \nabla u^{\varepsilon }_i\Vert _{L^{2}(Q_T)}\le C_T, \quad \forall i=1,\ldots ,m. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} \{u^{\varepsilon }_i\}_{\varepsilon \ge 0}~&\text {is bounded uniformly in}\, \varepsilon \,\text { in }~L^{\infty }(Q_T)\cap L^2(0,T;H^1_0(\Omega )),\\ \{\partial _tu^{\varepsilon }_i\}_{\varepsilon \ge 0}~&\text {is bounded uniformly in}\, \varepsilon \,\text {in}~L^2(0,T;H^{-1}(\Omega )). \end{aligned} \end{aligned}$$

The classical Aubin-Lions lemma gives the strong convergence

$$\begin{aligned} \begin{aligned} u_{i}^{\varepsilon }{\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} u_{i}~\text {strongly in}~L^{2}(0,T;L^{2}(\Omega )). \end{aligned} \end{aligned}$$

Consequently, for any $1\le p<\infty$,

$$\begin{aligned} \begin{aligned} u_{i}^{\varepsilon }{\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} u_{i}~\text {strongly in}~L^{p}(Q_T), \end{aligned} \end{aligned}$$

thanks to the uniform $L^{\infty }$-bound of $u_{i}^{\varepsilon }$. This is enough to pass to the limit in the weak formulation of (4.19)

$$\begin{aligned} \begin{aligned}&-\int _{\Omega }\varphi (\cdot ,0)u_{i,0}^{\varepsilon }dx -\int ^T_0\int _{\Omega }u_{i}^{\varepsilon }\partial _t\varphi dxdt+ \int ^T_0\int _{\Omega }d_i(x,t)\nabla u_{i}^{\varepsilon }\cdot \nabla \varphi dxdt\\&\quad = \int ^T_0\int _{\Omega }f_i^{\varepsilon }(x,t,u^{\varepsilon })\varphi dxdt, \end{aligned} \end{aligned}$$

to obtain that $u = (u_1,\ldots ,u_m)$ is a global weak solution to (1.13) and additionally

$$\begin{aligned} \Vert u_i\Vert _{L^{\infty }(Q_T)}\le C_T, \quad \forall i=1,\ldots ,m. \end{aligned}$$

The proof of uniform-in-time bounds follows similarly to that of Theorem 1.3, so we omit the details here. $\square$