Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects

Abstract

We introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.

1 Introduction

Reaction–diffusion equations arise in various applications in chemistry and biology (cf. [20, Ch. 2]) and form an important class of model problems in the study of systems of parabolic equations (see [24, Ch. 33]). Already at the stage of basic theories of solvability, a major challenge for the analysis of such systems consists in the presence of commonly superlinear source terms. While the possibility of blow up then is apparrent as long as suitably destabilizing reaction mechanisms are admitted (cf. e.g. [13]), even the requirement of dissipation of mass—which is sufficient to yield global existence and boundedness in the corresponding ODE systems—cannot preclude its occurrence, as impressively demonstrated by the counterexamples in [22]. Global classical solutions have, accordingly, been searched for and found under certain restrictive conditions.

In the context of boundary value problems for systems of the general form

$$\begin{aligned} \partial _t u_i = d_i \Delta u_i^{m_i} + f_i(u_1,...,u_N), \qquad i\in \{1,...,N\}, \end{aligned}$$

(1.1)

in the linear diffusion case when $m_1=...=m_N=1$ such results on global smooth solvability cover settings where boundedness of the first among two components is a priori known (from a sign of $f_1$, cf. [18]), where the diffusion coefficients are close to each other [2, 5], or where sources exhibit subquadratic growth [4], and recently ideas of [12] have successfully been extended to show global solvability for quadratic or slightly superquadratic reaction functions [3, 9, 26].

Another line of investigations pursues solutions in a weaker sense. Weak solutions can be constructed if $L^1$-bounds for the reaction terms $f_i(u)$ are known [19], or if the reaction functions are at most quadratic [21]. For nonlinear diffusion of porous medium type, the existence of weak solutions to (1.4) with Dirichlet boundary data is shown in [15] under the assumptions that

$$\begin{aligned} \left\{ \begin{array}{l} f_i\in W^{1,\infty }_{loc}([0,\infty )^N) \quad \text{ is } \text{ such } \text{ that } \quad \\ f_i(s_1,...s_{i-1},0,s_{i+1},...,s_N) \ge 0 \qquad \text{ for } \text{ all } (s_1,...,s_N)\in [0,\infty )^N\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{ and } i\in \{1,...,N\}, \end{array} \right. \end{aligned}$$

(1.2)

that with some $K \ge 0$ and $a\in (0,\infty )^N$ we have

$$\begin{aligned} \sum _{i=1}^N a_i f_i(s_1,...,s_N) \le K \cdot \bigg ( \sum _{i=1}^N a_i s_i+1\bigg ) \qquad&\text{ for } \text{ all } (s_1,...,s_N)\in [0,\infty )^N\nonumber \\&\text{ and } i\in \{1,...,N\}, \end{aligned}$$

(1.3)

and that either a priori $L^1$-bounds for the reaction terms are known, or $f_i(s_1,..,s_N) \ge - C \left( \sum _{j=1}^N s_j^{\beta _j} +1 \right)$ with $\beta _i<m_i+1$ for all $i\in \{1,\ldots ,N\}$. An analogous result has been achieved for the corresponding Neumann problem in [14] with a different proof and less restrictive conditions on the initial data, and in the cases when additionally all $m_i$ are sufficiently large compared to $\max _i \beta _i$, higher regularity and convergence of solutions were shown in [8].

The concept of renormalized solutions, that is, the idea that not u itself, but a transformed quantity $\rho (u)$ solves (a weak form) of the equation, makes it possible to bypass even further restrictions on the form of the system. This concept has been successfully introduced for the Boltzmann equation by DiPerna and Lions [7] and was employed for reaction diffusion equations with quadratic reaction functions and linear diffusion in [6]. The apparently most far-reaching application of this idea to reaction–diffusion systems (with linear diffusion) can be found in [10], where essentially no growth restriction on the $f_i$ is needed, but where the reaction function is supposed to obey a certain entropy condition. The term in the definition of solutions for whose treatment this entropy condition is essential arises from the choice of renormalization functions $\xi :[0,\infty )^N\rightarrow \mathbb {R}$ with compactly supported $D\xi$, which in particular depend on all solution components simultaneously.

Main results.    In the present manuscript, we intend to introduce an approach by which it becomes possible to avoid any requirement of the latter type, and it turns out that this can in fact be achieved by resorting to separate renormalization functions for each component $u_i$. Thereby, our main result, as stated in Theorem 1.1 below, partially answers the open problem [10, p.585] to find a similar notion of solution without requiring an entropy condition.

Specifically, we shall be concerned with the Neumann problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \partial _t u_i = d_i \Delta u_i^{m_i} + f_i(u_1,...,u_N), \qquad &{} x\in \Omega , \ t>0, \ i\in \{1,...,N\},\\ \partial _\nu u_i^{m_i}=0, \qquad &{} x\in \partial \Omega , \ t>0, \ i\in \{1,...,N\},\\ u_i(x,0)=u_{0i}(x), \qquad &{} x\in \Omega , \ i\in \{1,...,N\}, \end{array} \right. \end{aligned}$$

(1.4)

under the assumptions that (1.2) and (1.3) hold, and that

$$\begin{aligned} \left\{ \begin{array}{l} f_i(s_1,..,s_N) \ge - \phi _i(s_i) \cdot \bigg ( \sum \limits _{\begin{array}{c} j\in \{1,...,N\} \\ j\ne i \end{array}} s_j^{\beta _j} + 1 \bigg ) \quad \text{ for } \text{ all } (s_1,...,s_N)\in [0,\infty )^N \\ \qquad \qquad \qquad \text{ and } i\in \{1,...,N\},\\ \text{ with } \text{ some } \text{ nonnegative } \phi _i\in C^1([0,\infty )) \text{ such } \text{ that } \phi _i'>0 \text{ on } (0,\infty )\\ \text{ for } i\in \{1,...,N\}, \text{ and } \text{ some } \beta _i>0 \text{ such } \text{ that } \beta _i<m_i+1 \text{ for } \text{ all } i\in \{1,...,N\}. \end{array} \right. \end{aligned}$$

(1.5)

Here we recall that the quasipositivity condition in (1.2) is important in order to avoid negative concentrations, and that (1.3) is a slightly generalized mass dissipation condition, and includes some stoichiometric coefficients a. In addition to this, (1.5) signifies a growth condition for the negative parts of the reaction functions, where in the special case of linear diffusion, subquadratic growth is admissible. It is important to note, however, that this restriction only applies to the cross-absorptive effects: For $(f_i)_-$, the possible growth with respect to the i-th argument remains unrestricted.

As for the initial data in (1.4), throughout this paper we shall suppose that

$$\begin{aligned}&u_{0i}, i\in \{1,...,N\}, \text{ is } \text{ a } \text{ nonnegative } \text{ function } \text{ from } L^r(\Omega ) \text{ with } \text{ some } \nonumber \\&\left\{ \begin{array}{l@{\quad }lll} r\ge 1 &{}\text{ if } n=1, \\ r>1 &{} \text{ if } n=2, \\ r\ge \frac{2n}{n+2} &{} \text{ if } n\ge 3. \end{array} \right. \end{aligned}$$

(1.6)

Postponing the precise description of the solution concept to be pursued here to Sect. 2, let us introduce our main result obtained in this framework, and give a few examples of its application.

Theorem 1.1

Let $n\ge 1$ and $N\ge 1$ and $\Omega \subset \mathbb {R}^n$ a bounded domain with smooth boundary, and suppose that $d_1,...,d_N$ and $m_1,...,m_N$ are positive, and that $f_1,...,f_N$ belong to $W^{1,\infty }_{loc}([0,\infty )^N)$ and satisfy (1.3), (1.2) and (1.5) with some positive constant K. Then given any $u_{01},...,u_{0N}$ fulfilling (1.6), one can find nonnegative functions $u_i\in L^{m_i+1}_{loc}(\overline{\Omega }\times [0,\infty ))$ such that $(u_1,...,u_N)$ is a generalized solution of (1.4) in the sense of Definition 2.2.

Remark

(i):

The required smoothness of the domain is not the focus of our investigation and could be weakened – in fact, already for the present construction, $C^{2+\alpha }$ regularity for some $\alpha \in (0,1)$ – entering in the construction of classical approximate solutions – would be sufficient.

(ii):

Likewise, in order to avoid additional technicalities we do not investigate in detail here whether covering less regular sources, such as e.g. merely continuous $f_i$, $i\in \{1,...,N\}$, might be possible at the cost of an additional approximation argument in the context of the regularized versions (3.1) of (1.4) below.

(iii):

An interesting question left open here is how far the regularity requirements in (1.6) could further be relaxed, so as to require integrability of $(u_{i0})_{i\in \{1,...,N\}}$ only, for instance. As will become clear in the proof of Lemma 3.2 below, our currently pursued strategy will crucially rely on (1.6) in order to appropriately control certain initial data appearing in the course of a duality-based reasoning.

Application #1. A first application of Theorem 1.1 addresses the system

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }lll} \partial _t u_i = d_i \Delta u_i^{m_i} + (p_i-q_i) \cdot \bigg ( k_2 \prod _{j=1}^N u_j^{q_j} - k_1 \prod _{j=1}^N u_j^{p_j}\bigg ), \qquad &{} x\in \Omega , \ t>0, \ i\in \{1,...,N\},\\ \partial _\nu u_i^{m_i}=0, \qquad &{} x\in \partial \Omega , \ t>0, \ i\in \{1,...,N\},\nonumber \\ u_i(x,0)=u_{0i}(x), \qquad &{} x\in \Omega , \ i\in \{1,...,N\}, \end{array} \right. \\ \end{aligned}$$

(1.7)

which describes a general reversible reaction of the form

and for which we obtain the following.

Proposition 1.2

Let $N\ge 2$, and suppose that $k_1>0$ and $k_2>0$, and that for $i\in \{1,...,N\}$, $d_i>0$, $m_i>0$, $p_i\ge 1$ and $q_i\ge 1$ are such that for some $a\in (0,\infty )^N$

$$\begin{aligned} \sum _{i=1}^N a_i p_i=\sum _{i=1}^N a_i q_i, \end{aligned}$$

(1.8)

and that

$$\begin{aligned} \sum _{\begin{array}{c} j\in \{1,...,N\} \\ j\ne i \end{array}} \frac{p_j}{m_j+1} <1 \qquad \text{ for } \text{ all } i\in \{1,...,N\} \text{ such } \text{ that } p_i>q_i \end{aligned}$$

(1.9)

as well as

$$\begin{aligned} \sum _{\begin{array}{c} j\in \{1,...,N\} \\ j\ne i \end{array}} \frac{q_j}{m_j+1}<1 \qquad \text{ for } \text{ all } i\in \{1,...,N\} \text{ such } \text{ that } p_i<q_i. \end{aligned}$$

(1.10)

Then for any choice of $u_{01},...,u_{0n}$ complying with (1.6), the problem (1.7) admits a generalized solution in the sense of Definition 2.2.

Proof of Proposition 1.2

Writing $f_i(s_1,...,s_N):=(p_i-q_i) \cdot \Big (k_2 \prod _{j=1}^N s_j^{q_j} - k_1 \prod _{j=1}^N s_j^{p_j}\Big )$ for $i\in \{1,...,N\}$ and $(s_1,...,s_N) \in [0,\infty )^N$, we see that (1.2) is fulfilled and (1.3) follows since $\sum _{j=1}^N a_i f_i \equiv 0$ due to (1.8). Moreover, if e.g. $i\in \{1,...,N\}$ is such that $p_i>q_i$, then (1.9) enables us to pick numbers $\theta _j>1$, $j\in \{1,...,N\}\setminus \{i\}$, such that $p_j\theta _j <m_j+1$ for all $j\in \{1,...,N\}\setminus \{i\}$ and $\sum _{j\ne i} \frac{1}{\theta _j} <1$. An application of Young’s inequality thus shows that for any such i,

$$\begin{aligned}&f_i(s_1,...,s_N) \ge -(p_i-q_i) k_1 s_i^{p_i} \cdot \prod _{j\ne i} s_j^{p_j} \\&\quad \ge -(p_i-q_i) k_1 s_i^{p_i} \cdot \bigg ( \sum _{j\ne i} s_j^{p_j \theta _j} +1 \bigg ) \qquad \text{ for } \text{ all } (s_1,...,s_N) \in [0,\infty )^N, \end{aligned}$$

and complementing this by a similar reasoning for all $i\in \{1,...,N\}$ for which $p_i<q_i$, we readily obtain that (1.5) holds and Theorem 1.1 becomes applicable so as to yield the claim. $\square$

Proposition 1.2 corresponds to [15, Remark 2.10], where the existence of weak solutions is proved. The main difference is that there the summation in (1.9) and (1.10) extends over all $j\in \{1,\ldots N\}$.

For linear diffusion, weak solutions of (1.7) have been found in [23] if the reaction functions grow at most quadratically or if the diffusion coefficients are sufficiently close to each other. The same article also deals with their exponential convergence.

Application #2.    A second application of our general theory is concerned with the variant of (1.7) given by

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \partial _t u_1 = d_1 \Delta u_1^{m_1} + u_1^{\beta _1} g_2(u_2) - g_1(u_1) u_2^{\beta _2}, \qquad &{} x\in \Omega , \ t>0, \\ \partial _t u_2 = d_2 \Delta u_2^{m_2} - u_1^{\beta _1} g_2(u_2) + \lambda g_1(u_1) u_2^{\beta _2}, \qquad &{} x\in \Omega , \ t>0, \\ \partial _\nu u_1^{m_1}=\partial _\nu u_2^{m_2}=0, \qquad &{} x\in \partial \Omega , \ t>0, \\ u_1(x,0)=u_{01}(x), \quad u_2(x,0)=u_{02}(x), \qquad &{} x\in \Omega , \end{array} \right. \end{aligned}$$

(1.11)

and underlines the mildness of the assumptions in Theorem 1.1 by admitting widely arbitrary growth of the main ingredients $g_1$ and $g_2$ appearing herein:

Proposition 1.3

Let $d_1>0, d_2>0, m_1>0$, $m_2>0$ and $\lambda \in [0,1]$, let $\beta _1\in [1,m_1+1)$ and $\beta _2\in [1,m_2+1)$, and let $g_1\in C^1([0,\infty ))$ and $g_2\in C^1([0,\infty ))$ be such that $g_1(0)=g_2(0)=0$ and that $g_1$ and $g_2$ are positive on $(0,\infty )$. Then for any pair $(u_{01}, u_{02})$ satisfying (1.6), there exists a generalized solution of (1.11) in the spirit of Definition 2.2.

Proof of Proposition 1.3

Taking any nonnegative $\phi _i\in C^1([0,\infty ))$ such that $\phi _i'>0$ and $\phi _i \ge g_i$ on $(0,\infty )$ for $i\in \{1,2\}$, one can readily verify that for

$$\begin{aligned} f_1(s_1,s_2):= & {} s_1^{\beta _1} g_2(s_2) - g_1(s_1) s_2^{\beta _2} \quad \text{ and } \quad f_2(s_1,s_2):=-s_1^{\beta _1} g_2(s_2) +\lambda g_1(s_1) s_2^{\beta _2},\\&(s_1,s_2)\in [0,\infty )^2, \end{aligned}$$

we have

$$\begin{aligned} f_1(s_1,s_2)+f_2(s_1,s_2)= -(1-\lambda ) g_1(s_1) s_2^{\beta _2} \le 0 \qquad \text{ for } \text{ all } (s_1,s_2)\in [0,\infty )^2 \end{aligned}$$

as well as

$$\begin{aligned} f_1(s_1,s_2) \ge -g_1(s_1) s_2^{\beta _2} \ge -\phi _1(s_1) s_2^{\beta _2} \qquad \text{ for } \text{ all } (s_1,s_2)\in [0,\infty )^2 \end{aligned}$$

and, similarly,

$$\begin{aligned} f_{2}(s_1,s_2) \ge -\phi _2(s_2) s_1^{\beta _1} \qquad \text{ for } \text{ all } (s_1,s_2)\in [0,\infty )^2. \end{aligned}$$

The assumptions $\beta _i<m_i+1$, $i\in \{1,2\}$, therefore warrant applicability of Theorem 1.1 with the intended result. $\square$

Let us remark that since in Proposition 1.3 not only $f_1+f_2\le 0$, but also $\lambda f_1(s_1,s_2)+f_2(s_1,s_2)=(\lambda -1) s_1^{\beta _1}g_2(s_2)\le 0$ for $(s_1,s_2)\in [0,\infty )^2$, [15, Cor. 2.11] could be applied to the variant of (1.11) involving homogeneous Dirichlet boundary conditions (cf. [15, Remark 2.12]) so as to yield weak solutions for any $L^1$-initial data; said corollary, however, requires that $m_1,m_2<2$.

Application #3.    We shall next briefly address

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \partial _t u_1 = d_1 \Delta u_1^{m_1} + k_2 u_1^{q_1} u_2^{q_2} - k_1 u_1^{p_1} u_2^{p_2}, \qquad &{} x\in \Omega , \ t>0, \\ \partial _t u_2 = d_2 \Delta u_2^{m_2} - k_2 u_1^{q_1} u_2^{q_2} + k_1 u_1^{p_1} u_2^{p_2}, \qquad &{} x\in \Omega , \ t>0, \\ \partial _\nu u_1^{m_1}=\partial _\nu u_2^{m_2}=0, \qquad &{} x\in \partial \Omega , \ t>0, \\ u_1(x,0)=u_{01}(x), \quad u_2(x,0)=u_{02}(x), \qquad &{} x\in \Omega , \end{array} \right. \end{aligned}$$

(1.12)

for which without imposing any smallness condition on $q_2$ nor $p_1$ we obtain the following.

Corollary 1.4

Let $k_1,k_2,d_1,d_2,m_1$ and $m_2$ be positive, and let $p_1\ge 1, p_2\ge 1, q_1\ge 1$ and $q_2\ge 1$ be such that

$$\begin{aligned} q_1<m_1+1 \qquad \text{ and } \qquad p_2<m_2+1. \end{aligned}$$

Then for all $(u_{01}, u_{02})$ fulfilling (1.6), the problem (1.12) possesses a generalized solution in the sense of Definition 2.2.

Proof of Corollary 1.4

With $g_1(s)=k_1s^{p_1}$, $g_2(s)=k_2s^{q_2}$, $\beta _1=q_1$, $\beta _2=p_2$, this immediately results from Proposition 1.3. $\square$

Application #4.    As final example, let us consider the generalized Lotka–Volterra system

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \partial _t u_i = d_i \Delta u_i^{m_i} + \gamma _i u_i + \sum _{j=1}^N a_{ij} u_j^{\beta _{ij}}u_i^{\beta _{ji}}, \qquad &{} x\in \Omega , \ t>0, \ i\in \{1,...,N\},\\ \partial _\nu u_i^{m_i}=0, \qquad &{} x\in \partial \Omega , \ t>0, \ i\in \{1,...,N\},\\ u_i(x,0)=u_{0i}(x), \qquad &{} x\in \Omega , \ i\in \{1,...,N\}, \end{array} \right. ,\qquad \end{aligned}$$

(1.13)

which does not obey the typical entropy condition (that is required for the renormalized solutions in [10] and for classical solvability e.g. in [26]). In [9], global classical solutions are shown to exist for the classical Lotka–Volterra system ($\beta _{ij}=1$ for all i, j) with linear diffusion. If $m_i=1$ for all i and $\beta _{ij}=\beta$ for all i, j, then the result of [9] covers $2\beta < 2+\varepsilon$ (for sufficiently small $\varepsilon \in (0,1)$, see [9, Step 1, (7)]).

Within the generalized solvability framework considered here, the following consequence of Theorem 1.1 shows that here actually the entire range $\beta <2$ can be exhausted.

Proposition 1.5

Let $N\ge 2$, $d_i>0$, $m_i>0$ and $\gamma _i\in \mathbb {R}$ for $i\in \{1,\ldots ,N\}$, and suppose that for $i,j\in \{1,...,N\}$ the numbers $a_{ij}\in \mathbb {R}$ and $\beta _{ij}>0$ are such that $a_{ij}+a_{ji}\le 0$, and that

$$\begin{aligned} \text {if }i\ne j \text { and } a_{ij}<0, \quad \text { then } \beta _{ij}<m_i+1. \end{aligned}$$

(1.14)

Then for all initial data $u_{01},...,u_{0n}$ as in (1.6), (1.13) has a generalized solution in the sense of Definition 2.2.

Proof of Proposition 1.5

With $f_i(s)=\gamma _i s_i + \sum _{j=1}^N a_{ij} s_j^{\beta _{ij}}s_i^{\beta _{ji}}$, $i\in \{1,\ldots ,N\}$, (1.2) is clearly satisfied. As

$$\begin{aligned} \sum _{i=1}^N f_i(s)&= \sum _{i=1}^N \gamma _i s_i + \sum _{i=1}^N\sum _{j=1}^Na_{ij} s_j^{\beta _{ij}}s_i^{\beta _{ji}} \le \sum _{i=1}^N \gamma _i s_i + \sum _{i< j} (a_{ij}s_j^{\beta _{ij}}s_i^{\beta _{ji}}\\&\quad +a_{ji}s_i^{\beta _{ji}}s_j^{\beta _{ij}}) \le \max _i \gamma _i \sum _{i=1}^Ns_i \end{aligned}$$

due to the fact that $a_{ii}\le 0$ for all $i\in \{1,...,N\}$, we see that also (1.3) holds. Finally,

$$\begin{aligned} f_i(s)&\ge \gamma _i s_i + \sum \limits _{\begin{array}{c} j\in \{1,...,N\} \\ a_{ij}<0 \end{array}} a_{ij} s_j^{\beta _{ij}}s_i^{\beta _{ji}}\\&\ge - \bigg (|\gamma _i| + |a_{ii}|s_i^{\beta _{ii}} + \sum \limits _{\begin{array}{c} j\in \{1,...,N\}\setminus \{i\} \\ a_{ij}<0 \end{array}} |a_{ij}| s_j^{\beta _{ij}} \bigg )\bigg (s_i+\sum _{j=1}^Ns_i^{\beta _{ji}}\bigg )\\&\ge - \phi _i(s_i) \bigg (1+\sum \limits _{\begin{array}{c} j\in \{1,...,N\}\setminus \{i\} \\ a_{ij}<0 \end{array}} s_j^{\beta _{ij}}\bigg ) \end{aligned}$$

for all $s=(s_1,...,s_N)\in [0,\infty )^N$ if we set $\phi _i(s_i)=\max \{|\gamma _i|, |a_{ij}|\mid j\in \{1,\ldots N\}\}\cdot (1+s_i^{\beta _{ii})} (s_i+ \sum _{j=1}^Ns_i^{\beta _{ji}} )$ for any such s, so that, according to (1.14), (1.5) is fulfilled and Theorem 1.1 is applicable. $\square$

2 Solution concept

The first step toward the design of our solution concept is concerned with an appropriate supersolution feature required in each of the equations making up (1.4):

Definition 2.1

Suppose that for $i\in \{1,...,N\}$, $u_{0i}:\Omega \rightarrow \mathbb {R}$ and $u_i:\Omega \times (0,\infty ) \rightarrow \mathbb {R}$ are measurable and nonnegative. Then $(u_1,...,u_N)$ will be called a renormalized supersolution of (1.4) if for every $\rho \in C^\infty (\left[ 0,\infty ))\right.$ fulfilling $\rho `\in C_0^\infty (\left[ 0,\infty ))\right.$, $\rho `\le 0$ and $\rho ''\ge 0$, with

$$\begin{aligned} P_i^{(1)}(s)&{:=}\int _0^s \sigma ^\frac{m_i-1}{2} \sqrt{\rho ''(\sigma )} d\sigma \ \text{ and } \ P_i^{(2)}(s){:=}\int _0^s \sigma ^{m_i-1} \rho '(\sigma ) d\sigma ,\nonumber \\&\quad s\ge 0, \ i\in \{1,...,N\}, \end{aligned}$$

(2.1)

we have

$$\begin{aligned}&\rho '(u_i) f_i(u_1,...,u_N)\in L^1_{loc}(\overline{\Omega }\times [0,\infty )) \quad \text{ and } \nabla P_i^{(1)}(u_i) \in L^2_{loc}(\overline{\Omega }\times [0,\infty );\mathbb {R}^n)\nonumber \\&\quad \qquad \text{ for } \text{ all } i\in \{1,...,N\}, \end{aligned}$$

(2.2)

and if moreover

$$\begin{aligned} - \int _0^\infty \int _\Omega \rho (u_i)\varphi _t - \int _\Omega \rho (u_{0i}) \varphi (\cdot ,0)\le & {} - d_i m_i \int _0^\infty \int _\Omega \varphi |\nabla P_i^{(1)}(u_i)|^2\nonumber \\&+ d_1 m_i \int _0^\infty \int _\Omega P_i^{(2)}(u_i) \Delta \varphi \nonumber \\&+ \int _0^\infty \int _\Omega \rho '(u_i) f_i(u_1,...,u_N)\varphi \qquad \end{aligned}$$

(2.3)

for all $i\in \{1,...,N\}$ and each nonnegative $\varphi \in C_0^\infty (\overline{\Omega }\times [0,\infty ))$ fulfilling $\partial _\nu \varphi =0$ on $\partial \Omega \times (0,\infty )$.

Remark

• (i) In the above situation, both integrals on the left of (2.3) as well as the second integral on the right-hand side therein exist due to the readily verified fact that $\rho$ and $P_i^{(2)}$, $i\in \{1,...,N\}$, are bounded on $[0,\infty )$.

• (ii) The supersolution property in [15, Prop. 3.6] is obtained upon the choice of $\rho (x)=x$ (inadmissible in Definition 2.1), integration by parts in the integral involving $P_i^{(2)}$ (and addition of a corresponding integrability requirement) and, finally, exchange of $\le$ by $\ge$ (resulting from the change of sign of $\rho '$).

• (iii) The reason for dealing with supersolutions, that is, an inequality instead of an equality in (2.3), lies in the treatment of the integral containing $|\nabla P_i^{(1)}(u_i)|^2$, which will be estimated by means of lower semicontinuity, cf. (3.28).

• (iv) The definition of $P_i^{(1)}$, crucial in making use of the comparatively weak limit information for gradients, c.f. iii), contains $\sqrt{\rho ''}$, thereby causing the requirement $\rho ''\ge 0$ – and, consequently, due to compact support of $\rho '$ also $\rho '\le 0$ –, whereas comparable definitions in other systems only ask for the renormalization to be smooth with compactly supported derivative.

As discussed in several previous related approaches toward generalized solvability on the basis of supersolution features of the above flavor [28, 29], supplementing Definition 2.1 by a mere requirement on mass control is already sufficient to create a notion of solvability which within classes of suitably smooth functions indeed reduces to classical ones (see, e.g, [17] and [27] for detailed reasonings in this regard):

Definition 2.2

By a generalized solution of (1.4) we mean a vector $(u_1,...,u_N)$ of nonnegative measurable functions on $\Omega \times (0,\infty )$ such that $(u_1,...,u_N)$ is a renormalized supersolution of (1.4) in the sense of Definition 2.1, that with some $a\in (0,\infty )^N$

$$\begin{aligned} u_1,...,u_N \quad \text{ and } \quad \sum _{i=1}^N a_i f_i(u_1,...,u_N)\quad \text{ belong } \text{ to } L^1_{loc}(\overline{\Omega }\times [0,\infty )), \end{aligned}$$

(2.4)

and that

$$\begin{aligned}&\int _\Omega \bigg (\sum _{i=1}^N a_i u_i(\cdot ,t)\bigg ) \le \int _\Omega \bigg ( \sum _{i=1}^N a_i u_{0i} \bigg ) + \int _0^t \int _\Omega \bigg ( \sum _{i=1}^N a_i f_i(u_1,...,u_N)\bigg )\nonumber \\&\text{ for } \text{ a.e. } t>0. \end{aligned}$$

(2.5)

3 Approximate systems

In order to construct such solutions through an essentially standard type of approximation, for $\varepsilon \in (0,1)$ we consider

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle \partial _t u_{i\varepsilon }= d_i \Delta (u_{i\varepsilon }+\varepsilon )^{m_i} + \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|}, \qquad &{} x\in \Omega , \ t>0, \ i\in \{1,...,N\},\\ \partial _\nu u_{i\varepsilon }=0, \qquad &{} x\in \partial \Omega , \ t>0, \ i\in \{1,...,N\},\nonumber \\ u_{i\varepsilon }(x,0)=u_{0i\varepsilon }(x), \qquad &{} x\in \Omega , \ i\in \{1,...,N\}, \end{array} \right. \\ \end{aligned}$$

(3.1)

where

$$\begin{aligned} \left\{ \begin{array}{l} (u_{0i\varepsilon })_{\varepsilon \in (0,1)} \subset C^1(\overline{\Omega }) \quad \text{ is } \text{ such } \text{ that } u_{0i\varepsilon } \ge 0 \text{ for } \text{ all } i\in \{1,...,N\}, \text{ that }\\ u_{0i\varepsilon } \rightarrow u_{0i} \quad \text{ in } L^1(\Omega ) \text{ and } \text{ a.e. } \text{ in } \Omega \text{ as } \varepsilon \searrow 0 \text{ for } \text{ all } i\in \{1,...,N\}, \text{ and } \text{ that } \nonumber \\ \sup _{\varepsilon \in (0,1)} \Vert u_{0i\varepsilon }\Vert _{L^r(\Omega )} <\infty \end{array} \right. \\ \end{aligned}$$

(3.2)

with $r\ge 1$ taken from (1.6).

Due to boundedness of the reaction term therein and nondegeneracy of the diffusion, by [1, Theorems 14.4 and 14.6] (for local existence) and [16, Theorems V.7.3 and V.7.2] (for a priori bounds ensuring extensibility to globally defined solutions), for each fixed $\varepsilon \in (0,1)$ the problem (3.1) indeed admits a global classical solution

$$\begin{aligned} u_{i\varepsilon }\in C^0(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\qquad \text {for all } i\in \{1,\ldots ,N\}, \end{aligned}$$

which, moreover, is nonnegative.

General assumption. Throughout the sequel, we shall suppose that the assumptions of Theorem 1.1 and (3.2) are satisfied, and given $\varepsilon \in (0,1)$ we let $(u_{1\varepsilon },...,u_{N\varepsilon })$ denote the global classical solution of (3.1).

The following basic observation concerning $L^1$-boundedness of these solutions is a fairly immediate consequence of (1.3).

Lemma 3.1

For all $T>0$ there exists $C(T)>0$ such that for all $i\in \{1,...,N\}$ and any $\varepsilon \in (0,1)$ we have

$$\begin{aligned} \Vert u_{i\varepsilon }(\cdot ,t)\Vert _{L^1(\Omega )} \le C(T) \qquad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$

(3.3)

Proof

By integrating in (3.1), we see that since $\partial _\nu u_{i\varepsilon }=0$ on $\partial \Omega \times (0,\infty )$, due to (1.3) we have

$$\begin{aligned} \frac{d}{dt} \sum _{i=1}^N a_i \int _\Omega u_{i\varepsilon }= & {} \int _\Omega \frac{1}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \sum _{i=1}^N a_i f_i(u_{1\varepsilon },...,u_{N\varepsilon })\\\le & {} \int _\Omega \frac{1}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot K \bigg \{ \sum _{i=1}^N a_i u_{i\varepsilon }+1\bigg \} \\\le & {} K \sum _{j=1}^N a_i \int _\Omega u_{i\varepsilon }+ K|\Omega | \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

because $u_{i\varepsilon }\ge 0$ for all $i\in \{1,...,N\}$. By an ODE comparison, this shows that

$$\begin{aligned} \sum _{i=1}^N a_i \int _\Omega u_{i\varepsilon }\le \bigg \{ \sum _{i=1}^N a_i \int _\Omega u_{0i\varepsilon } + |\Omega | \bigg \} \cdot e^{Kt} \qquad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1) \end{aligned}$$

and hence, again by nonnegativity of $u_{i\varepsilon }$ for $i\in \{1,...,N\}$, establishes (3.3) due to (3.2). $\square$

The following estimate rests on a duality-based reasoning inspired by a corresponding argument from [15]. An important difference is given by the change of boundary conditions: Where [15] dealt with Dirichlet boundary data and thus could use the solution of Poisson’s equation as test function, the non-invertibility of the Laplacian with homogeneous Neumann boundary data leads us to employ the solution of a Helmholtz equation instead. An alternative approach of working with the Neumann Laplacian after subtracting the mean value has been followed in [14], but it seems unclear how far strategies of this type can be applied so as to successfully cover the present setting.

Lemma 3.2

For all $T>0$ there exists $C(T)>0$ such that for all $i\in \{1,...,N\}$,

$$\begin{aligned} \int _0^T \int _\Omega u_{i\varepsilon }^{m_i+1} \le C(T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$

(3.4)

Proof

Following [15, Proof of Theorem 2.7], we first observe that according to (3.1) and (1.3), writing

$$\begin{aligned}&w_{\varepsilon }(\cdot ,t):= e^{-Kt} \cdot \sum _{i=1}^N a_i u_{i\varepsilon }(\cdot ,t)\\&\qquad \text{ and } \qquad z_{\varepsilon }(\cdot ,t):=\int _0^t e^{-Ks} \cdot \bigg \{ \sum _{i=1}^N a_i d_i(u_{i\varepsilon }+\varepsilon )^{m_i} \bigg \} \end{aligned}$$

for $\varepsilon \in (0,1)$ and $t\ge 0$, we have

$$\begin{aligned} \partial _t w_{\varepsilon }= & {} e^{-Kt} \Delta \bigg \{ \sum _{i=1}^N a_i d_i (u_{i\varepsilon }+\varepsilon )^{m_i} \bigg \} + e^{-Kt} \sum _{i=1}^N \frac{a_if_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|}\\&\quad - K e^{-Kt} \sum _{i=1}^N a_i u_{i\varepsilon }\\\le & {} e^{-Kt} \Delta \bigg \{ \sum _{i=1}^N a_i d_i (u_{i\varepsilon }+\varepsilon )^{m_i} \bigg \} + K e^{-Kt} \quad \text{ in } \Omega \times (0,\infty ) \quad \text{ for } \text{ all } \varepsilon \in (0,1) \end{aligned}$$

and hence

$$\begin{aligned} w_{\varepsilon }(\cdot ,t) \le w_{\varepsilon }(\cdot ,0) + \Delta z_{\varepsilon }(\cdot ,t) +1 \quad \text{ in } \Omega \quad \text{ for } \text{ all } t>0 \text{ and } \varepsilon \in (0,1), \end{aligned}$$

(3.5)

because $\int _0^t K e^{-Ks} ds = 1-e^{-Ks}\le 1$ for all $t>0$. Upon multiplication by $\partial _t z_{\varepsilon }\ge 0$ and integration over $\Omega \times (0,T)$ for $T>0$, as in [15] we obtain that since $z_{\varepsilon }(\cdot ,0)\equiv 0$,

$$\begin{aligned} \int _0^T \int _\Omega w_{\varepsilon }\partial _t z_{\varepsilon }\le & {} \int _\Omega w_{\varepsilon }(\cdot ,0) z_{\varepsilon }(\cdot ,T) - \frac{1}{2} \int _\Omega |\nabla z_{\varepsilon }(\cdot ,T)|^2 + \int _\Omega z_{\varepsilon }(\cdot ,T) \nonumber \\\le & {} \int _\Omega \Big (w_{\varepsilon }(\cdot ,0)+1\Big ) z_{\varepsilon }(\cdot ,T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$

(3.6)

where the right-hand side is now estimated in a way slightly deviating from that in [15] due to the different boundary conditions considered here. In fact, by nonnegativity of $w_{\varepsilon }$ the inequality in (3.5) implies that for each fixed $T>0$ and arbitrary $\varepsilon \in (0,1)$,

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta z_{\varepsilon }(\cdot ,T) + z_{\varepsilon }(\cdot ,T) \le w_{\varepsilon }(\cdot ,0)+1+z_{\varepsilon }(\cdot ,T) \quad \text{ in } \Omega \qquad \text{ and } \\ \partial _\nu z_{\varepsilon }(\cdot ,T)=0 \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

so that since the Helmholtz operator $-\Delta +1$ admits a comparison principle under homogeneous Neumann boundary conditions, we obtain that

$$\begin{aligned} z_{\varepsilon }(\cdot ,T) \le \overline{z_{\varepsilon }}\qquad \text{ in } \Omega , \end{aligned}$$

(3.7)

where $\overline{z_{\varepsilon }}$ denotes the solution of

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta \overline{z_{\varepsilon }}+ \overline{z_{\varepsilon }}= w_{\varepsilon }(\cdot ,0)+1+z_{\varepsilon }(\cdot T) \quad \text{ in } \Omega , \\ \partial _\nu \overline{z_{\varepsilon }}=0 \quad \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$

Now without loss of generality assuming that the number $r\ge 1$ in (1.6) satisfies $r<2$, we take $r'\in (2,\infty ]$ such that $\frac{1}{r}+\frac{1}{r'}=1$, and employ a Sobolev embedding theorem and elliptic regularity theory [11] to find $c_1>0$ and $c_2>0$ such that for all $\varepsilon \in (0,1)$,

$$\begin{aligned} \Vert \overline{z_{\varepsilon }}\Vert _{L^{r'}(\Omega )}&\le c_1\Vert \overline{z_{\varepsilon }}\Vert _{W^{2,r}(\Omega )} \le c_2\Vert w_{\varepsilon }(\cdot ,0)+1+z_{\varepsilon }(\cdot ,T)\Vert _{L^{r}(\Omega )} \nonumber \\&\le c_2 \Vert w_{\varepsilon }(\cdot ,0)+1\Vert _{L^r(\Omega )} + c_2\Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^r(\Omega )}, \end{aligned}$$

(3.8)

because the restrictions on r in (1.6) warrant that $2-\frac{n}{r}\ge -\frac{n}{r'}$ if $n\ge 3$, and that $2-\frac{n}{r}>0$ if $n\le 2$. Here we note that according to Lemma 3.1 we know that if we let $\theta \in (0,1)$ small enough such that $\theta m_i \le 1$ for all $i\in \{1,...,N\}$, then we can find $c_3(T)>0$ fulfilling

$$\begin{aligned} \int _\Omega z_{\varepsilon }^\theta (\cdot ,T)\le & {} T^\theta \sup _{t\in (0,T)} \int _\Omega \bigg \{ \sum _{i=1}^Na_i d_i (u_{i\varepsilon }(\cdot ,t)+\varepsilon )^{m_i} \bigg \}^\theta \\\le & {} \Big \{ NT \max _{i\in \{1,...,N\}}a_i d_i \Big \}^\theta \max _{i\in \{1,...,N\}}\sup _{t\in (0,T)}\int _\Omega (u_{i\varepsilon }(\cdot ,t)+\varepsilon )^{\theta m_i} \\\le & {} \Big \{ NT \max _{i\in \{1,...,N\}} a_i d_i \Big \}^\theta \max _{i\in \{1,...,N\}}\sup _{t\in (0,T)}\int _\Omega (u_{i\varepsilon }(\cdot ,t)+1) \\\le & {} c_3(T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$

whence using that then $\theta<1<r<2<r'$ we may invoke Hölder’s and Young’s inequality to see that with $\lambda =(\frac{1}{\theta }-\frac{1}{r})/(\frac{1}{\theta }-\frac{1}{r'})\in (0,1)$ and some $c_4>0$ we have

$$\begin{aligned} c_2\Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^r(\Omega )}\le & {} c_2 \Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )}^{\lambda }\Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{\theta }(\Omega )}^{1-\lambda }\\\le & {} \frac{1}{2} \Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )} + c_4\Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^\theta (\Omega )} \\\le & {} \frac{1}{2} \Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )} + c_3^\frac{1}{\theta }(T) c_4 \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$

In view of (3.7) and the nonnegativity of $z_{\varepsilon }$, (3.8) thus implies that

$$\begin{aligned} \Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )}&\le \Vert \overline{z_{\varepsilon }}\Vert _{L^{r'}(\Omega )} \le c_2 \Vert w_{\varepsilon }(\cdot ,0)+1\Vert _{L^r(\Omega )} \\&\quad + \frac{1}{2}\Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )} + c_3^\frac{1}{\theta }(T) c_4 \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$

so that in (3.6) we can use the Hölder inequality to estimate

$$\begin{aligned} \int _\Omega \Big (w_{\varepsilon }(\cdot ,0)+1\Big )z_{\varepsilon }(\cdot ,T)\le & {} \Vert w_{\varepsilon }(\cdot ,0)+1\Vert _{L^r(\Omega )} \Vert z_{\varepsilon }(\cdot ,T)\Vert _{L^{r'}(\Omega )} \\\le & {} \Vert w_{\varepsilon }(\cdot ,0)+1\Vert _{L^r(\Omega )} \cdot \Big \{ 2c_2 \Vert w_{\varepsilon }(\cdot ,0)+1\Vert _{L^r(\Omega )} + 2c_3^\frac{1}{\theta }(T) c_4 \Big \} \end{aligned}$$

for all $\varepsilon \in (0,1)$. As $\sup _{\varepsilon \in (0,1)} \Vert w_{\varepsilon }(\cdot ,0)\Vert _{L^r(\Omega )}$ is finite according to the hypothesis (3.2), (3.6) thereby entails the existence of $c_5(T)>0$ such that

$$\begin{aligned} \int _0^T \int _\Omega e^{-2Kt} \cdot \bigg \{ \sum _{i=1}^N d_i(u_{i\varepsilon }+\varepsilon )^{m_i} \bigg \} \cdot \bigg \{ \sum _{i=1}^N u_{i\varepsilon }\bigg \} \le c_5(T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$

from which (3.4) readily follows. $\square$

We next rely on (1.5) in deriving the following estimates for gradients and reaction terms. Testing (3.1) by $-\frac{1}{\phi _i(u_{i\varepsilon })}$, namely, enables us to successfully combine (1.5) with (3.4). In order to obtain a bound for, e.g., $|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|$ instead of $|-\frac{1}{\phi _i(u_{i\varepsilon })}f_i(u_{1\varepsilon },...,u_{N\varepsilon })|$, we here restrict our attention to sets of the form $\{u_{i\varepsilon }\le M\}$, where $|-\frac{1}{\phi _i(u_{i\varepsilon })}|$ can be estimated from below by a positive constant.

Lemma 3.3

Let $M>0$ and $T>0$. Then one can find $C(M,T)>0$ such that

$$\begin{aligned} \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} (u_{i\varepsilon }+\varepsilon )^{m_i-1} |\nabla u_{i\varepsilon }|^2 \le C(M,T)\nonumber \\ \text{ for } \text{ all } i\in \mathbb {N} \text{ and } \varepsilon \in (0,1) \end{aligned}$$

(3.9)

and

$$\begin{aligned} \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \le C(M,T) \nonumber \\ \text{ for } \text{ all } i\in \mathbb {N} \text{ and } \varepsilon \in (0,1). \end{aligned}$$

(3.10)

Proof

For fixed $i\in \{1,...,N\}$, we take $\phi _i\in C^1([0,\infty ))$ as in (1.5) and define

$$\begin{aligned} \Phi _i(s):=-\int _1^{s+1} \frac{d\sigma }{\phi _i(\sigma )}, \qquad s\ge 0. \end{aligned}$$

Then since $\phi _i$ is nondecreasing, we have

$$\begin{aligned} 0 \ge \Phi _i(s) \ge - c_{1i} s \qquad \text{ for } \text{ all } s\ge 0 \end{aligned}$$

(3.11)

with $c_{1i}:=\frac{1}{\phi _i(1)}>0$, and moreover $\Phi _i'(s)=-\frac{1}{\phi _i(s+1)}, \ s\ge 0$, satisfies

$$\begin{aligned} 0 \le - \Phi _i'(s) \le \frac{1}{\phi _i(s)} \qquad \text{ for } \text{ all } s\ge 0. \end{aligned}$$

(3.12)

Now using (3.1), for $\varepsilon \in (0,1)$ and $t>0$ we compute

$$\begin{aligned} \frac{d}{dt} \int _\Omega \Phi _i(u_{i\varepsilon })= & {} \int _\Omega \Phi _i'(u_{i\varepsilon }) \cdot \bigg \{ d_i m_i \nabla \cdot \Big ((u_{i\varepsilon }+\varepsilon )^{m_i-1} \nabla u_{i\varepsilon }\Big )\\&\quad + \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \bigg \} \\= & {} - d_i m_i \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \Phi _i''(u_{i\varepsilon }) |\nabla u_{i\varepsilon }|^2\\&\quad + \int _\Omega \Phi _i'(u_{i\varepsilon }) \cdot \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|}, \end{aligned}$$

so that splitting $f_i=|f_i|-2(f_i)_-$, upon further integration we find that

$$\begin{aligned}&d_i m_i \int _0^T \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \Phi _i''(u_{i\varepsilon }) |\nabla u_{i\varepsilon }|^2\nonumber \\&\quad + \int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \cdot \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \nonumber \\&= \int _\Omega \Phi _i(u_{0i\varepsilon }) - \int _\Omega \Phi _i(u_{i\varepsilon }(\cdot ,T)) \nonumber \\&\quad + 2\int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \qquad \text{ for } \text{ all } T>0,\nonumber \\ \end{aligned}$$

(3.13)

because $\Phi _i'$ is nonpositive. Now on the right-hand side of (3.13) we use (1.5) to see that thanks to (3.12),

$$\begin{aligned} 2\int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|}\le & {} 2\int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \cdot \phi _i(u_{i\varepsilon }) \\&\quad \cdot \bigg \{ \sum _{j\ne i} u_{j\varepsilon }^{\beta _j}+1\bigg \} \\\le & {} 2\sum _{j\ne i} \int _0^T \int _\Omega u_{j\varepsilon }^{\beta _j} + 2|\Omega |T \\&\quad \qquad \text{ for } \text{ all } T>0, \end{aligned}$$

whence recalling Lemma 3.2 we obtain $c_{2i}(T)>0$ such that for all $\varepsilon \in (0,1)$,

$$\begin{aligned} 2\int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \le c_{2i}(T). \end{aligned}$$

(3.14)

As

$$\begin{aligned} \int _\Omega \Phi _i(u_{0i\varepsilon }) - \int _\Omega \Phi _i(u_{i\varepsilon }(\cdot ,T)) \le c_{1i} \int _\Omega u_{i\varepsilon }(\cdot ,T) \qquad \text{ for } \text{ all } T>0 \end{aligned}$$

by (3.11), in view of Lemma 3.1 we thus conclude from (3.13) that there exists $c_{3i}(T)>0$ with the property that

$$\begin{aligned}&d_i m_i \int _0^T \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \Phi _i''(u_{i\varepsilon }) |\nabla u_{i\varepsilon }|^2 + \int _0^T \int _\Omega |\Phi _i'(u_{i\varepsilon })| \nonumber \\&\quad \cdot \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \le c_{3i}(T) \end{aligned}$$

(3.15)

for all $\varepsilon \in (0,1)$. Noting that for fixed $M>0$ and any $i\in \{1,...,N\}$ the numbers

$$\begin{aligned}&c_{4i}:=\min _{s\in [0,M]} \Phi _i''(s) = \min _{s\in [0,M]} \frac{\phi _i'(s+1)}{\phi _i^2(s+1)} \qquad \text{ and }\\&c_{5i}:=\min _{s\in [0,M]} |\Phi _i'(s)| = \frac{1}{\phi _i(M+1)} \end{aligned}$$

are both positive according to our hypotheses on $\phi _i$ and $\phi _i'$, from (3.15) we readily infer (3.9) and (3.10) if we let $C(M,T):=\max \limits \Big \{ \max _{i\in \{1,...,N\}} \frac{c_{3i}(T)}{d_i m_i c_{4i}} \, , \, \max _{i\in \{1,...,N\}} \frac{c_{3i}(T)}{c_{5i}} \Big \}$, for instance. $\square$

While the bound in Lemma 3.2 is sufficient for concluding relative compactness of $\{u_{i\varepsilon }\mid \varepsilon \in (0,1)\}$ in some weak topology, we are additionally interested in possible pointwise convergence of $u_{i\varepsilon _j}$ along some sequence $(\varepsilon _j)_{j\in \mathbb {N}}\searrow 0$. We thus strive to derive a suitable strong compactness property in $L^2(\Omega \times (0,T))$, at least of a power of $u_{i\varepsilon }$ which has been cut off at large values so as to ensure accessibility to the estimates of Lemma 3.3.

Lemma 3.4

Given $\zeta \in C_0^\infty ([0,\infty ))$, for $i\in \{1,...,N\}$ let $\rho _i(s):=s^{\kappa _i}\zeta (s)$, $s\ge 0$, where $\kappa _i:=\max \{\frac{m_i+1}{2} \, , \, 2\}$. Then

$$\begin{aligned} \Big (\rho _i(u_{i\varepsilon })\Big )_{\varepsilon \in (0,1)} \text{ is } \text{ relatively } \text{ compact } \text{ in } L^2(\Omega \times (0,T)) \text{ for } \text{ all } T>0. \end{aligned}$$

(3.16)

Proof

Let us first make sure that for each $T>0$,

$$\begin{aligned} \Big (\rho _i(u_{i\varepsilon })\Big )_{\varepsilon \in (0,1)} \text{ is } \text{ bounded } \text{ in } L^2((0,T);W^{1,2}(\Omega )). \end{aligned}$$

(3.17)

To see this, we note that due to the compactness of $\mathrm{supp} \, \rho _i$ it is clear that $(\rho _i(u_{i\varepsilon }))_{\varepsilon \in (0,1)}$ is bounded in $L^\infty (\Omega \times (0,\infty ))$. Therefore, (3.17) results upon the observation that if we fix $M>0$ such that $\zeta \equiv 0$ on $(M,\infty )$, then by Young’s inequality and (3.9) we see that for all $T>0$ there exists $c_1(T)>0$ such that

$$\begin{aligned} \int _0^T \int _\Omega |\nabla \rho _i(u_{i\varepsilon })|^2= & {} \int _0^T \int _\Omega (\rho _i'(u_{i\varepsilon }))^2 |\nabla u_{i\varepsilon }|^2 \\= & {} \int _0^T \int _\Omega \Big ( \kappa _i u_{i\varepsilon }^{\kappa _i-1} \zeta (u_{i\varepsilon }) + u_{i\varepsilon }^{\kappa _i} \zeta '(u_{i\varepsilon })\Big )^2 |\nabla u_{i\varepsilon }|^2 \\\le & {} 2\kappa _i^2 \Vert \zeta \Vert _{L^\infty ((0,\infty ))}^2 \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} u_{i\varepsilon }^{2\kappa _i-2} |\nabla u_{i\varepsilon }|^2 \\&+ 2\Vert \zeta '\Vert _{L^\infty ((0,\infty ))}^2 \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} u_{i\varepsilon }^{2\kappa _i} |\nabla u_{i\varepsilon }|^2 \\\le & {} 2\kappa _i^2 \Vert \zeta \Vert _{L^\infty ((0,\infty ))}^2 M^{2\kappa _i-m_i-1} \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} u_{i\varepsilon }^{m_i-1} |\nabla u_{i\varepsilon }|^2 \\&+ 2\Vert \zeta '\Vert _{L^\infty ((0,\infty ))}^2 M^{2\kappa _i-m_i+1} \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} u_{i\varepsilon }^{m_i-1} |\nabla u_{i\varepsilon }|^2 \\\le & {} c_1(T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$

where we have used that $2\kappa _i-m_i+1 \ge 2\kappa _i-m_i-1\ge 0$ by hypothesis.

We next fix any integer $k\ge 1$ such that $k>n$, and claim that then for all $T>0$,

$$\begin{aligned} \Big (\partial _t \rho _i(u_{i\varepsilon })\Big )_{\varepsilon \in (0,1)} \text{ is } \text{ bounded } \text{ in } L^1 \Big ( (0,T);(W^{k,2}(\Omega ))^\star \Big ). \end{aligned}$$

(3.18)

To verify this in quite a straightforward manner, we pick $\psi \in C^\infty (\overline{\Omega })$ and use (3.1) to see that for each $t>0$ and any $\varepsilon \in (0,1)$,

$$\begin{aligned}&\bigg | \int _\Omega \partial _t \rho _i(u_{i\varepsilon }(\cdot ,t)) \cdot \psi \bigg | \nonumber \\&= \Bigg | \int _\Omega \rho _i'(u_{i\varepsilon }) \psi \cdot \bigg \{ d_i m_i \nabla \cdot \Big ( (u_{i\varepsilon }+\varepsilon )^{m_i-1} \nabla u_{i\varepsilon }\Big ) + \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \bigg \} \Bigg | \nonumber \\&=\Bigg | -d_i m_i \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \rho _i''(u_{i\varepsilon }) |\nabla u_{i\varepsilon }|^2 \psi - d_i m_i \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \rho _i'(u_{i\varepsilon }) \nabla u_{i\varepsilon }\cdot \nabla \psi \nonumber \\&\quad + \int _\Omega \rho _i'(u_{i\varepsilon }) \cdot \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \psi \Bigg | \nonumber \\\le & {} d_i m_i \Vert \rho _i''\Vert _{L^\infty ((0,\infty ))} \cdot \bigg \{ \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} (u_{i\varepsilon }+\varepsilon )^{m_i-1} |\nabla u_{i\varepsilon }|^2 \bigg \} \cdot \Vert \psi \Vert _{L^\infty (\Omega )} \nonumber \\&\quad + d_i m_i \cdot \bigg \{ \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} (u_{i\varepsilon }+\varepsilon )^{m_i-1} |\nabla u_{i\varepsilon }|^2 + |\Omega |\sup _{s\ge 0} (s+\varepsilon )^{m_i-1} |\rho _i'(s)|^2 \bigg \} \Vert \nabla \psi \Vert _{L^\infty (\Omega )}\nonumber \\&\quad + \Vert \rho _i'\Vert _{L^\infty ((0,\infty ))} \cdot \bigg \{ \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \bigg \} \cdot \Vert \psi \Vert _{L^\infty (\Omega )}, \end{aligned}$$

(3.19)

where finiteness of both $\Vert \rho '\Vert _{L^\infty ((0,\infty ))}$ and $\Vert \rho _i''\Vert _{L^\infty ((0,\infty ))}$ is asserted by our restriction that $\kappa _i\ge 2$. Here we observe that in the case $m_i\ge 1$ we have

$$\begin{aligned} (s+\varepsilon )^{m_i-1} |\rho _i'(s)|^2 \le (M+1)^{m_i-1} \Vert \rho _i'\Vert _{L^\infty ((0,\infty ))}^2, \end{aligned}$$

whereas if $m_i<1$ then

$$\begin{aligned} (s+\varepsilon )^{m_i-1} |\rho _i'(s)|^2\le & {} s^{m_i-1} |\rho _i'(s)|^2 \\= & {} s^{m_i-1} \cdot \Big |\kappa _i s^{\kappa _i-1} \zeta (s) + s^{\kappa _i} \zeta '(s)\Big |^2 \\\le & {} 2\kappa _i^2 s^{2\kappa _i+m_i-3} \zeta ^2(s) + 2s^{2\kappa _i+m_i-1} |\zeta '(s)|^2 \\\le & {} 2\kappa _i^2 M^{2\kappa _i+m_i-3} \Vert \zeta \Vert _{L^\infty ((0,\infty ))}^2 + 2M^{2\kappa _i+m_i-1} \Vert \zeta '\Vert _{L^\infty ((0,\infty ))}^2, \end{aligned}$$

because again by definition of $\kappa _i$, we have $2\kappa _i+m_i-1\ge 2\kappa _i+m_i-3 \ge 0$. As furthermore $W^{k,2}(\Omega ) \hookrightarrow W^{1,\infty }(\Omega )$ due to our restriction that $k>n$, from (3.19) we thus infer the existence of $c_2>0$ such that for all $t>0$ and any $\varepsilon \in (0,1)$,

$$\begin{aligned} \Big \Vert \partial _t \rho _i(u_{i\varepsilon }(\cdot ,t))\Big \Vert _{(W^{k,2}(\Omega ))^\star }\le & {} c_2 \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} (u_{i\varepsilon }+\varepsilon )^{m_i-1} |\nabla u_{i\varepsilon }|^2 + c_2 \\&+ c_2 \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|}, \end{aligned}$$

which in light of (3.9) and (3.10) establishes (3.18) upon a time integration.

We finally only need to combine (3.17) with (3.18) to conclude that (3.16) is a consequence of an Aubin-Lions type lemma ([25, Cor. 4]). $\square$

Based on this compactness statement, we may conclude the existence of a limit object.

Lemma 3.5

There exist $(\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)$ and nonnegative functions $u_1,...,u_N$ defined on $\Omega \times (0,\infty )$ such that $\varepsilon _j\searrow 0$ as $j\rightarrow \infty$, and such that for all $i\in \{1,...,N\}$,

$$\begin{aligned} u_{i\varepsilon }\rightarrow u_i \qquad \text{ a.e. } \text{ in } \Omega \times (0,\infty ) \end{aligned}$$

(3.20)

and

$$\begin{aligned} u_{i\varepsilon }\rightarrow u_i \qquad \text{ in } L^p_{loc}(\overline{\Omega }\times [0,\infty )) \quad \text{ for } \text{ all } p\in [1,m_i+1) \end{aligned}$$

(3.21)

as well as

$$\begin{aligned} u_{i\varepsilon }\rightharpoonup u_i \qquad \text{ in } L^{m_i+1}_{loc}(\overline{\Omega }\times [0,\infty )) \end{aligned}$$

(3.22)

as $\varepsilon =\varepsilon _j\searrow 0$.

Proof

In Lemma 3.4 choosing $\zeta =\zeta _l$, with $\zeta _l \in C_0^\infty ([0,\infty ))$ satisfying $\zeta _{l}\equiv 1$ in [0, l] for $l\in \mathbb {N}$, by means of a straightforward extraction procedure relying on the relative compactness of $\{u_{i\varepsilon }^2\zeta _l(u_{i\varepsilon })\mid \varepsilon \in (0,1)\}$ in $L^2(\Omega \times (0,T))$ for any $T>0$ we obtain $(\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)$ and $u=(u_1,...,u_N):\Omega \times (0,\infty )\rightarrow \mathbb {R}^N$ such that for all $i\in \{1,...,N\}$ we have $u_{i}\ge 0$ and $u_{i\varepsilon }\rightarrow u_i$ a.e. in $\Omega \times (0,\infty )$ as $\varepsilon =\varepsilon _j\searrow 0$. Since for each $i\in \{1,...,N\}$ and all $T>0$ we know from Lemma 3.2 that $(u_{i\varepsilon })_{\varepsilon \in (0,1)}$ is bounded in $L^{m_i+1}(\Omega \times (0,T))$, and that hence $(u_{i\varepsilon }^p)_{\varepsilon \in (0,1)}$ is uniformly integrable over $\Omega \times (0,T)$ for all $p\in [1,m_i+1)$, by reflexivity of $L^{m_i+1}(\Omega \times (0,T))$ and the Vitali convergence theorem we readily infer that on passing to a further subsequence if necessary we can also achieve simultaneous validity of (3.21) and (3.22). $\square$

Our next goal is to show that the functions just constructed actually form a solution. We begin by confirming that they enjoy a renormalized supersolution property in the style of Definition 2.1. The most crucial ingredient in our verification of this – and actually the reason for dealing with supersolutions – becomes apparent in (3.28), which is enlisted to control the integral involving the gradient (of $P_i^{(1)}(u)$) from above by means of lower semicontinuity.

Lemma 3.6

Let $u_1,...,u_N$ be as given by Lemma 3.5. Then $u=(u_1,...,u_N)$ forms a renormalized supersolution of (1.4) in the sense of Definition 2.1.

Proof

We fix $i\in \{1,...,N\}$ and a nonincreasing convex $\rho \in C^\infty ([0,\infty ))$ such that $\rho ' \in C_0^\infty ([0,\infty ))$, and use (3.1) to see that for all nonnegative $\varphi \in C_0^\infty (\overline{\Omega }\times [0,\infty ))$ such that $\frac{\partial \varphi }{\partial \nu }=0$ on $\partial \Omega \times (0,\infty )$, we have

$$\begin{aligned}&- \int _0^\infty \int _\Omega \rho (u_{i\varepsilon }) \varphi _t - \int _\Omega \rho (u_{0i\varepsilon }) \varphi (\cdot ,0) = + \int _0^\infty \int _\Omega \partial _t \rho (u_{i\varepsilon }) \varphi \nonumber \\&\quad = \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \varphi \cdot \bigg \{d_i m_i \nabla \cdot \Big ( (u_{i\varepsilon }+\varepsilon )^{m_i-1} \nabla u_{i\varepsilon }\Big )\nonumber \\&\qquad + \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \bigg \} \nonumber \\&\quad = -d_i m_i \int _0^\infty \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \rho ''(u_{i\varepsilon }) |\nabla u_{i\varepsilon }|^2 \varphi \nonumber \\&\qquad - d_i m_i \int _0^\infty \int _\Omega (u_{i\varepsilon }+\varepsilon )^{m_i-1} \rho '(u_{i\varepsilon }) \nabla u_{i\varepsilon }\cdot \nabla \varphi \nonumber \\&\qquad + \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \nonumber \\&\quad = - d_i m_i \int _0^\infty \int _\Omega |\nabla P_{i\varepsilon }^{(1)} (u_{i\varepsilon })|^2 \varphi + d_i m_i \int _0^\infty \int _\Omega P_{i\varepsilon }^{(2)} (u_{i\varepsilon }) \Delta \varphi \nonumber \\&\qquad + \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{f_i(u_{1\varepsilon },...,u_{N\varepsilon })}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \qquad \text{ for } \text{ all } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$

(3.23)

where we have set

$$\begin{aligned}&P_{i\varepsilon }^{(1)}(s):=\int _0^s (\sigma +\varepsilon )^\frac{m_i-1}{2} \sqrt{\rho ''(\sigma )} d\sigma \nonumber \\&\quad \text{ and } \quad P_{i\varepsilon }^{(2)}(s):=\int _0^s (\sigma +\varepsilon )^{m_i-1} \rho '(\sigma ) d\sigma , \qquad s\ge 0, \end{aligned}$$

(3.24)

for $i\in \{1,...,N\}$ and $\varepsilon \in (0,1)$. Here we note that if we take $M>0$ and $T>0$ large enough fulfilling $\mathrm{supp} \, \rho ' \subset [0,M]$ and $\mathrm{supp} \, \varphi \subset \overline{\Omega }\times [0,T]$, then for all $\varepsilon \in (0,1)$,

$$\begin{aligned}&\int _0^\infty \int _\Omega \Big | \sqrt{\varphi } \nabla P_{i\varepsilon }^{(1)} (u_{i\varepsilon }) \Big |^2 \le \Vert \rho ''\Vert _{L^\infty ((0,\infty ))} \Vert \varphi \Vert _{L^\infty (\Omega \times (0,\infty ))} \\&\quad \int _0^T \int _\Omega \chi _{\{u_{i\varepsilon }\le M\}} (u_{i\varepsilon }+\varepsilon )^{m_i-1} |\nabla u_{i\varepsilon }|^2, \end{aligned}$$

whence again employing Lemma 3.3 we infer that with $(\varepsilon _j)_{j\in \mathbb {N}}$ as provided by Lemma 3.5, we can find a subsequence $(\varepsilon _{j_k})_{k\in \mathbb {N}}$ such that

$$\begin{aligned} \sqrt{\varphi } \nabla P_{i\varepsilon }^{(1)} (u_{i\varepsilon }) \rightharpoonup z \quad \text{ in } L^2(\Omega \times (0,\infty );\mathbb {R}^n) \qquad \text{ as } \varepsilon =\varepsilon _{j_k} \searrow 0 \end{aligned}$$

(3.25)

for some $z\in L^2(\Omega \times (0,\infty );\mathbb {R}^n)$. On the other hand, since (3.24) entails that $P_{i\varepsilon }^{(1)} \rightarrow P_i^{(1)}$ in $L^\infty _{loc}([0,\infty ))$ as $\varepsilon \searrow 0$, with $P_i^{(1)}$ given by (2.1), and since moreover

$$\begin{aligned} |P_{i\varepsilon }^{(1)}(s)|\le & {} \Vert \rho ''\Vert _{L^\infty ((0,\infty ))}^\frac{1}{2} \int _0^M (\sigma +\varepsilon )^\frac{m_i-1}{2} d\sigma \nonumber \\= & {} \Vert \rho ''\Vert _{L^\infty ((0,\infty ))}^\frac{1}{2} \cdot \frac{(M+\varepsilon )^\frac{m_i+1}{2}-\varepsilon ^\frac{m_i+1}{2}}{\frac{m_i+1}{2}} \nonumber \\\le & {} \Vert \rho ''\Vert _{L^\infty ((0,\infty ))}^\frac{1}{2} \cdot \frac{2(M+1)^\frac{m_i+1}{2}}{m_i+1} \qquad \text{ for } \text{ all } s\ge 0 \text{ and } \varepsilon \in (0,1),\nonumber \\ \end{aligned}$$

(3.26)

from (3.20) and the dominated convergence theorem it follows that

$$\begin{aligned} P_{i\varepsilon }^{(1)}(u_{i\varepsilon }) \rightarrow P_i^{(1)}(u_i) \quad \text{ in } L^1(\Omega \times (0,T)) \qquad \text{ as } \varepsilon =\varepsilon _j\searrow 0. \end{aligned}$$

(3.27)

Therefore, a standard argument shows that in (3.25) we must have $z=\sqrt{\varphi } \nabla P_i^{(1)}(u_i)$ a.e. in $X_{\delta }=\{\varphi >\delta \}$ for all $\delta >0$ (where (3.27) and (3.25) enable us to pass to the limit in the definition of the weak gradient of $P_{i\varepsilon }^{(1)}(u_{i\varepsilon })$: $-\int _{X_{\delta }} P_i^{(1)}(u_i)\nabla \psi \leftarrow -\int _{X_{\delta }} \nabla \psi P_{i\varepsilon }^{(1)}(u_{i\varepsilon }) = \int _{X_{\delta }} \psi \nabla P_{i\varepsilon }^{(1)}(u_{i\varepsilon }) \rightarrow \int _{X_{\delta }} \psi \frac{z}{\sqrt{\varphi }}$ for each $\psi \in C_0^{\infty }(X_\delta )$), and hence actually $\sqrt{\varphi } \nabla P_{i\varepsilon }^{(1)} (u_{i\varepsilon }) \rightharpoonup \sqrt{\varphi } \nabla P_i^{(1)}(u_i)$ in $L^2_{loc}(\Omega \times (0,\infty ))$ as $\varepsilon =\varepsilon _{j_k}\searrow 0$, so that by lower semicontinuity of the norm in $L^2(\Omega \times (0,\infty ))$ with respect to weak convergence,

$$\begin{aligned} d_i m_i \int _0^\infty \int _\Omega |\nabla P_i^{(1)}(u_i)|^2 \varphi \le \liminf _{\varepsilon =\varepsilon _{j_k}\searrow 0} \bigg \{ d_i m_i \int _0^\infty \int _\Omega |\nabla P_{i\varepsilon }^{(1)}(u_{i\varepsilon })|^2 \varphi \bigg \}.\qquad \end{aligned}$$

(3.28)

Next addressing the integrals in (3.23) exclusively containing zero-order expressions with respect to $u_{i\varepsilon }$, we first observe that clearly

$$\begin{aligned} |\rho (s)| \le \Vert \rho '\Vert _{L^\infty ((0,\infty ))} \cdot M +|\rho (0)| \qquad \text{ for } \text{ all } s\ge 0, \end{aligned}$$

and that furthermore, by (3.24), similarly to (3.26) we can estimate

$$\begin{aligned} |P_{i\varepsilon }^{(2)}(s)|\le & {} \Vert \rho '\Vert _{L^\infty ((0,\infty ))} \int _0^M (\sigma +\varepsilon )^{m_i-1} d\sigma \\\le & {} \Vert \rho '\Vert _{L^\infty ((0,\infty ))} \cdot \frac{(M+1)^{m_i}}{m_i} \qquad \text{ for } \text{ all } s\ge 0 \text{ and } \varepsilon \in (0,1). \end{aligned}$$

Therefore, three applications of the dominated convergence theorem on the basis of (3.20) and (3.2) show that if we take $P_i^{(2)}$ from (2.1) then

$$\begin{aligned} \int _0^\infty \int _\Omega \rho (u_{i\varepsilon }) \varphi _t \rightarrow \int _0^\infty \int _\Omega \rho (u_i)\varphi _t \end{aligned}$$

(3.29)

and

$$\begin{aligned} \int _\Omega \rho (u_{0i\varepsilon }) \varphi (\cdot ,0) \rightarrow \int _\Omega \rho (u_{0i}) \varphi (\cdot ,0) \end{aligned}$$

(3.30)

as well as

$$\begin{aligned} d_i m_i \int _0^\infty \int _\Omega P_{i\varepsilon }^{(2)}(u_{i\varepsilon }) \Delta \varphi \rightarrow d_i m_i \int _0^\infty \int _\Omega P_i^{(2)}(u_i) \Delta \varphi \end{aligned}$$

(3.31)

as $\varepsilon =\varepsilon _j\searrow 0$, the latter because in addition obviously $P_{i\varepsilon }^{(2)} \rightarrow P_i^{(2)}$ in $L^\infty _{loc}([0,\infty ))$ as $\varepsilon \searrow 0$.

Finally, in the crucial rightmost summand in (3.23) containing the reactive contribution, we once more rewrite $f_i=|f_i|-2(f_i)_-$ and note that fixing any $\delta >0$ such that $(1+\delta )\beta _j \le m_j+1$ for all $j\in \{1,...,N\} \setminus \{i\}$, again relying on (1.5) we can estimate

$$\begin{aligned}&\int _0^T \int _\Omega \bigg | \rho '(u_{i\varepsilon }) \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_{-}}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \bigg |^{1+\delta } \nonumber \\&\le \int _0^T \int _\Omega |\rho '(u_{i\varepsilon })|^{1+\delta } \cdot \bigg \{ \phi _i(u_{i\varepsilon }) \bigg (\sum _{j\ne i} u_{j\varepsilon }^{\beta _j} +1\bigg ) \bigg \}^{1+\delta } \cdot \varphi ^{1+\delta } \\&\le \Big \{ \Vert \rho '\Vert _{L^\infty ((0,\infty ))} \cdot \Vert \phi _i\Vert _{L^\infty ((0,M))} \cdot \Vert \varphi \Vert _{L^\infty (\Omega \times (0,\infty ))} \Big \}^{1+\delta }\\&\quad \cdot N^{1+\delta } \cdot \bigg ( \sum _{j\ne i} \int _0^T \int _\Omega u_{j\varepsilon }^{(1+\delta )\beta _j} +|\Omega |T\bigg ) \end{aligned}$$

for all $\varepsilon \in (0,1)$. In view of Lemma 3.2, by positivity of $\delta$ this implies uniform integrability of $\Big (\rho '(u_{i\varepsilon }) \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \Big )_{\varepsilon \in (0,1)}$ over $\Omega \times (0,T)$ and hence entails, when combined with (3.20), that

$$\begin{aligned}&- 2 \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \nonumber \\&\quad \rightarrow -2 \int _0^\infty \int _\Omega \rho '(u_i) (f_i(u_1,...,u_N))_- \varphi \end{aligned}$$

(3.32)

as $\varepsilon =\varepsilon _j\searrow 0$. Since apart from that, by nonnegativity of both $-\rho '$ and $\varphi$ we can invoke Fatou’s lemma to see that, again thanks to (3.20),

$$\begin{aligned}&- \int _0^\infty \int _\Omega \rho '(u_i) |f_i(u_1,...,u_N)| \varphi \nonumber \\&\quad \le \liminf _{\varepsilon =\varepsilon _j\searrow 0} \bigg \{ - \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \bigg \}, \end{aligned}$$

(3.33)

upon collecting (3.28)–(3.33) we altogether conclude from (3.23) that

$$\begin{aligned}&d_i m_i \int _0^\infty \int _\Omega |\nabla P_i^{(1)} (u_i) |^2 \varphi - \int _0^\infty \int _\Omega \rho '(u_i) |f_i(u_1,...,u_N)| \varphi \\&\quad \le \liminf _{\varepsilon =\varepsilon _{j_k}\searrow 0} \bigg \{ d_i m_i \int _0^\infty \int _\Omega \Big |\nabla P_{i\varepsilon }^{(1)}(u_{i\varepsilon })|^2 \varphi \\&\quad \quad - \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{|f_i(u_{1\varepsilon },...,u_{N\varepsilon })|}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \bigg \} \\&\quad = \liminf _{\varepsilon =\varepsilon _{j_k}\searrow 0} \bigg \{ \int _0^\infty \int _\Omega \rho (u_{i\varepsilon }) \varphi _t + \int _\Omega \rho (u_{0i\varepsilon })\varphi (\cdot ,0) \\&\qquad + d_i m_i \int _0^\infty \int _\Omega P_{i\varepsilon }^{(2)} (u_{i\varepsilon }) \Delta \varphi \\&\quad \quad - 2 \int _0^\infty \int _\Omega \rho '(u_{i\varepsilon }) \cdot \frac{(f_i(u_{1\varepsilon },...,u_{N\varepsilon }))_-}{1+\varepsilon \sum _{j=1}^N |f_j(u_{1\varepsilon },...,u_{N\varepsilon })|} \cdot \varphi \bigg \} \\&\quad = \int _0^\infty \int _\Omega \rho (u_i) \varphi _t + \int _\Omega \rho (u_{0i})\varphi (\cdot ,0) \\&\qquad + d_i m_i \int _0^\infty \int _\Omega P_i^{(2)} (u_i) \Delta \varphi - 2 \int _0^\infty \int _\Omega \rho '(u_i) (f_i(u_1,...,.u_N))_- \varphi , \end{aligned}$$

which is equivalent to the desired inequality (2.3). The integrability requirements in (2.2) are evident by-products of the above considerations. $\square$

But also the subsolution property encoded in (2.5) is fulfilled:

Lemma 3.7

The function $u=(u_1,...,u_N)$ from Lemma 3.5 satisfies (2.4) and (2.5) of Definition 2.2.

Proof

According to (3.21), we can pick a null set $N\subset (0,\infty )$ such that with $(\varepsilon _j)_{j\in \mathbb {N}}$ taken from Lemma 3.5, for each $t\in (0,\infty )\setminus N$ we have $u_{i\varepsilon }(\cdot ,t)\rightarrow u_i(\cdot ,t)$ in $L^1(\Omega )$ for all $i\in \{1,...,N\}$ and hence

$$\begin{aligned} \int _\Omega \sum _{i=1}^N a_i u_{i\varepsilon }(\cdot ,t) \rightarrow \int _\Omega \sum _{i=1}^N a_i u_i(\cdot ,t) \end{aligned}$$

(3.34)

as $\varepsilon =\varepsilon _j\searrow 0$, where $a\in (0,\infty )^N$ is taken from (1.3). We next let $F_\varepsilon (s_1,...,s_N):=\sum _{i=1}^N \frac{a_i f_i(s_1,...,s_N)}{1+\varepsilon \sum _{j=1}^N |f_j(s_1,...,s_N)|}$ and $F(s_1,...,s_N):=\sum _{i=1}^N a_i f_i(s_1,...,s_N)$ for $(s_1,...,s_N)\in \mathbb {R}^N$ and $\varepsilon \in (0,1)$, and then obtain from (1.3) that

$$\begin{aligned} \Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_+ \le K \sum _{i=1}^N a_i u_{i\varepsilon }+ K \quad \text{ in } \Omega \times (0,\infty ) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$

In view of (3.21) applied to $p:=1$, a version of the dominated convergence theorem thus ensures that

$$\begin{aligned}&\int _0^t \int _\Omega \Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_+ \rightarrow \int _0^t \int _\Omega F_+(u_1,...,u_N) \nonumber \\&\quad \text{ for } \text{ all } t>0 \qquad \text{ as } \varepsilon =\varepsilon _j\searrow 0, \end{aligned}$$

(3.35)

because clearly $\Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_+ \rightarrow F_+(u_1,...,u_N)$ a.e. in $\Omega \times (0,\infty )$ as $\varepsilon =\varepsilon _j\searrow 0$ by (3.20). Since from (3.1) we know that for all $t>0$ and $\varepsilon \in (0,1)$ we have

$$\begin{aligned}&\int _\Omega \sum _{i=1}^N a_i u_{i\varepsilon }(\cdot ,t) + \int _0^t \int _\Omega \Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_- = \int _\Omega \sum _{i=1}^N a_i u_{0i\varepsilon } \\&\quad + \int _0^t \int _\Omega \Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_+, \end{aligned}$$

where by (3.20) also $\Big (F_\varepsilon (u_{1\varepsilon },...,u_{N\varepsilon })\Big )_- \rightarrow F_-(u_1,...,u_N)$ a.e. in $\Omega \times (0,\infty )$ as $\varepsilon =\varepsilon _j\searrow 0$, and where $\int _\Omega \sum _{i=1}^N a_i u_{0i\varepsilon } \rightarrow \int _\Omega \sum _{i=1}^N a_i u_{0i}$ as $\varepsilon \searrow 0$ due to (3.2), invoking Fatou’s lemma we infer by means of (3.34) and (3.35) that

$$\begin{aligned}&\int _\Omega \sum _{i=1}^N a_i u_i(\cdot ,t) + \int _0^t \int _\Omega F_-(u_1,...,u_N) \le \int _\Omega \sum _{i=1}^N a_i u_{0i} \\&\quad + \int _0^t \int _\Omega F_+(u_1,...,u_N) \qquad \text{ for } \text{ all } t\in (0,\infty )\setminus N. \end{aligned}$$

For any such t, this firstly implies that $F(u_1,\ldots ,u_N)$ belongs to $L^1(\Omega \times (0,t))$, and secondly entails that (2.5) holds. $\square$

The previous two lemmata already demonstrate that u is a generalized solution in the sense of Definition 2.2:

Proof of Theorem 1.1

We take $u_1,...,u_N$ as given by Lemma 3.5 and then only need to combine Lemma 3.6 with Lemma 3.7. $\square$