Global classical solutions to reaction-diffusion systems in one and two dimensions

TL;DR

This paper proves the global existence and polynomial growth bounds of classical solutions to reaction-diffusion systems in one and two dimensions, using simplified methods and applying results to chemical systems with exponential convergence to equilibrium.

Contribution

It provides a simplified proof of global existence for reaction-diffusion systems in low dimensions, extending previous results with new techniques and applications.

Findings

Global classical solutions exist in 1D and 2D under entropy conditions.

Solutions grow at most polynomially in time in the L-infinity norm.

Chemical systems with complex balance converge exponentially to equilibrium.

Abstract

The global existence of classical solutions to reaction-diffusion systems in dimensions one and two is proved. The considered systems are assumed to satisfy an {\it entropy inequality} and have nonlinearities with at most cubic growth in 1D or at most quadratic growth in 2D. This global existence was already proved in [T. Goudon and A. Vasseur, Ann. Sci. \'Ecole Norm. Sup. (4) 43 (2010), no. 1, 117--142] by a De Giorgi method. In this paper, we give a simplified proof by using a modified Gagliardo-Nirenberg inequality and the regularity of the heat operator. Moreover, the classical solution is proved to have $L^{\infty}$-norm growing at most polynomially in time. As an application, solutions to chemical reaction-diffusion systems satisfying the so-called complex balance condition are proved to converge exponentially to equilibrium in $L^{\infty}$-norm.

Equations

$${\partial }_t{u}_i-d_i\Delta u_i$$

$$\nabla u_i\cdot \nu$$

$$u_i(x,0)$$

$$f_i(u_1,\dots ,u_{i-1},0,u_{i+1},\dots ,u_N)\ge 0\text{ for all }i=1,\dots ,N.$$

$$\sum _{i=1}^N{f}_i(u)({\mu }_i+\log u_i)\le 0\text{ for all }u\in {\mathbb{R}}_{+}^N.$$

$$|f_i(u)|\le K(|u{|}^{\mu }+1),$$

$$\mu \le 3\text{ when }d=1,$$

$$\mu \le 2\text{ when }d=2.$$

$$2S{O}_2+O_2\leftrightharpoons 2S{O}_3$$

$${\partial }_t{u}_1-d_1\Delta u_1$$

$${\partial }_t{u}_2-d_2\Delta u_2$$

$${\partial }_t{u}_3-d_3\Delta u_3$$

$$\sum _{i=1}^3{f}_i(u)({\mu }_i+\log u_i)=-(u_1^2{u}_2-u_3^2)(\log (u_1^2{u}_2)-\log (u_3^2))\le 0$$

$$\Vert u_i(t){\Vert }_{L^{\infty }(\Omega )}\le C_T\text{ for all }0<t\le T\text{ and all }i=1,\dots ,N,$$

$$\sum _{i=1}^N{f}_i(u)({\mu }_i+\log u_i)\le K_1\sum _{i=1}^N{u}_i+K_2\text{ for all }u\in {\mathbb{R}}_{+}^N,$$

$$\underset{t\ge 0}{\sup }\Vert u_i(t){\Vert }_{L^1}\le C_0,\underset{t\ge 0}{\sup }\Vert u_i(t)\log u_i(t){\Vert }_{L^1}\le C_0$$

$$\frac{d}{dt}$$

$$={\int }_{\Omega }\sum _{i=1}^N({\mu }_i+\log u_i){\partial }_t{u}_{i}dx$$

$$=-{\int }_{\Omega }\sum _{i=1}^N{d}_i\frac{|\nabla u_i{|}^2}{u_i}dx+{\int }_{\Omega }\sum _{r=1}^N({\mu }_i+\log u_i)f_i(u)dx$$

$$\le -{\int }_{\Omega }\sum _{i=1}^N{d}_i\frac{|\nabla u_i{|}^2}{u_i}dx\le 0.$$

$${\int }_{\Omega }\sum _{i=1}^N(u_i(t)\log u_i(t)-u_i(t)+{\mu }_i{u}_i(t))dx\le {\int }_{\Omega }\sum _{i=1}^N(u_{i,0}\log u_{i,0}-u_{i,0}+{\mu }_i{u}_{i,0})dx.$$

$${\int }_{\Omega }\sum _{i=1}^N(u_i(t)\log u_i(t)-u_i(t)+1)dx$$

$$\le C_1+N|\Omega |+\underset{i=1,\dots ,N}{\max }|{\mu }_i|{\int }_{\Omega }\sum _{i=1}^N{u}_i(t)dx.$$

$${\int }_{\Omega }\sum _{i=1}^N(u_i(t)\log u_i(t)-u_i(t)+1)dx\ge K{\int }_{\Omega }\sum _{i=1}^N{u}_i(t)dx-e^{K}N|\Omega |+N|\Omega |$$

$${\int }_{\Omega }\sum _{i=1}^N{u}_i(t)dx\le \frac{1}{K}(C_1+e^{K}N|\Omega |)$$

$$\Vert f{\Vert }_{L^4}^4\le \epsilon \Vert f{\Vert }_{H^1(\Omega )}^2\Vert f\log |f|{\Vert }_{L^1}^2+C_{\epsilon }\Vert f{\Vert }_{L^1}\text{ when }d=1,$$

$$\Vert f{\Vert }_{L^3}^3\le \epsilon \Vert f{\Vert }_{H^1(\Omega )}^2\Vert f\log |f|{\Vert }_{L^1}+C_{\epsilon }\Vert f{\Vert }_{L^1}\text{ when }d=2.$$

$$\Omega \{|f|\ge N\}:=\{x\in \Omega :|f(x)|\ge N\}.$$

$$\Vert f{\Vert }_{L^4}^4\le {\left(\Vert \chi (f){\Vert }_{L^4}+\Vert |f|-\chi (f){\Vert }_{L^4}\right)}^4\le 8(\Vert \chi (f){\Vert }_{L^4}^4+\Vert |f|-\chi (f){\Vert }_{L^4}^4)$$

$$\Vert |f|-\chi (f){\Vert }_{L^4}^4={\int }_{\Omega }||f|-\chi (f){|}^{4}dx\le (2N{)}^3{\int }_{\Omega \{|f|\le 2N\}}|f|dx\le (2N{)}^3\Vert f{\Vert }_{L^1}.$$

$$\Vert \chi (f){\Vert }_{L^4}^4\le C_4\Vert \chi (f){\Vert }_{H^1(\Omega )}^2\Vert \chi (f){\Vert }_{L^1}^2$$

$$\Vert \chi (f){\Vert }_{H^1(\Omega )}^2=\Vert {\chi }^{\prime }(f)\nabla f{\Vert }_{L^2}^2+\Vert \chi (f){\Vert }_{L^2}^2\le 4\Vert f{\Vert }_{H^1(\Omega )}^2,$$

$$\Vert \chi (f){\Vert }_{L^1}^2\le {\left({\int }_{\Omega \{|f|\ge N\}}|f|dx\right)}^2\le (\log N{)}^{-2}\Vert f\log |f|{\Vert }_{L^1}^2.$$

$$\Vert f{\Vert }_{L^4}^4\le 16C_4(\log N{)}^{-2}\Vert f{\Vert }_{H^1(\Omega )}^2\Vert f\log |f|{\Vert }_{L^1}^2+4(2N{)}^3\Vert f{\Vert }_{L^1}.$$

Full text

Classification AMS 2010: 35K57, 35B40, 35Q92, 80A30, 80A32

Keywords: Reaction-diffusion systems; Global classical solutions; Entropy estimates; Chemical reaction networks.

1. Introduction

Let $\Omega\subset\mathbb{R}^{d}$ with $d=1,2$ be a bounded domain with smooth boundary $\partial\Omega$ in case $d=2$ (e.g. $\partial\Omega$ is of class $C^{2+\epsilon}$ for $\epsilon>0$). In this paper, we study the global existence of classical soltuions to nonlinear reaction-diffusion system for $u=(u_{1},\ldots,u_{N}):\Omega\times\mathbb{R}_{+}\to\mathbb{R}^{N}$

[TABLE]

where $d_{i}>0$ for $i=1,\ldots,N$ are diffusion coefficients, $\nu$ is the outward normal on $\partial\Omega$, the nonlinearities $f_{i}:\mathbb{R}^{N}\to\mathbb{R}$ are assumed satisfy some properties which will be specified later.

Reaction-diffusion systems (RD systems) of type (1) are used to model many phenomena in natural sciences such as physics, chemistry or biology. The global existence of (classical, strong, weak) solutions to systems of type (1) is thus of importance, and has been extensively studied in literature and become nowadays a classical topic (see e.g. [1, 18, 21, 24, 29, 17] and references therein). However, it still poses many open problems, since it is usually difficult to obtain suitable a priori estimates of solutions to general RD systems (maximum principle fails to apply to RD systems except very special cases).

This paper studies global existence of classical solutions to RD system (1) in one and two dimensions, where the nonlinearities $f_{i}:\mathbb{R}^{N}\to\mathbb{R}$ are assumed to be locally Lipschitz continuous and satisfy the following conditions:

• •

(Positivity preserving) For all $u\in\mathbb{R}_{+}^{N}$ it holds

[TABLE]

• •

(Entropy inequality) There exists $\mu_{1},\ldots,\mu_{N}\in\mathbb{R}$, such that

[TABLE]

• •

(Growth condition) For all $u\in\mathbb{R}^{N}$ it holds for all $i=1,\ldots,N$

[TABLE]

for some constant $K>0$, where the growth rate $\mu$ satisfies

[TABLE]

and

[TABLE]

Here $|u|=\sum_{i=1}^{N}|u_{i}|$ for all $u\in\mathbb{R}^{N}$.

Example 1**.**

We give an example of a reaction-diffusion system satisfying ($\mathbf{P}$), ($\mathbf{E}$) and ($\mathbf{G}$). Consider the following reversible chemical reaction involving sulfur dioxide, oxygen and sulfur trioxide

[TABLE]

where the forward and backward reaction rate constants are assumed to be one. Denote by $u_{1},u_{2},u_{3}$ the concentrations of $SO_{2}$, $O_{2}$ and $SO_{3}$ respectively. The corresponding reaction-diffusion system for $u=(u_{1},u_{2},u_{3})$ reads as

[TABLE]

with homogeneous Neumann boundary condition $\nabla u_{i}\cdot\nu=0$ and initial data $u_{i}(\cdot,0)=u_{i,0}(\cdot)$. Here $d_{1},d_{2},d_{3}>0$ are positive diffusion coefficients. It is obvious that $f_{i}(u)$ satisfies the positivity preserving property ($\mathbf{P}$) and the growth condition ($\mathbf{G}$) in one dimension. Moreover, with $\mu_{1}=\mu_{2}=\mu_{3}=0$ we have

[TABLE]

since the function $\log x$ is increasing. That means the entropy inequality ($\mathbf{E}$) is satisfied.

Condition ($\mathbf{P}$) has a simple interpretation: when the $i$-th concentration is zero then it cannot be consumed in the reaction. This assumption is sometimes called quasi-positivity and it helps to obtain the positivity of solutions provided initial data are positive (see e.g. [25]). The entropy inequality ($\mathbf{E}$) provides a control on solutions in $L\log L$-norm (see Lemma 5), and it is guaranteed in many physical systems, see e.g. chemical reaction networks satisfying a complex balanced condition in Section 3. Loosely speaking, ($\mathbf{E}$) means that the free energy of the corresponding system is dissipating. Condition ($\mathbf{G}$) is the only ”real” restriction of systems under consideration in this paper. This restriction on the growth of nonlinearities is necessary for obtaining from $L\log L$-bound (implied by ($\mathbf{E}$)) suitable a priori estimates, which in turn lead to global classical solutions. Possible extensions to higher orders or higher dimensions, for example $\mu=2$ and $d\geq 3$, remains as an open problem.

Systems of type (1) with the conditions ($\mathbf{P}$), ($\mathbf{E}$) has been extensively investigated (see e.g. [3, 5, 8, 14, 12, 19, 20, 29]). We refer the reader to the review paper [26] for a detailed discussion. It is worth mentioning that the global existence of classical solution to (1) is widely open in general. For weaker notions of solutions, we refer the reader to [15] where the author showed the global existence of renormalised solutions under the conditions ($\mathbf{P}$) and ($\mathbf{E}$).

To put our work into context, let us recall that in [3], by using a duality method the authors have proved the global existence of classical solutions to (1) with quadratic nonlinearities, i.e. $|f_{i}(u)|\leq C(|u|^{2}+1)$, in one and two dimensions. When the dimension is three or higher, the classical solution is proved global provided the diffusion coefficients are closed to each other. We remark that, since the duality method is independent of dimensions, the arguments in [3] are not directly applicable to systems with cubic nonlinearities (even in one dimension). It’s also worth noting that the global classical solutions to quadratic systems in higher dimensions (without the assumption on diffusion coefficients) had remained open until the very recent preprint [4], in which the authors solved the problem (even for slightly super quadratic - depending on the dimension - systems) by utilising the De Giorgi method.

On the other hand, it was proved in [20] that system (1) with conditions ($\mathbf{P}$), ($\mathbf{E}$) and ($\mathbf{G}$) has global classical solutions. However, the proof therein was based on the famous De Giorgi method, and apparently did not provide any bounds (w.r.t. time) on the classical solutions.

In this paper, with the help of a modified Gagliardo-Nirenberg’s inequality and the regularity of the heat operator, we prove by simple arguments that system (1) with conditions ($\mathbf{P}$), ($\mathbf{E}$) and ($\mathbf{G}$) has a unique global classical solution. One advantage of our results is that the $L^{\infty}$-norm of the solution grows at most polynomially in time. This is usually called slowly growing a-priori bounds, see e.g. [6, 32]. This result is very helpful, especially when the asymptotic behaviour of solutions in weaker norm, say $L^{1}$-norm, is already established (see Corollary 11 for an application to chemical reaction networks). Note finally that similar ideas were also used in [19, 27, 30] to obtain global solutions for systems with quadratic nonlinearities in two dimensions.

The main result of this paper is the following.

Theorem 2**.**

Let $\Omega\subset\mathbb{R}^{d}$ with $d=1,2$ be a bounded domain with smooth boundary $\partial\Omega$ in case $d=2$ (e.g. $\partial\Omega$ is of class $C^{2+\epsilon}$ with $\epsilon>0$). Assume the diffusion coefficients are positive, i.e. $d_{i}>0$ for all $i=1,\ldots,N$, and the nonlinearities are locally Lipschitz and satisfy ($\mathbf{P}$), ($\mathbf{E}$) and ($\mathbf{G}$). Then for any initial data $(u_{i,0})\in L^{\infty}(\Omega)^{N}$, system (1) has a unique classical solution on $(0,\infty)$. Moreover, the $L^{\infty}$-norm of this solution grows at most polynomially in time, i.e. for all $T>0$

[TABLE]

in which $C_{T}$ is a constant depends at most polynomially w.r.t. $T$.

Remark 3**.**

The entropy condition ($\mathbf{E}$) can in fact be weakened as

[TABLE]

with positive constants $K_{1},K_{2}>0$. See e.g. [27] for more details.

Remark 4**.**

The $L^{\infty}(\Omega)$ initial data, which is chosen for the sake of simplicity, is certainly not optimal. For example, with more careful analysis, the initial data can be relaxed to $L^{p}(\Omega)$ for some $p>2$ (when one only wishes to have the bounds of solution after some time $t_{0}>0$).

The rest of this paper is organised as follows: In the next section, we give the proof of Theorem 2, and then its application to complex balanced systems is described in Section 3.

Notations: For simplicity, we will use the following set of notations:

• •

The norm in $L^{p}(\Omega)$ with $1\leq p\leq\infty$ is denoted by $\|\cdot\|_{L^{p}}$.

• •

For any $T>0$ we write $Q_{T}=\Omega\times(0,T)$ and $\|\cdot\|_{L^{p}(Q_{T})}:=\|\cdot\|_{L^{p}(0,T;L^{p}(\Omega))}$ for $p\geq 1$.

• •

We will denote by $C_{i}$ a varying constant (possibly) depending on the domain $\Omega$, the diffusion coefficients, etc., but independent of time $t$.

2. Proof of main results

The local existence of classical solutions to (1) with locally Lipschitz nonlinearities is standard (see e.g. [1, 29]). Moreover, thanks to the positivity preserving property ($\mathbf{P}$) of the nonlinearities, the solution is positive as long as it exists, see e.g. [25].

To prove that the classical solution is global, we first show in Lemma 5 a uniform-in-time bound for solution in $L^{1}$- and $L\log L$-norms. These bounds, in a combination with a modified Gagliardo-Nirenberg inequality in Lemma 6, lead then to some $L^{p}$-integrability of solutions in Lemma 7 for suitable $p$. Finally this integrability, with the help from smoothing effect of the heat operator in Lemma 8, implies the bound of solutions in $L^{\infty}$.

Lemma 5** (Entropy estimate).**

Assume that ($\mathbf{E}$) holds. Then we have the following a priori estimate for any classical solution to (1),

[TABLE]

here $C_{0}$ is a constant independent of $t$.

Proof.

From the entropy inequality ($\mathbf{E}$) we have

[TABLE]

Hence,

[TABLE]

Denote by $C_{1}$ the right hand side, we rewrite this inequality as

[TABLE]

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

[TABLE]

with $K=2\max_{i=1,\ldots,N}|\mu_{i}|$. Therefore we obtain the estimate

[TABLE]

which, together with the positivity of the solution, leads to the uniform in tim $L^{1}$-bound. The bound of $\|u_{i}(t)\log u_{i}(t)\|_{L^{1}}$ follows immediately from (3) and $x\log x\geq x-1$ for all $x\geq 0$. ∎

Lemma 6** (A modified Gagliardo-Nirenberg inequality).**

For any $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that for all $f\in H^{1}(\Omega)$

[TABLE]

and

[TABLE]

Proof.

The proof follows from the ideas in [2] where the authors obtained a similar version (with estimates for $L^{3}$-norm) in two dimensions. We only give a proof in case $d=1$ since the proof in case $d=2$ is similar.

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

[TABLE]

First we write

[TABLE]

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

[TABLE]

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

[TABLE]

for some constant $C_{4}>0$. On the one hand

[TABLE]

and on the other hand

[TABLE]

By combining (4)–(8) we obtain

[TABLE]

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

Lemma 7**.**

Assume that ($\mathbf{P}$), ($\mathbf{E}$) and ($\mathbf{G}$) hold. Then for any $T>0$, we have for all $i=1,\ldots,N$

[TABLE]

and

[TABLE]

with $\epsilon>0$ arbitrary, where $C_{T}$ is a constant grows at most polynomially w.r.t. $T$.

Proof.

Recall that $C_{i}$ denotes a various constant depending on the domain $\Omega$, the diffusion coefficients, the constant $K$ in ($\mathbf{G}$), and constant $C_{0}$ in Lemma 5, but independent of time $t$. Note that all constants $C_{i}$ can be explicitly computed.

We first prove (10). Multiplying the equation

[TABLE]

by $u_{i}$ in $L^{2}(\Omega)$ we have

[TABLE]

Hence, by summing over $i=1,\ldots,N$ and using Lemma 6, it follows that

[TABLE]

Choosing $\varepsilon$ small enough we get

[TABLE]

for some $C_{6}>0$. By adding $2\sum_{i=1}^{N}d_{i}\|u_{i}\|_{L^{2}}^{2}$ to both sides and using the one dimensional embedding inequality $\|u_{i}\|_{L^{\infty}}^{2}\leq C_{7}\|u_{i}\|_{H^{1}(\Omega)}^{2}$, it follows that

[TABLE]

Thus, using one more $\|u_{i}\|_{L^{\infty}}^{2}\geq\frac{1}{|\Omega|}\|u_{i}\|_{L^{2}}^{2}$, it leads to

[TABLE]

and consequently an integration on $(0,T)$ gives

[TABLE]

From this we have $u_{i}\in L^{\infty}(0,T;L^{2}(\Omega))$ and $u_{i}\in L^{2}(0,T;L^{\infty}(\Omega))$ for all $i=1,\ldots,N$. Finally, by an interpolation

[TABLE]

we obtain (10).

Consider now the case $d=2$. With computations similar to (12) we get

[TABLE]

Applying Lemma 6 with $d=2$ to $\|u_{i}\|_{L^{3}}^{3}$, and using the bounds $\sup\limits_{t\geq 0}\|u_{i}(t)\|_{L^{1}}\leq C_{0}$ and $\sup\limits_{t\geq 0}\|u_{i}(t)\log u_{i}(t)\|_{L^{1}}\leq C_{0}$ in Lemma 5 lead to

[TABLE]

By adding both sides with $\sum_{i=1}^{N}d_{i}\|u_{i}\|_{L^{2}}^{2}$ and using the two-dimensional embedding $H^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$ for any $2\leq p<+\infty$, we end up with

[TABLE]

Interpolation inequality and Young’s inequality give

[TABLE]

Inserting this into (13) and using the $L^{1}$-bound of $u_{i}$ we finally obtain, after integrating on $(0,T)$,

[TABLE]

Hence $u_{i}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;L^{p}(\Omega))$. Therefore the interpolation

[TABLE]

and the fact that $1\leq p<+\infty$ is arbitrary give us the desired estimate (11).

∎

We need the following the regularity of solutions to a heat equation with homogeneous Neumann boundary condition. The proof can be found in [3, Lemma 3.3].

Lemma 8** (Regularity of heat kernel).**

[3]** Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary $\partial\Omega$ (e.g. $\partial\Omega$ is of class $C^{2+\epsilon}$ with $\epsilon>0$). Consider the heat equation

[TABLE]

where $\delta>0$, with homogeneous Neumann boundary condition $\nabla y\cdot\nu=0$ on $\partial\Omega$, and initial data $y(\cdot,0)=y_{0}\in L^{\infty}(\Omega)$. Assume that right hand side $f\in L^{p}(Q_{T})$ with $p\geq 1$.

• (i)

If $p<(d+2)/2$ then

[TABLE]

• (ii)

If $p\geq(d+2)/2$ then

[TABLE]

Here $C_{T}$ is a constant depending on domain $\Omega$, the integrability $p$ and the diffusion coefficient $\delta$, and especially depending at most polynomially on $T$.

Remark 9**.**

Note that the regularity of solution to heat equation presented in Lemma 8 is classical (see e.g. [21]). The novelty of the lemma is to provide the bound $\|y\|_{L^{s}(Q_{T})}\leq C_{T}$ in which $C_{T}$ is a constant depending at most polynomially in $T$.

We are now ready to give a proof of Theorem 2.

Proof.

We will prove that for $i=1,\ldots,N$ it holds $\|u_{i}(t)\|_{L^{\infty}}<\infty$ for all $t>0$ and thus confirms the global existence of classical solution.

In this proof, we will always denote by $C_{T}$ a constant depending polynomially in $T>0$.

If $d=1$ and $\mu=3$ then it follows from $\|u_{i}\|_{L^{4}(Q_{T})}\leq C_{T}$ in Lemma 7 and ($\mathbf{G}$) that

[TABLE]

By applying Lemma 8 (i) to

[TABLE]

with $d=1$ and $p=4/3$ we have for all $i=1,\ldots,N$

[TABLE]

Using again the cubic growth ($\mathbf{G}$) we obtain for $i=1,\ldots,N$,

[TABLE]

Thus, we apply Lemma 8 (ii) to get

[TABLE]

Hence, $\|f_{i}(u)\|_{L^{r}(Q_{T})}\leq C_{T}$ for all $1\leq r<+\infty$. One more bootstrap gives us the desired result

[TABLE]

If $d=2$ and $\mu=2$ then we have

[TABLE]

due to ($\mathbf{G}$) and $\|u_{i}\|_{L^{4-\epsilon}}(Q_{T})\leq C$ in Lemma 7. Applying Lemma 8 (i) to $\partial_{t}u_{i}-d_{i}\Delta u_{i}=f_{i}(u)$ leads to

[TABLE]

Since $\epsilon>0$ is arbitrary, we obtain in fact $\|u_{i}\|_{L^{s}(Q_{T})}\leq C_{T}$ for all $1\leq s<+\infty$. Now Lemma 8 (ii) is applicable and finally provides the estimate $\|u_{i}(t)\|_{L^{\infty}}\leq C_{T}$ for all $0<t\leq T$.

The uniqueness of classical solution follows immediately from the $L^{\infty}$-bound and the fact that the nonlinearities $f_{i}(u)$ are locally Lipschitz. ∎

Remark 10**.**

From Lemma 7 we obtain in particular $\|u_{i}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq C_{T}$ by using entropy bound in Lemma 5, the restriction on dimension $d=1,2$ and the growth ($\mathbf{G}$). If the bound $L^{\infty}(0,T;L^{2}(\Omega))$ can be obtained from some other way (without using ($\mathbf{G}$)) then we can in fact improve the results of Theorem 2 to $\mu<5$ for $d=1$ and $\mu<3$ for $d=2$. The interested reader is referred to [31, Section 5] for more details.

3. Applications to complex balanced systems

One advantage of Theorem 2 is that it does not only provides the global existence of a classical solutions, but also gives a control on the growth (w.r.t to time) of $L^{\infty}$-norm of the solution. This becomes very helpful in the situation that we describe in the following.

A direct application of Theorem 2 can be seen in chemical reaction network theory. Consider $N$ chemical substances $S_{1},\ldots,S_{N}$ reacting in a network consisting of $R$ reactions, in which the $r$-th reaction is of the form

[TABLE]

Here $k_{r}>0$ is the reaction constant rate, $y_{r,i},y_{r,i}^{\prime}\in\{0\}\cup[1,\infty)$ are stoichiometric coefficients. By using the notation $y_{r}=(y_{r,1},\ldots,y_{r,N})$ and $y_{r}^{\prime}=(y_{r,1},\ldots,y_{r,N})$ we can rewrite the $r$-th reaction as

[TABLE]

in which we call $y_{r}$ a reactant and $y_{r}^{\prime}$ a production, and both of them are named complex. Denote by $\mathfrak{C}=\{y_{r},y_{r}^{\prime}\}_{r=1,\ldots,R}$ the set of all complexes. Note that each complex $y\in\mathfrak{C}$ can be both a reactant and a production (in possibly different reactions). Assume now that the following reaction network is taken place is a bounded domain $\Omega\subset\mathbb{R}^{d}$ and each substance $S_{i}$ diffuses with a constant rate $d_{i}>0$. Hence, by applying the law of mass action we obtain the reaction-diffusion system for the concentrations $u_{1},\ldots,u_{N}$ of $S_{1},\ldots,S_{N}$ respectively,

[TABLE]

where $\nu$ is the outward normal on $\partial\Omega$. System (14) is called complex balanced if there exists a strictly positive equilibrium $u_{\infty}=(u_{1,\infty},\ldots,u_{N,\infty})\in\mathbb{R}^{N}_{+}$ such that at $u_{\infty}$ the total in-flow and total out-flow at any complex $y\in\mathfrak{C}$ are balanced, i.e.

[TABLE]

For more details concerning complex balanced systems, the interested reader is referred to [7, 10, 11]. Note that (14) can also have boundary equilibrium $u^{*}$, that is $u^{*}$ satisfies (15) and $u^{*}\in\partial\mathbb{R}^{N}_{+}$. The convergence to equilibrium for systems of type (14) was extensively studied recently, see e.g. [7, 13, 23, 28] and references therein. In particular, it was proved in [13] that if (14) is complex balanced and has no boundary equilibria, then any renormalised solution (see [15]) converges in $L^{1}$-norm exponentially to the unique strictly positive complex balanced equilibrium $u_{\infty}$. Thus, by applying Theorem 2 we can show that the classical solution (in cases globally exists) in fact converges to equilibrium in $L^{\infty}$-norm exponentially.

Corollary 11**.**

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth enough boundary $\partial\Omega$ (e.g. $\partial\Omega$ is of class $C^{2+\epsilon}$ for $\epsilon>0$). Assume either

[TABLE]

or

[TABLE]

recalling that $d$ is the spatial dimension and $|y|=|y_{1}|+\ldots+|y_{N}|$ for $y\in\mathbb{R}^{N}$. Moreover, assume that system (14) is complex balanced and has no boundary equilibria. Then for any nonnegative initial data $u_{0}\in L^{\infty}(\Omega)$, the system (14) has a unique global classical solution which converges exponentially in $L^{\infty}$-norm to the corresponding complex balanced equilibrium $u_{\infty}$, i.e.

[TABLE]

where $C$ and $\lambda$ are positive constants.

Remark 12**.**

In the recent work [16] Fischer proved the weak-strong uniqueness result for (14), that is a renormalised solution to (14) identifies with the classical solution in the time interval where the latter exists. By using the global existence and uniqueness of classical solution in Corollary 11, we consequently obtain the uniqueness of renormalised solution to (14).

Proof.

Since system (14) is complex balanced and has no boundary equilibria, it follows from [13] that any renormalised solution converges exponentially to equilibrium $u_{\infty}$ in $L^{1}$-norm, i.e.

[TABLE]

for $C,\gamma>0$. From Theorem 2 and the growth conditions of nonlinearities we have $\|u_{i}(T)\|_{L^{\infty}}\leq C_{T}$. Now for each $1<p<\infty$, by using an interpolation estimate we have

[TABLE]

for some $\theta\in(0,1)$ and $0<\theta^{\prime}<(1-\theta)\gamma$, thus we get the exponential convergence in $L^{p}$-norm for all $1<p<\infty$. To obtain the convergence in $L^{\infty}$-norm, we first note that due to $\|u_{i}\|_{L^{\infty}(Q_{T})}\leq C_{T}$ we have $\|f_{i}(u)\|_{L^{\infty}(Q_{T})}\leq C_{T}$ and consequently

[TABLE]

thus it follows from [22] in particular that $\|\partial_{t}u_{i}\|_{L^{2}(Q_{T})}\leq C_{T}$. Hence

[TABLE]

where $\alpha>0$ is some fixed constant depending on the growth of the nonlinearity $f_{i}(u)$. On the other hand, due to the smoothing effect of the heat operator, for any small $0<\tau$ we have $u_{i}(\cdot,\tau)\in H^{1}(\Omega)$. Now consider the equation $\partial_{t}u_{i}-d_{i}\Delta u_{i}=f_{i}(u)$ with initial time at $\tau$ and initial data $u_{i}(\cdot,\tau)$ in $H^{1}(\Omega)$ we have the following estimate (see e.g. [9, Theorem 7.1.5])

[TABLE]

Therefore, using the usual Gagliardo-Nirenberg inequality (both in the cases $d=1$ and $d=2$) we obtain for some $\theta\in(0,1)$ that

[TABLE]

for some $0<\lambda<\gamma(1-\theta)$, due to the fact that $C_{T}$ grows at most polynomially in $T$. Combining this with $u_{i}\in L^{\infty}(0,\tau;L^{\infty}(\Omega))$ we can finish the proof of Corollary 11. ∎

Acknowledgements: The author would like to thank Prof. Laurent Desvillettes and Prof. Klemens Fellner for fruitful discussion, which leads to this work. This work is partially supported by International Training Program IGDK 1754 and NAWI Graz.