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The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. (English) Zbl 1259.35015

The authors are interested in the instantaneous limit for reaction-diffusion systems which contain a fast irreversible reaction of type \(A+B\rightarrow P,\) but also additional slow processes like other chemical reactions or macroscopic convection terms, where the point is to allow for quadratic growth of these. More precisely, in a bounded open subset \(\Omega\subset \mathbb{R}^{n}\) with a smooth boundary \(\partial\Omega\) the following system of reaction-diffusion equations, as a prototype model, is considered: \[ \begin{cases} \partial_{t}u_{1}-d_{1}\Delta u_{1}=-ku_{1}u_{2}+f_{1}(t,u)\\ \partial_{t}u_{2}-d_{2}\Delta u_{2}=-ku_{1}u_{2}+f_{2}(t,u)\\ \partial_{t}u_{i}-d_{i}\Delta u_{i}=f_{i}(t,u)\,\,\,(i=3,\dots,n)\\ \partial_{n}u_{i|\partial\Omega}=0,\,\,\,u_{i|t=0}=u_{i,0}.\\ \end{cases}\tag{1} \] Here \(u=(u_{1},\dots,u_{n})\) where \(u_{1}\) and \(u_{2}\) denote the molar concentrations of \(A\) and \(B,\) respectively, and for \(i\geq 3\) the quantity \(u_{i}\) is the concentration of further components which are involved in the chemical reaction network. The \(d_{i}\) \((i=1,\dots,n)\) are positive diffusion coefficients, \(k>0\) is the rate constant of the irreversible reaction and \(u_{i,0}\) are the initial concentrations which the authors assume to be nonnegative and at least integrable.
It is assumed that the nonlinearity \(f:[0,T]\times \mathbb R_{+}^{n}\rightarrow \mathbb R^{n}\) is jointly continuous and locally Lipschitz continuous in the second variable with special attention to the case where \(f\) has at most quadratic growth, i.e. \(|f(t,u)|\leq K(\psi_{0}+|u|^{2})\) with some \(K>0,\) \(\psi_{0}\in L^{1}(Q_{T})\). Further, \(f\) allows \(L^{2}\)-control of the total mass, i.e.
\[ \begin{cases} \langle f(t,u),e\rangle \leq L(\phi_{0}+\langle u,e\rangle)\,\,\text{with some}\,\, L>0,e\gg 0,\text{and}\\ \phi_{0}\in L^{1}(Q_{T}),\quad\Phi_{0}\in L^{2}(Q_{T}),\,\,\text{where}\,\,\Phi_{0}(t,x)=\int_{0}^{t}\phi_{0}(s,x)ds,\\ \end{cases} \]
where the notation \(e\gg 0\) is short for \(e\in \mathbb{R}^{n}\) with \(e_{i}>0\) for all \(i.\) Moreover the authors are interested in nonnegative concentrations, when \(f\) is quasi-positive, which means \(f_{i}(t,u)\geq 0\) whenever \(u\in \mathbb{R}_{+}^{n}\) satisfies \(u_{i}=0.\)
Setting \(v=u_{1}-u_{2}\) and \(w=(u_{3},\dots,u_{n}),\) the limit system for \(k\rightarrow\infty\) reads as \[ \begin{cases} \partial_{t}v-\Delta \phi(v)=g(t,v,w)\\ \partial_{t}w-D\Delta w=h(t,v,w)\\ \partial_{n}\phi(v)_{|\partial\Omega}=0,\,\,\,\partial_{n}w_{|\partial\Omega}=0,\,\,\,v_{|t=0}=v_{0},\,\,\,w_{|t=0}=w_{0}.\\ \end{cases}\tag{2} \]
Here \(\phi\) is the function of the form \(D=\mathrm{diag}(d_{3},\dots,d_{n})\) with \(d_{i}>0\), \(g(t,v,w)=f_{1}(t,v^{+},v^{-},w)-f_{2}(t,v^{+},v^{-},w)\) and \(h(t,v,w)=(f_{3}(t,v^{+},v^{-},w),\dots,f_{n}(t,v^{+},v^{-},w)).\) Before looking at the limit \(k\rightarrow +\infty,\) the authors show that (1) has a weak solution on \(Q_{T}=(0,T)\times\Omega\) for every set on initial concentrations such that \(u_{i,0}\in L^{2}(\Omega;\mathbb{R}_{+}).\)
The main result of the article is the proof that, given a sequence \(k\rightarrow\infty,\) every sequence \((u^{k})\) of weak solution to (1) has a subsequence converging in \(L^{2}(Q_{T})\) to \((v^{+},v^{-},w),\) where \((u,w)\) is a weak solution of (2).

MSC:

35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
92E20 Classical flows, reactions, etc. in chemistry
35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
35R35 Free boundary problems for PDEs
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