Interior estimates for a class of reaction-diffusion systems from \(L^ 1\) a priori estimates
type: Item
instance of: scholarly article
title: Interior estimates for a class of reaction-diffusion systems from \(L^ 1\) a priori estimates
zbMATH Open document ID: 0780.35052
DOI: 10.1016/0022-0396(92)90093-3
author: Selwyn L. Hollis
author: J. J. Morgan
published in: Journal of Differential Equations
publication date: 1993-01-16T00:00:00Z
review text: The authors consider first a system of parabolic differential inequalities \[ u_{kt}\leq {\mathcal L}_ k u_ k+f_ k\text{ in } (0,T)\times \Omega, \qquad u_ k\geq 0 \text{ on } [0,T)\times \overline{\Omega} \] (\({\mathcal L}_ k\) a second order elliptic operator) and show that for bounded subdomains \(\Omega'\subset \Omega''\subset\Omega\) satisfying certain conditions and \(T<\infty\) an \(L^ 1\)-inequality \(\| u\|_{1,(0,T)\times \Omega''}<\infty\) implies an \(L^ p\)-inequality \(\| u\|_{p,(0,T)\times \Omega'}<\infty\) for all \(1<p<\infty\), and furthermore, additional inequalities as \(t\to \infty\) in the case \(T=\infty\). These results are then applied to the parabolic system \[ \begin{aligned} v_{1_ t} &= d_ 1\Delta v_ 1+a_ 1 v_ 1+ b_ 1 v_ 2- c(x)f(v_ 1,v_ 2),\\ v_{2_ t} &= d_ 2 \Delta v_ 2+ a_ 2 v_ 1+ b_ 2 v_ 2+ c(x) f(v_ 1,v_ 2),\\ v_ 1&=\beta_ 1,\quad v_ 2=\beta_ 2,\\ v_ 1 &= v_{1_ 0}, \quad v_ 2= v_{2_ 0}, \end{aligned} \qquad \begin{aligned} t&> 0,\;x\in\Omega,\\ t&>0,\;x\in\Omega,\\ t&>0,\;x\in\partial\Omega,\\ t&=0,\;x\in\Omega,\end{aligned} \] where \(f\) is nonnegative and locally Lipschitz with \(f(0,z)=f(z,0)=0\) for \(z\geq 0\), and \(c(x)\) may change sign (for example, the special case \(a_ i= b_ i=0\), \(c(x)=1\), \(f(v_ 1,v_ 2)=v_ 1 v^ \gamma_ 2\), which has some literature, is covered). If \(T_{\max}\), the maximal interval of existence of the solution \((v_ 1,v_ 2)\), is finite, then \(v_ 1+v_ 2\) is bounded on \((0,T_{\max})\times \Omega'\), while for \(T_{\max}=\infty\) the maximum norm \(\| v_ 1(t,\cdot)+ v_ 2(t,\cdot)\|_{\infty,\Omega'} \to 0\) as \(t\to\infty\). There are other results, in particular an application to a one-dimensional flame propagation model, which leads to global existence.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35K50
zbMATH DE Number: 89290
zbMATH Keywords: interior estimates
zbMATH Keywords: parabolic differential inequalities
zbMATH Keywords: one-dimensional flame propagation
zbMATH Keywords: global existence
MaRDI profile type: Publication
full work available at URL: 0022-0396(92)90093-3
OpenAlex ID: W2024654730
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description: scientific article; zbMATH DE number 89290