Interior estimates for a class of reaction-diffusion systems from \(L^ 1\) a priori estimates
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.
scientific article; zbMATH DE number 89290
scientific article