Differential and Integral Equations

Steady state solutions for balanced reaction diffusion systems on heterogeneous domains

W. E. Fitzgibbon, S. L. Hollis, and J. J. Morgan

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We consider a class of semilinear diffractive-diffusion systems of the form $$ \begin{cases} -d_{A_{i}}\Delta u_i = f_{A_{i}}(u) & x\in A \\ -d_{B_{i}}\Delta \tilde u_i = f_{B_{i}}(\tilde u) &x\in B \\ u_i=\tilde u_i & \hbox{on } \partial A \\ d_{A_{i}} { \partial u_i\over \partial\eta_A} = d_{B_{i}} { \partial \tilde u_i \over \partial \eta _A}& \hbox{on } \partial A \\ \tilde u_i = g_i& \hbox{on } \partial B\backslash \partial A \end{cases} \tag{P} $$ where $A$ and $B$ are smooth bounded domains in $\Re^n$ such that there exists a smooth bounded domain $\Omega\subseteq \Re^n$ so that $A$ is a strict subdomain of $\Omega$ and $\bar A\cup B =\Omega$. We assume that $d_{A_{i}}$, $d_{B_{i}}>0$, each $g_i$ is nonnegative and smooth, and $f_A = (f_{A_{i}})$ and $f_B=(f_{B_{i}})$ are locally Lipschitz vector fields which are quasi-positive, nearly balanced, and polynomial bounded. We prove that these conditions guarantee the existence of a nonnegative solution of (P) for the case of $n=2$. In addition, for the case of $n=3$, we show that nonnegative solutions of (P) exist provided that $f_A$, $f_B$ satisfy a quadratic intermediate sum property. In particular, our results imply that, for space dimensions $n=2,3$, if (P) arises from standard balanced quadratic mass action kinetics, then nonnegative solutions of (P) are guaranteed. We apply our results to two multicomponent chemical models.

Article information

Differential Integral Equations, Volume 12, Number 5 (1999), 637-660.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B45: A priori estimates 35J55 35R05: Partial differential equations with discontinuous coefficients or data


Fitzgibbon, W. E.; Hollis, S. L.; Morgan, J. J. Steady state solutions for balanced reaction diffusion systems on heterogeneous domains. Differential Integral Equations 12 (1999), no. 5, 637--660. https://projecteuclid.org/euclid.die/1367255389

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