## Differential and Integral Equations

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

### Steady state solutions for balanced reaction diffusion systems on heterogeneous domains

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 29 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367255389

**Mathematical Reviews number (MathSciNet)**

MR1697249

**Zentralblatt MATH identifier**

1064.35514

**Subjects**

Primary: 35K57: Reaction-diffusion equations

Secondary: 35B45: A priori estimates 35J55 35R05: Partial differential equations with discontinuous coefficients or data

#### Citation

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