On the existence of stationary and periodic solutions for a class of autonomous reaction-diffusion systems

Abstract

We examine the autonomous reaction-diffusion system $u_t=\lambda_1u_{xx}+f(u,v)u-v$, $v_t=\lambda_2v_{xx}+f(u,v)v+u$ for $t>0$ with Dirichlet boundary conditions on $I=(0,1)$, where $\lambda_1,\lambda_2$ are positive constants. If $\lambda_1$ and $\lambda_2$ coincide, then the zero solution is the only stationary solution. If the zero solution is unstable and $f$ satisfies some monotonicity condition, then periodic motion can occur. In this case, we actually construct a periodic solution and show, in addition, that it has certain properties. If $\lambda_1\ne\lambda_2$, nontrivial stationary solutions might exist---we prove the existence for some values of $\lambda_1,\lambda_2$. On the other hand, we show that no such nontrivial stationary solution exists if the ratio $\lambda_1/\lambda_2$ is contained in an interval around $1$ which we will state explicitly.