## Advances in Differential Equations

- Adv. Differential Equations
- Volume 5, Number 10-12 (2000), 1319-1340.

### 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.

#### Article information

**Source**

Adv. Differential Equations, Volume 5, Number 10-12 (2000), 1319-1340.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651225

**Mathematical Reviews number (MathSciNet)**

MR1785677

**Zentralblatt MATH identifier**

0987.35075

**Subjects**

Primary: 35K57: Reaction-diffusion equations

Secondary: 35B10: Periodic solutions 35B45: A priori estimates 35D05 35K20: Initial-boundary value problems for second-order parabolic equations

#### Citation

Büger, Matthias. On the existence of stationary and periodic solutions for a class of autonomous reaction-diffusion systems. Adv. Differential Equations 5 (2000), no. 10-12, 1319--1340. https://projecteuclid.org/euclid.ade/1356651225