On the existence of stationary and periodic solutions for a class of autonomous reaction-diffusion systems (Q5954421)

The author investigates the following two component reaction-diffusion system: NEWLINE\[NEWLINE u_t=\lambda_1u_{xx}+f(u,v)u-v,\;\;v_t=\lambda_2v_{xx}+f(u,v)v+u, \tag{1} NEWLINE\]NEWLINE where \(x\in I:=(0,1)\), equipped by Dirichlet boundary conditions. It is proved that, if \(|\lambda_1-\lambda_2|\leq\delta\) is small enough (\(\delta\) depends on \(f\)) and the nonlinearity \(f\) satisfies some natural assumptions, then the zero solution is a unique stationary solution of (1). Moreover, if this equilibrium is unstable, \(\lambda_1=\lambda_2\) and \(f\) satisfies some additional monotonicity assumptions, then (1) possesses nontrivial time periodic solutions. It is also shown that the nontrivial stationary solutions may exist if the diffusion rates \(\lambda_1\) and \(\lambda_2\) are essentially different.