## Asymptotic stability in the presence of convection.(English)Zbl 0712.35012

The nonlinear parabolic system $u_ t+au_ x+g(x,u)=Du_{xx},\quad x\in (0,1),\quad t>0$ with initial-boundary conditions $u(x,0)=\Phi (x),\quad x\in (0,1);\quad u(0,t)-Pu(0,t)=\alpha,\quad u(1,t)=0,\quad t>0$ is considered. Here $$a=const$$, P and D are diagonal matrices with constant positive entries. It is assumed that the steady problem $aU_ x+g(x,U)=DU_{xx},\quad x\in (0,1);\quad U(0)-PU(0)=\alpha,\quad U(1)=0,$ has a smooth solution and that the initial perturbation U(x)-$$\Phi$$ (x) is sufficiently small. A sufficient condition on the nonlinearity g(x,u) for the asymptotic stability of the steady solution $\| u(x,t)-U(x)\| \leq \| \Phi (x)-U(x)\| \exp (-\sigma t)$ is established.
Reviewer: L.Kaljakin

### MSC:

 35B35 Stability in context of PDEs 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K50 Systems of parabolic equations, boundary value problems (MSC2000)