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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)
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