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Supersolutions and stabilization of the solution of a nonlinear parabolic system. (English) Zbl 0693.35018

Summary: Let us consider a nonlinear parabolic system of the following type: \[ (S)\quad \partial u/\partial t-div(| \nabla u|^{p-2}\nabla u)=\partial H/\partial u(x,u,v),\quad \partial v/\partial t-div(| \nabla v|^{p-2}\nabla v)=\partial H/\partial v(x,u,v) \] with Dirichlet boundary conditions and initial data. We construct sub- supersolutions of (S), and by use of them, we prove that, for \(t_ n\to +\infty\), the solution of (S) converges to some solution of the elliptic system associated with (S).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B35 Stability in context of PDEs
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