Qualitative results for parabolic equations involving the p-Laplacian under dynamical boundary conditions
Résumé
We discuss comparison principles, the asymptotic behaviour, and the occurrence of blow up phenomena for nonlinear parabolic problems involving the p-Laplacian operator of the form ∂ t u = ∆ p u + f (t, x, u) in Ω for t > 0, σ∂ t u + |∇u| p−2 ∂ ν u = 0 on ∂Ω for t > 0, u(0, •) = u 0 in Ω, where Ω is a bounded domain of R N with Lipschitz boundary, and where ∆ p u := div |∇u| p−2 ∇u is the p-Laplacian operator for p > 1. As for the dynamical time lateral boundary condition σ∂ t u + |∇u| p−2 ∂ ν u = 0 the coefficient σ is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearity f (•, •, u) = λ|u| q−2 u will be investigated by means of the evolution of associated norms.
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