On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping

This paper deals essentially with a nonlinear degenerate evolution equation of the formKu″-Δu+∑j=1nbj∂u′/∂xj+uσu=0supplemented with nonlinear boundary conditions of Neumann type given by∂u/∂ν+h·, u′=0. Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.

On a nonlinear degenerate evolution equation with strong damping

In this paper we consider the nonlinear degenerate evolution equation with strong damping,(*) {K(x,t)utt−Δu−Δut+F(u)=0 in Q=Ω×]0,T[u(x,0)=u0, (ku′)(x,0)=0 in Ωu(x,t)=0 on ∑=Γ×]0,T[whereKis a function withK(x,t)≥0,K(x,0)=0andFis a continuous real function satisfying(**) sF(s)≥0, for all s∈R, Ωis a bounded domain ofRn, with smooth boundaryΓ. We prove the existence of a global weak solution for (*).