On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping

Abstract

This paper deals essentially with a nonlinear degenerate evolution equation of the form $K{u}^{″}-\mathrm{\Delta }u+{\sum }_{j=1}^n{b}_j\left(\partial u^{\prime }/\partial x_j\right)+{\left|u\right|}^{\sigma }u=0$ supplemented with nonlinear boundary conditions of Neumann type given by $\partial u/\partial \nu +h\left(·, u^{\prime }\right)=0$. 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.