Nonlinear Degenerate Evolution Equations and Partial Differential Equations of Mixed Type

Abstract

The Cauchy problem for the evolution equation $Mu'(t) + N(t,u(t)) = 0$ is studied, where M and $N(t, \cdot )$ are, respectively, possibly degenerate and nonlinear monotone operators from a vector space to its dual. Sufficient conditions for existence and for uniqueness of solutions are obtained by reducing the problem to an equivalent one in which M is the identity but each $N(t, \cdot )$ is multivalued and accretive in a Hilbert space. Applications include weak global solutions of boundary value problems with quasilinear partial differential equations of mixed Sobolev-parabolic-elliptic type, boundary conditions with mixed space-time derivatives, and those of the fourth or fifth type. Similar existence and uniqueness results are given for the semilinear and degenerate wave equation $Bu''(t) + F(t,u'(t)) + Au(t) = 0$, where each nonlinear $F(t, \cdot )$ is monotone and the nonnegative B and positive A are self-adjoint operators from a reflexive Banach space to its dual.