Implicit Degenerate Evolution Equations and Applications

Abstract

The initial-value problem is studied for evolution equations in Hilbert space of the general form $\frac{d}{{dt}}\mathcal{A}(u) + \mathcal{B}(u) \mathrel\backepsilon f,$ where $\mathcal{A}$ and $\mathcal{B}$ are maximal monotone operators. Existence of a solution is proved when $\mathcal{A}$ is a subgradient and either $\mathcal{A}$ is strongly monotone or $\mathcal{A}$ is coercive; existence is established also in the case where $\mathcal{A}$ is strongly monotone and $\mathcal{B}$ is subgradient. Uniqueness is proved when one of $\mathcal{A}$ or $\mathcal{A}$ is continuous self -adjoint and the sum is strictly monotone; examples of nonuniqueness are given. Applications are indicated for various classes of degenerate nonlinear partial differential equations or systems of mixed elliptic-parabolic-pseudo-parabolic types and problems with nonlocal nonlinearity.