Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces

摘要

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: ¶_V yt (u¢(t)) + ¶_V j(u(t)) + B(t, u(t)) ' f(t){partial_V psi^t (u{^prime}(t)) + partial_V varphi(u(t)) + B(t, u(t)) ni f(t)} in V*, 0 < t < T, u(0) = u ₀, where ¶_V yt, ¶_V j: V ® 2V*{partial_V psi^t, partial_V varphi : V to 2^{V^*}} denote the subdifferential operators of proper, lower semicontinuous and convex functions yt, j: V ® (-¥, +¥]{psi^t, varphi : V to (-infty, +infty]}, respectively, for each t Î [0,T]{t in [0,T]}, and f : (0, T) → V* and u₀ Î V{u_0 in V} are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T) × V into V*. We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., B º 0{B equiv 0}) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.