Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:231

Article

Akagi, Goro

关键词: doubly nonlinear evolution equation;    subdifferential;    reflexive Banach space;    elliptic-parabolic problem;   

DOI  :  10.1016/j.jde.2006.04.006

来源: Elsevier

【 摘 要 】

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt + delta(V)p(t) (u(t)) there exists f (t), v(t) is an element of delta H psi(u(t)), 0 < t < T, where delta H psi (respectively, delta(V)p(t)) denotes the subdifferential operator of a proper lower semicontinuous functional psi (respectively, p(t) explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (-infinity, +infinity)] and f is given. To do so, we suppose that V hooked right arrow H equivalent to H* hooked right arrow V* compactly and densely, and we also assume smoothness in t, boundedness and coercivity of p(t) in an appropriate sense, but use neither strong monotonicity nor boundedness of delta(H)psi The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem. (c) 2006 Elsevier Inc. All rights reserved.

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