Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces

The existence of strong solutions of Cauchy problem for the following evolution equation du (t)/dt + partial derivativephi(1) (u(t)) - partial derivativephi(2)(u(t)) epsilon f(t) is considered in a real reflexive Banach space V, where partial derivativephi(1) and partial derivativephi(2) are subdifferential operators from V into its dual V*. The study for this type of problems has been done by several authors in the Hilbert space setting. The scope of our study is extended to the V-V* setting. The main tool employed here is a certain approximation argument in a Hilbert space and for this purpose we need to assume that there exists a Hilbert space H such that V subset of H equivalent to H* subset of V* with densely defined continuous injections. The applicability of our abstract framework will be exemplified in discussing the existence of solutions for the nonlinear heat equation: u(1)(x, t) -Delta(p)u(x, t) - \u\(q-2)u(x, t) = f(x, t), x epsilon Omega, t > 0, u\(deltaOmega) = 0, where Omega is a bounded domain in R-N. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q < p* (Sobolev's critical exponent) for all initial data u(0) epsilon W-0(1,p)(Omega). This fact has been conjectured but left as an open problem through many years. (C) 2004 Elsevier Inc. All rights reserved.