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+∂ 1(u(t))-∂ 2(u(t)) f(t) is considered in a real reflexive Banach space V, where ∂ 1 and ∂ 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 H≡H* 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: where Ω is a bounded domain in RN. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q