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 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 This fact has been conjectured but left as an open problem through many years.