Evolution equations and subdifferentials in Banach spaces

Abstract

Sufficient conditions for the existence of strong solutions to the Cauchy problem are given for the evolution equation $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t)) \in f(t)$ in V*, where $\partial\upsilon^1$ is the so-called subdifferential operator from a Banach space V into its dual space V* ($i$ = 1,2). Studies for this equation in the Hilbert space framework has been done by several authors. However the study in the V -V* setting is not pursued yet. Our method of proof relies on some approximation arguments in a Hilbert space. To carry out this procedure, it is assumed that there exists a Hilbert space H satisfying $V \subset H \-= H$* $\subset V$* with densely defined continuous injections. As an application of our abstract theory, the initial-boundary value problem is discussed for the nonlinear heat equation: $ut(x, t)-\Delta_p u(x, t)-|u|^(q-2) u(x, t) = f(x, t), x \in \Omega, u|_(\partial\Omega) = 0, t \>= 0$, where ­$\Omega$ is a bounded domain in $\mathbb(R)^N$. In particular, the local existence of solutions is assured under the so-called subcritical condition, i.e., $q < p$*, where $p$* denotes Sobolev’s critical exponent, provided that the initial data $u_0$ belongs to $W_0^(1,p)(\­Omega)$.