On a Class of First Order Evolution Inequalities Arising in Heat Conduction with Memory

Abstract

This paper deals with the evolution problem \[u' + a_0 u + \int_0^t {a(t - s)u(s)ds} + \int_0^t {b(t - s)Bu} (s)ds + \partial \varphi (u) \ni f,\quad u(0) = u_0 ,\] on an arbitrary interval $[0,T]$, where $a_0 \in \mathbb{R}$ and $a,b:[0,T] \to \mathbb{R}$ are given functions; B is a linear bounded mapping from a Hilbert space V into its dual, while $\partial \varphi $ denotes the subdifferential mapping of a proper, convex and lower semicontinuous functional $\varphi :V \to ( - \infty , + \infty ]$, We prove an existence and uniqueness theorem for a solution u to the above problem. The existence is established by combining the Galerkin method with a regularization of the functional $\varphi $. An application of the abstract result to a Volterra integrodifferential equation arising from the theory of heat conduction in materials with memory is also given.