An existence result for semilinear equations in viscoelasticity: the case of regular kernels

This paper is concerned with the abstract semilinear integro-differential equation \begin{equation}\label{1} \ddot u(t)+Au(t)+ \int_0^tB(t-s)u(s)ds=F(u(t))+f(t)\qquad t\ge 0\,, \end{equation} that may be regarded as a model problem for some elastic systems with memory, like the ones considered in \cite{D1,D2,RHN}. Here, $A$ is a positive operator on a Hilbert space $X$ with domain $D(A)$, $B(t)$ is a self-adjoint linear operator on $X$ with domain $D(B(t))\supset D(A)$, commuting with $A$, and $F$ is a locally Lipschitz map defined on the domain of $\sqrt{A}$. The linear version of (\ref{1}) , that is for $F\equiv 0$, can be reduced to an integral form and then solved applying known existence results, see \cite{Pruss}. Such a procedure, however, requires smooth initial conditions and provides no maximal regularity results that are needed to study the nonlinear problem. The purpose of the present paper is twofold. First, we shall complete the above mentioned procedure with the derivation of suitable maximal regularity estimates for the resolvent of the linear problem. Then, we shall apply the properties of such a resolvent to obtain a local existence result for (\ref{1}) by standard fixed-point arguments. An important aspect of our analysis is that we require no sign condition on $B(t)$. Instead, we assume that $B(\cdot)y$ is absolutely continuous in $t$ for any $y\in D(A)$. We plan to study the case of singular kernels in a forthcoming paper. The outline of this paper is the following. In section 2 we recall some preliminaries and prove our maximal regularity result for the resolvent of the linear problem. Section 3 is devoted to the solution of the Cauchy problem for linear equations, while, in section 4, we obtain local existence results for the nonlinear problem. Finally, in section 5, we describe a typical system in viscoelasticity that can be studied by our abstract approach.