A Volterra integro-differential equation ``without initial conditions'' (Q2747822)

Let \(V\) and \(H\) be two Hilbert spaces, \((. , .)\) the scalar product of \(H\), \(|\cdot|\) the norm in \(H\) and \(\|\cdot\|\) the norm in \(V\). We assume that \(V\subset H\) where the embedding is dense and continuous. In particular there is \(c>0\) such that \(c|v|\leq\|v\|\), \(v\in V\). NEWLINENEWLINENEWLINELet \(a:V\times V\to\mathbb{R}\), \(k:[0,+\infty) \times V\times V\to\mathbb{R}\) where \(a\) and \(k(s, . , .)\), \(s\geq 0\), are bilinear symmetric continuous forms on \(V\) verifying the following conditions:NEWLINENEWLINENEWLINE\((\text{H}_1)\) \(k(s, . , .)\in W^{1, 1}(0,+ \infty)\), \(s\geq 0\),NEWLINENEWLINENEWLINE\((\text{H}_2)\) \(\exists c\in L^1(0,+ \infty)\), \(\forall (s,u,v) \in[0, +\infty) \times V\times V\), \(|k(s,u,v) |\leq c(s)\|u\|\|v\|\),NEWLINENEWLINENEWLINE\((\text{H}_3)\) the derivative \(\dot k(. ,u,v)\), \((u,v)\in V\times V\), with respect to \(s\), of \(k(. ,u,v)\), \((u,v)\in V\times V\), satisfies \(\dot k(s,u,u)\geq 0\), a.e. \(s\geq 0\), \(u\in V\).NEWLINENEWLINENEWLINE\((\text{H}_4)\) the bilinear continuous symmetric form NEWLINE\[NEWLINEb(u,v)= a(u,v)+ \int^{+\infty}_0 k(s,u, v)dsNEWLINE\]NEWLINE is coercive, that is there exists \(\gamma>0\) such that \(b(u,u)\geq \gamma \|u\|^2\) for \(u\in V\).NEWLINENEWLINENEWLINEThe goal of this paper is to examine the following abstract linear second order integro-differential Volterra problem of variational type and with infinite delay NEWLINE\[NEWLINE\bigl(\ddot u(t),v\bigr)+ a\bigl(u(t), v\bigr)+\int^t_{-\infty} k\bigl(t-\tau, u(\tau),v\bigr) d\tau= \bigl(F(t), v\bigr) \tag{1}NEWLINE\]NEWLINE a.e. \(t\in T\), \(v\in V\), where \(T\in\mathbb{R}\), \(u\) is unknown for any time \(t\leq T\) and \(F\in L^1(-\infty,T;H)\). A solution \(u\) of (1) is said to be a classical solution if \(u\) verifies (1) and \(u\in W^{2,1}(-\infty,T;H) \cap W^{1,1} (-\infty,T;V)\). The authors prove that if \((\text{H}_1)\)--\((\text{H}_4)\) hold and \(u\) is a classical solution of (1) then \(u\) is unique. If \(F\in W^{1,1} (-\infty, T;H)\) and the functions NEWLINE\[NEWLINEt\mapsto \int^t_{-\infty} \bigl|F(\tau)\bigr|d\tau,\;t\mapsto \int^t_{-\infty} \bigl|\dot F (\tau) \bigr|d\tauNEWLINE\]NEWLINE belong to \(L^1(-\infty,T)\) then there exists a unique classical solution of (1).NEWLINENEWLINENEWLINEFinally, the authors give an existence and uniqueness result for the solution of the linear viscoelasticity equation as an application of the theorem of existence of a classical solution of problem (1).