Identification of the memory kernel in the strongly damped wave equation by a flux condition
The authors consider an abstract parabolic integrodifferential problem which is motivated by an initial and boundary value problem for a semilinear strongly damped wave equation with memory effects. The problem is formulated as follows. Let \(X\) be a Banach space and let \(A\) be a sectorial operator in \(X\), with domain \(D(A)\). Further, let~\(B\) be a linear operator of domain \(D(A)\) and let \(f\) be a nonlinear perturbation. Then, given \(p\in(1,\infty)\) and \(T>0\), and knowing function \(G\in W^{1,p}(0,T;X)\) and initial value \(U_0\in D(A)\), one looks for \(\tau\in(0,T]\) and \(U\in W^{2,p}(0,\tau;X)\cap W^{1,p}(0,\tau;D(A))\), \(h\in L^p(0,\tau)\), satisfying the system \[ U'(t)=AU(t)+h*BU(t)+f(U(t))+G(t),\quad t\in(0,\tau),\quad U(0)=U_0, \] where \[ (h*f)(t):=\int_0^t h(t-s)f(s)\,ds. \] The supplementary information, necessary to find \(h\) together with~\(U\), is given by the condition \[ \Phi_0(U(t))+\Phi_1(h*U(t))=g(t),\quad t\in(0,\tau), \] where \(g\in W^{1,p}(0,T)\) and~\(\Phi_0\), \(\Phi_1\) are functionals defined in~\(D(A)\). Moreover, \(\Phi_1\neq 0\). Applying common techniques from analytic semigroup theory, the authors formulate and prove first the local in time existence and uniqueness theorem. Then, assuming that the nonlinearity \(f\) has bounded Fréchet-derivative, they proceed with the global in time existence and uniqueness result.
abstract parabolic integrodifferential equation
strongly damped wave equation with memory
scientific article; zbMATH DE number 5530379