Regularization of initial inverse problem for strongly damped wave equation

In this paper, we consider the problem of finding the function u( t), t. 0, T , from the final data u( T) =. and ut ( T) =. utt = aAut + Au + F t, u , where a > 0, A is a linear, unbounded, self- adjoint and positive definite operator. This problem is known as the inverse initial problem for non- linear strongly damped wave and is ill- posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose new quasi- boundary value method to solve the non- linear problem, i. e. for > 0 replacing (.,., F),. H x H x R by I , a, . , I , a, . , I , a, F , with the operator I , a, will be defined later and. ,. satisfies ( 1.8). Moreover, we show that the regularized solutions converge to the exact solution strongly with respect to t. [ 0, T] under a priori assumption on the exact solution in Gevrey space.