AN INVERSE PROBLEM FOR PSEUDOPARABOLIC EQUATION WITH MEMORY TERM AND DAMPING

Abstract

In this paper, we study the inverse problem of determining, along with solution u(x,t) of a pseudo-parabolic equation with memory (convolution term) and a damping term, also an unknown coefficient f(t) determining the external effect (the free term). In the investigating inverse problem, the overdetermination condition is given in integral form, which represents the average value of a solution tested with some given function over all the domain. By reducing the considering inverse problem to an equivalent nonlocal direct problem. The applicability of the Faedo-Galerkin method to the inverse problem is analyzed. The damping term  a|u|^q-2 u  affects as nonlinear source in the case a>0, and an absorption, if a<0. In all these cases, we establish the conditions on the range of exponent q, the dimension d, and the data of the problem for the global and local in time existence and uniqueness of a weak solution of the studing problem.