Abstract
Inverse problems of determining the right-hand side of a differential equation arise in the mathematical modeling of many physical phenomena, when an external source or some of its parameters acting to the motion of the process are unknown or unacceptable for measurement, for example, the source is in a high-temperature environment or underground, etc. This paper deals to study the solvability of an inverse problem for a nonlinear pseudo-parabolic equation (sometimes they called Sobolev-type equations) with p-Laplacian and damping term with variable exponent. The inverse problem consists of determining a coefficient of the right-hand side depending only on time. Additional information for this investigated inverse problem is given as an integral overdetermination condition. Under the suitable conditions on the exponents and on the data the global and local in time existence of a weak solutions to the inverse problem are established. The existence of a weak solution proved by the Faedo-Galerkin method. The global and local in time a priori estimates for the Galerkin approximations are obtained. On the basis of a priori estimates and by using compactness theorems and the monotonicity method, the convergence of the Galerkin approximations to the solution of the initial inverse problem is proved. Key words: inverse problem, pseudoparabolic equations, existence, weak solution.