The study of nonlinear equations of mathematical physics, including inverse problems, is currently relevant. This work is devoted to the fundamental problem of investigating the qualitative properties of the inverse problem for pseudoparabolic equations (also called Sobolev-type equations) with a sufficiently smooth boundary. In the article, the Galerkin method proves the existence of a weak solution to the inverse problem in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. Using Galerkin approximations, you can get a top-down estimate of the existence of the solution. A local and global theorem on the existence of a solution are obtained. We consider the problems of asymptotic behavior of solutions at, as well as blow-up in finite time. Sufficient conditions for $$t\rightarrow \infty$$  the "blow-up" of the solution in a finite time are obtained, and a lower estimate of the blow-up of the solution is obtained.