An inverse source problem for a nonlinear pseudoparabolic equation with p-Laplacian diffusion and damping term

Abstract

In this work, we study the unique solvability of the inverse problem that consists of determining a weak solution u(x, t) to the nonlinear pseudoparabolic equation with p−Laplacian and damping term

u_t − Δu_t − div (|∇_u|^p−2 ∇u) = γ |u|^σ−2 u + f (t)g(x, t)

and the coefficient f (t) of right-hand side, which depends on t. Due to the presence of an unknown coefficient in such problems supposes that there is an additional condition along with the initial and boundary conditions. In the present work, an additional condition is represented by the integral overdetermination condition, which represents the average value of a solution tested with some given function over all the domain.

The investigating inverse problem considered in two cases: the coefficient of the damping term γ |u|^σ−2 u is a positive (nonlinear source term) or negative (an absorption). In both cases, we establish the global and local in time existence and uniqueness of a weak solution to the inverse problem under suitable conditions on the exponents p, σ, the dimension d, and the data of the problem.