An inverse mixed problem for an integro-differential equation with a multidimensional Benney–Luke operator and nonlinear maximums

In this paper, we examine the unique generalized solvability and construct solutions to a nonlinear multidimensional inverse mixed problem for a nonlinear fourth-order Benney–Luke integro-differential equation with a degenerate kernel and nonlinear maximums. Sufficient coefficient conditions for the unique solvability of the problem are established. We prove that the solution of the direct mixed problem continuously depends on the initial functions and the overdetermination function. Our research is based on the Fourier method of separation of variables, the method of contraction mappings, the method of successive approximations, and the method of integral and sum inequalities.