We study the existence and the uniqueness of the weak solution of an inverse problem for a semilinear higher order ultraparabolic equation with Lipschitz nonlinearity. The main aim is to determine the weak solution of the equation and some functions that depend on the time variable, appearing on the right-hand side of the equation. The overdetermination conditions introduced are of integral type. In order to prove the solvability of this problem in Sobolev spaces we use the Galerkin method and the method of successive approximations.
We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order \[ D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)), \] where $z_{(x,y)} \: [-\tau ,0] \times [0,h] \rightarrow \mathbb{R}$ is a function defined by $z_{(x,y)}(t,s) = z(x+t, y+s)$, $(t,s) \in [-\tau ,0] \times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.