A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem

The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $$u_t - \Delta u + c(\bar u(x,T))u = f(x,t),$$ where $\bar u(x,t) = \alpha (t)u(x,t) + \int_0^t {\beta (\tau )u(x,\tau )d\tau } $ for given functions $\alpha (t)$ and $\beta (t)$ , is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems.