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 $\bar u(x,t)= \alpha(t)u(x,t)+\int^t_0\beta(\tau)u(x,\tau)\,d\tau$ 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.