On the determination of the coefficient of a nonlinear term in a parabolic equation

A second-order parabolic equation with an inhomogeneity of the form q(x0; t)u1+α(x; t), where x = (x0; xn) = (x1; · · · ; xn−1; xn) and α > 0, is considered in both direct and inverse problems. The inverse problem focuses on determining the coefficient q(x0; t) under heterogeneity. The trace of the direct problem solution is provided at the hyperplane xn = 0 as supplementary information. Conditions for the unique solvability of the direct problem are established. For the inverse problem, local existence and uniqueness theorems are proven.