On the solution of the heat equation with nonlinear unbounded memory

summary:The paper deals with the question of global solution $u,\tau$ to boundary value problem for the system of semilinear heat equation for $u$ and complementary nonlinear differential equation for $\tau$ ("thermal memory"). Uniqueness of the solution is shown and the method of successive approximations is used for the proof of existence of a global solution provided the condition $(\Cal P)$ holds. The condition $(\Cal P)$ is verified for some particular cases (e. g.: bounded nonlinearity, homogeneous Neumann problem (even for unbounded nonlinearities), apriori estimate of the solution holds)