Global existence of solutions of semilinear heat equation with nonlinear memory condition

We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as $t\to \mathrm{\infty }\mathrm{.}$