Influence of variable coefficients on global existence of solutions of semilinear heat equations with nonlinear boundary conditions

We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t\to \mathrm{\infty }\mathrm{.}$