Global existence and blow-up analysis for parabolic equations with nonlocal source and nonlinear boundary conditions

We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions: (g(u)), = E,i 1 (alj (x)ux)xj + yi um (fp ul dx)P Ku' in D x (0,r), EiNj i di (x)uxivj = h(u) on apx ail, u(x,0)= uo(x)> 0 in T, where p and yi are some nonnegative constants, m, I, y2, and r are some positive constants, D c I@N (N > 2) is a bounded convex region with smooth boundary a D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.