GLOBAL EXISTENCE AND BLOW-UP PROBLEMS FOR QUASI-LINEAR PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY-CONDITIONS

This paper deals with the solutions of nonlinear parabolic equations u(t) = del(a(u)delu) with nonlinear boundary conditions partial derivative u/partial derivative n = b(u) on partial derivative OMEGA x (0, T), where a(u) and b(u) are positive and nondecreasing C1 functions for u > 0. It is shown that if integral + infinity ds/b(s) < + infinity or integral + infinity ds/b(s) = + infinity and integral + infinity ds/(a(s)b(s)b'(s) + a(s)b(s) + a'(s)b2(s)) < + infinity, then the solution blows up in finite time, and the solution exists globally if integral + infinity ds/b(s) = + infinity ds/(a(s)b(s)b'(s) + a(s)b(s) + a'(s)b2(s)) = + infinity.