The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption.

Abstract

We consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term: partial derivative(t)u - (vertical bar u(x)vertical bar(P-2)u(x))(x) + u(-beta) X{u > 0} = f (u,x,t), in I x (0, T), with the homogeneous zero boundary condition on I = (x(1), x(2)), an open bounded interval in R. Through this paper, we assume that p > 2 and beta is an element of (0, 1). To show the local existence result, we prove first a sharp pointwise estimate for vertical bar u(x vertical bar) I. One of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t is an element of(0, infinity), the so called global solutions, or by the contrary a finite time blow-up tau(0) > 0 arises such that lim (t ->tau 0) vertical bar vertical bar u(t)vertical bar vertical bar(L infinity(I)) = +infinity. Moreover, we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough. Finally, we show that the condition f (0, x, t) = 0, for all(x, t) is an element of I x (0, infinity) is a necessary and sufficient condition for the existence of solution of equations of this type.