NECESSARY AND SUFFICIENT CONDITIONS FOR COMPLETE BLOW-UP AND EXTINCTION FOR ONE-DIMENSIONAL QUASI-LINEAR HEAT-EQUATIONS

We characterize the occurrence of complete or incomplete blow-up (and extinction) for a general quasilinear heat equation of the form (HE) u(t) = (phi(u))(xx) +/- f(u) in R x (0, T) in terms of the constitutive functions phi and f. We assume that phi'(u) > 0 for u > 0 and that f(u) greater than or equal to 0. For the positive sign + before f(u) in (HE), with f(u) superlinear as u --> infinity, blow-up occurs in finite time: sup(x)u(x, t) --> infinity as t --> T < infinity. For the negative sign, we consider the case of singular absorption: flu) --> infinity as u --> 0. Then initially positive solutions vanish at some point in finite time (extinction), and a singularity in the equation occurs there. An important aspect of blow-up or extinction problems is the possibility of having a nontrivial extension of the solution for t > T, i.e., after the singularity occurs. If such continuation exists, we say that the blow-tip (extinction) is incomplete; otherwise it is called complete. Our characterization is based on the qualitative behaviour of the family of travelling-wave solutions and a proper use of the Intersection-Comparison argument. The analysis applies to other nonlinear models, like the equations with gradient-dependent diffusivity.