BLOW-UP FOR QUASI-LINEAR HEAT-EQUATIONS DESCRIBED BY MEANS OF NONLINEAR HAMILTON-JACOBI EQUATIONS

It is well-known that the nonnegative solutions of the semilinear heat equation u(t) = Delta u + (1 + u)(log (1 + u))(beta), with beta > 1, blow up in a Finite time T (depending on the initial data, assumed to be large enough). This equation is interesting because it exhibits in different beta-ranges the three most typical blow-up behaviours for so lutions of nonlinear parabolic equations. Indeed, we consider radialy symmetric solutions and show that for beta > 2 single-point blow-up oc curs, for beta < 2 blow-up is global, and for beta = 2 we have regiona l blow-up. Moreover, the analysis shows that the precise asymptotic be haviour is described by a nonconstant self-similar blow-up solution of the first-order Hamilton-Jacobi equation U-t = \del U\(2)/1 + U + (1 + U)(log (1 + U))(beta). This means that both equations are asymptotic ally equivalent near blow-up. This type of asymptotic ''degeneracy'' o f a parabolic equation into a first-order equation is actually proved for a more general class of quasilinear heat equations. (C) 1996 Acade mic Press, Inc.