REGIONAL BLOW-UP IN A SEMILINEAR HEAT-EQUATION WITH CONVERGENCE TO A HAMILTON-JACOBI EQUATION

The authors investigate the asymptotic behaviour of blowing-up solutio ns u = u(x, t) greater-than-or-equal-to 0 to the semilinear parabolic equation with source u(t) = u(xx) + (1 + u)log2(1 + u) or x is-an-elem ent-of R, t > 0, with nonnegative and radial symmetric initial data u0 (Absolute value of x) that are nonincreasing in Absolute value of x. A ny nontrivial solution u to this problem blows up in a finite time T > 0. It is remarkable that the blow-up behaviour of u as t approaches T can be described by the exact blow-up solutions of the quasilinear Ha milton-Jacobi equation U(t) = (U(x))2/1 + U + (1 + U) log2(1 + U), wit h the same blow-up time T. These explicit profiles are only approximat e solutions for the problem. The authors prove that the blow-up set B of the solution satisfies meas (B) greater-than-or-equal-to 2pi, and u nder some additional hypothesis on the initial function it is shown th at B is just the interval [-pi, pi] and the rescaled blow-up shape con sists of one hump with formula cos2(x/2). The proofs rely on the knowl edge of a family of explicit solutions, the method of intersection com parison, some dynamical systems ideas, and a stability analysis for so lutions of the Hamilton-Jacobi equation.