Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation

The authors investigate the asymptotic behaviour of blowing-up solutions u = u(x, t) ≥ 0 to the semilinear parabolic equation with source ut = uxx + (1 + u) log2 (1 + u) for x ∈ R, t > 0, with nonnegative and radial symmetric initial data u0(|x|) that are nonincreasing in |x|. Any 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 Hamilton-Jacobi equation Ut = (Ux)2/1 + U + (1 + U) log2 (1 + U), with the same blow-up time T. These explicit profiles are only approximate solutions for the problem.