BLOW-UP IN SEMILINEAR PARABOLIC EQUATIONS WITH WEAK DIFFUSION

Finite time blow-up in the semilinear reactive-diffusive parabolic equ ation phi(t) = mu phi(xx) + e(phi) is examined in the limit of weak di ffusion mu << 1, for a Cauchy initial-value problem with phi(x, t = 0) = phi(i)(x) in which phi(i)(x) possesses a smooth global maximum. An asymptotic description of the evolution of phi is obtained from the in itial time through blow-up using singular perturbation techniques. Nea r blow-up,an exact self-similar focusing structure for phi, identical to that previously associated with non-diffusive thermal runaway, is s hown to be appropriate. However, in an exponentially small layer close to the blow-up time, the focusing structure must be modified to ensur e a uniformly valid solution. This modification uncovers the asymptoti cally self-similar focusing structure previously recognized for blow-u p in equations of the form phi(t) = phi(xx) + e(phi). In contrast to p revious studies, however, the structure arises here as a natural conse quence of removing the non-uniformity in the expansions which occurs e xponentially close to blow-up when the effects of diffusion have to be reinstated. Identical weak-diffusion limit asymptotics can be applied to a variety of semilinear or quasilinear parabolic equations that ex hibit finite time blow-up in order to reveal the associated focusing s tructure.