Blow-up in semilinear parabolic equations with weak diffusion

Abstract

Finite time blow-up in the semilinear reactive-diffusive parabolic equation φ₁=µφ_xx+e^φ is examined in the limit of weak diffusion μ<<1, for a Cauchy initial-value problem with φ(x, t=0)=φ_i(x) in which φ_i(x) possesses a smooth global maximum. An asymptotic description of the evolution of φ is obtained from the initial time through blow-up using singular perturbation techniques. Near blow-up, an exact self-similar focusing structure for φ, identical to that previously associated with non-diffusive thermal runaway, is shown to be appropriate. However, in an exponentially small layer close to the blow-up time, the focusing structure must be modified to ensure a uniformly valid solution. This modification uncovers the asymptotically self-similar focusing structure previously recognized for blow-up in equations of the form φ₁=φ_xx+e^φ. In contrast to previous studies, however, the structure arises here as a natural consequence of removing the non-uniformity in the expansions which occurs exponentially 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 exhibit finite time blow-up in order to reveal the associated focusing structure.