Behaviour of interfaces in a diffusion-absorption equation with critical exponents

Abstract

We consider the Cauchy problem for the porous medium equation with strong absorptionwith continuous compactly supported initial data u(x, 0) = u 0 (x) 0 in the critical case m + p = 2 of the range of parameters m > 1, p < 1. We study the regularity of solutions and interfaces and compare the results with the purely diffusive case u t = (u m ) x x . Important differences are found in the interface behaviour and equations, in the occurrence of turning points and inflection points of the interfaces, and in the fact that bounded solutions extinguish in finite time. All these phenomena are examined and described. The critical case studied here allows for a comparatively simpler and richer analysis of the qualitative behaviour of non-linear parabolic equations involving finite propagation (m > 1) and strong absorption ( p < 1).

The problem Of blow-up in nonlinear parabolic equations

“…If p > −m, such a maximal (weak) continuation exists and the Cauchy problem with compactly supported initial data makes sense. The asymptotic behaviour of such extinction vanishing behaviour was established to be approximately self-similar [86] in the critical case p = 2 − m. The interface propagation in this case is studied in [83], where analyticity is proved and a countable spectrum of turning and inflection interface patterns is constructed with the local spatial shape governed by Kummer's polynomials. It turned out that the singular interface propagation is the most interesting in the subcritical range p ∈ (−m, 2 − m), where the interface equation was proved to be of the second order [82], unlike the first-order Darcy's law for the PME.…”

Bifurcations of self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption

“…Such scenarios have been examined in this range of exponents previously in [6,7]. For the special case when m + n = 2 it has been shown, in [9], that solutions can exhibit reversing interfaces but cannot display a "waiting time" where an interface remains static for some finite time. The behaviour of solutions local to the extinction time has also been examined in the limiting case when m + n = 1 in [10,11] and in the case m > 1 and n < 1 in [5].…”

Properties of solutions to porous medium problems with different sources and boundary conditions

“…When Ω is a bounded and smooth domain of R N , N ≥ 1, and under Dirichlet boundary conditions, in [10] is proved that for 1 < p < m the problem admits global solutions for all u 0 such that u m−1 0 ∈ H 1 0 (Ω), while for m < p < m(1 + (2/N )) + (2/N ) specific initial data produce unbounded solutions (see also [31]). It is also worth to mention that [13] and [14] focus on results dealing with regularity and asymptotic behavior of solutions when g(u, |∇u|) = −u p , with p > 0, defined in the whole space R N , with N ≥ 1.…”