Thermal avalanche for blowup solutions of semilinear heat equations

Abstract

We consider the semilinear heat equation u t = u + u p both in R N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p s where p s is the Sobolev exponent. This problem has solutions with finite-time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T , while it becomes unbounded as t T . In order to understand the situation for t > T , we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well-known, the limit solution undergoes complete blowup: after it blows up at t = T , the continuation is identically infinite for all t > T .We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T − and t = T + . This is called the "avalanche." We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T , the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern.

Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up

“…If such a continuation exists, blow-up is said to be incomplete; otherwise, it is called complete. Complete blow-up was first studied for problems where the nonlinearity occurs in the equation as a reaction term, u t = u xx + f (u), see [1], [9], [10], [11], [12], [13], [18]. For the scalar version of the present problem complete blow-up is proved in [7], see also [17].…”

“…Since blow-up takes place only at one point and there is complete blow-up, at t = T an instantaneous propagation of the blow-up singularity to the whole spatial domain takes place, what is called an avalanche, see [17], [18].…”

Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent

“…The construction of such travelling waves is inspired by the technique used in the so-called KPP problems [14], which has developed a wide literature; see, e.g., [2], [22] for applications to porous media, and [18] for blow-up problems. We thus begin with a phase-plane analysis, proving the existence of the desired travelling waves.…”

A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems

“…More precisely, if p > p S , n 10, and Ω is a ball, then there exist smooth radial data u 0 such that T (u 0 ) < ∞ butū(x, t) is a classical solution of (1.1) for all t ∈ (0, ∞) \ K, where K is a finite set (see [14,15]). Complete blow-up for a(x) ≡ 1 was studied by many other authors, see [20,22,[24][25][26]29,36], for example.…”