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.
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