Continuation of blowup solutions of nonlinear heat equations in several space dimensions

✍ Scribed by Victor A. Galaktionov; Juan L. Vazquez

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 394 KB
• Volume: 50
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199701)50:1<1::aid-cpa1>3.0.co;2-h

✦ Synopsis

The possible continuation of solutions of the nonlinear heat equation in

after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N -2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations.

We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation

We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction).