CONTINUATION OF BLOWUP SOLUTIONS OF NONLINEAR HEAT-EQUATIONS IN SEVERAL SPACE DIMENSIONS

The possible continuation of solutions of the nonlinear heat equation in R(N) x R(+) u(t) = Delta u(m) + u(p) with m > 0, p > 1, after the b lowup time is studied and the different continuation modes are discuss ed in terms of the exponents m and p. Thus, for m + p less than or equ al to 2 we find a phenomenon of nontrivial continuation where the regi on {x : u(x, t) = infinity} is bounded and propagates with finite spee d. This we call incomplete blowup. For N greater than or equal to 3 an d 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 blowu p 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 equati ons. We apply the same technique of analysis to the problem of continu ation after the onset of extinction for example, for the equation u(t) = Delta u(m) - u(p), m > 0. We find that no continuation exists if p + m less than or equal to 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). (C) 1997 John Wiley & Sons, Inc.