Victor A. Galaktionov

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 R+ ut = u m + u p with m> 0 ;p > 1 ; 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 regionfx : u(x;t )= 1g 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 ut = u m u p ;m > 0 :

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

AbstractWe consider nonnegative solutions of initial-boundary value problems for parabolic equationsut=uxx, ut=(um)xxand

Blow-up for quasilinear heat equations with critical Fujita's exponents

u_t = (\left| {u_x } \right|^{m - 1} u_x )_x

Third-Order Nonlinear Dispersive Equations: Shocks, Rarefaction, and Blowup Waves

(m>1) forx>0,t>0 with nonlinear boundary conditions−ux=up,−(um)x=upand

Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations

- \left| {u_x } \right|^{m - 1} u_x = u^p

Extinction for a quasilinear heat equation with absorption. I: Technique of intersection comparison

forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp0,pc(withp0<pc)such that forp∃(0,p0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>pcsmall data solutions exist globally in time while large data solutions are nonglobal. We havepc=2,pc=m+1 andpc=2m for each problem, whilep0=1,p0=1/2(m+1) andp0=2m/(m+1) respectively.