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Blow-up behavior in nonlocal vs local heat equations

Philippe Souplet — 2000

Banach Center Publications

We present some recent results on the blow-up behavior of solutions of heat equations with nonlocal nonlinearities. These results concern blow-up sets, rates and profiles. We then compare them with the corresponding results in the local case, and we show that the two types of problems exhibit "dual" blow-up behaviors.

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet;  Slim Tayachi — 2001

Colloquium Mathematicae

Consider the nonlinear heat equation (E): u t - Δ u = | u | p - 1 u + b | ∇ u | q . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C ₁ ( T - t ) - 1 / ( p - 1 ) ≤ | | u ( t ) | | ∞ ≤ C ₂ ( T - t ) - 1 / ( p - 1 ) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality u t - u x x ≥ u p . More general inequalities of the form u t - u x x ≥ f ( u ) with, for instance, f ( u ) = ( 1 + u ) l o g p ( 1 + u ) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary...

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta;  Anibal Rodriguez-Bernal;  Philippe Souplet — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u : either the space derivative u x blows up in finite time (with u itself remaining bounded), or u is global and converges in C 1 norm to the unique steady state. The main difficulty is to prove C 1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

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