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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.

Blow-up for a localized singular parabolic equation with weighted nonlocal nonlinear boundary conditions

Youpeng Chen, Baozhu Zheng (2015)

Annales Polonici Mathematici

This paper deals with the blow-up properties of positive solutions to a localized singular parabolic equation with weighted nonlocal nonlinear boundary conditions. Under certain conditions, criteria of global existence and finite time blow-up are established. Furthermore, when q=1, the global blow-up behavior and the uniform blow-up profile of the blow-up solution are described; we find that the blow-up set is the whole domain [0,a], including the boundary, in contrast to the case of parabolic equations...

Blow-up of a nonlocal p-Laplacian evolution equation with critical initial energy

Yang Liu, Pengju Lv, Chaojiu Da (2016)

Annales Polonici Mathematici

This paper is concerned with the initial boundary value problem for a nonlocal p-Laplacian evolution equation with critical initial energy. In the framework of the energy method, we construct an unstable set and establish its invariance. Finally, the finite time blow-up of solutions is derived by a combination of the unstable set and the concavity method.

Blow-up of nonnegative solutions to quasilinear parabolic inequalities

Stanislav I. Pohozaev, Alberto Tesei (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We investigate critical exponents for blow-up of nonnegative solutions to a class of parabolic inequalities. The proofs make use of a priori estimates of solutions combined with a simple scaling argument.

Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou (2016)

Annales Polonici Mathematici

This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation u t - d i v ( | ∇ u m | p - 2 ∇ u m ) = f ( u ) with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

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