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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...

Breakdown in finite time of solutions to a one-dimensional wave equation.

Mokhtar Kirane, Salim A. Messaoudi (2000)

Revista Matemática Complutense

We consider a special type of a one-dimensional quasilinear wave equation wtt - phi (wt / wx) wxx = 0 in a bounded domain with Dirichlet boundary conditions and show that classical solutions blow up in finite time even for small initial data in some norm.

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