Continuity of solutions of a singular parabolic equation

Sacks, Paul

doi:10.1016/0362-546x(83)90092-5

“…However, the best known regularity of the temperature u for (1.7) is just continuity (see [8,14,37,30]), and the codimension-one property of the moving front is still open.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global weak solutions of non-isothermal front propagation problem

Su

¹

,

Burger

²

2007

Electron. Res. Announc. Amer. Math. Soc.

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“…This problem has a unique solution at least on a bounded interval ( [16]). We suppose that p satisfies the following hypothesis:…”

Section: Asymptotic Behavior Of the Solutionmentioning

confidence: 99%

“…In this paper, a numerical scheme is proposed to solve the reaction diffusion problem: find a nonnegative function u defined on Ω, a smooth domain R d (d ≥ 1) and such that u t − ∆u m = αu This problem and analogous problems have been studied from a theoretical point of view by several authors: Aronson-Crandall-Peletier [2], Berryman-Holland [3], Friedman-Lacey [4], Friedman-McLeod [5], [6], Levine-Sacks [13], Sabinina [15], and Sacks [16], [17], [18]. M.-N.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of a fast diffusion equation

Roux

¹

,

Maingé

²

1999

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Abstract. In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: ut − ∆u m = αu p 1 in Ω, a smooth bounded domain in R d (d ≥ 1) with the zero lateral boundary condition and with a positive initial condition, m ∈ ]0, 1[ (fast diffusion problem), α ≥ 0 and p 1 ≥ m. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

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“…A denotes the operator −∆ of domain D(A) = H 1 0 (Ω) ∩ H 2 (Ω). This problem has a unique solution at least on a bounded interval ( [16]). We suppose that p satisfies the following hypothesis:…”

Section: Asymptotic Behavior Of the Solutionmentioning

confidence: 99%

“…where m ∈ ]0, 1[ (fast diffusion problem), α > 0 and p 1 ≥ m. This problem and analogous problems have been studied from a theoretical point of view by several authors: Aronson-Crandall-Peletier [2], Berryman-Holland [3], Friedman-Lacey [4], Friedman-McLeod [5], [6], Levine-Sacks [13], Sabinina [15], and Sacks [16], [17], [18]. M.-N.…”

Section: Introductionmentioning

confidence: 99%

Untitled

1999

Math. Comp.

0

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Abstract. In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: ut − ∆u m = αu p 1 in Ω, a smooth bounded domain in R d (d ≥ 1) with the zero lateral boundary condition and with a positive initial condition, m ∈ ]0, 1[ (fast diffusion problem), α ≥ 0 and p 1 ≥ m. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

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Continuity of solutions of a singular parabolic equation

Cited by 183 publications

References 3 publications

Global weak solutions of non-isothermal front propagation problem

Global weak solutions of non-isothermal front propagation problem

Numerical solution of a fast diffusion equation

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