The smoothing property for a class of doubly nonlinear parabolic equations

Ebmeyer, Carsten; Urbano, José Miguel

doi:10.1090/s0002-9947-05-03790-6

Cited by 10 publications

(3 citation statements)

References 20 publications

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“…Indeed, when m = 1 we recover the range 2N N +1 < p < 2 considered in this paper. For basic results on existence, boundedness and regularity results for equation (DNLE) we refer to [13,15,22,32,33,36,41,46,62,68,57,58,71,77,78]. As far as asymptotic behaviour is concerned, nonnegative integrable solutions behave like Barenblatt for large times, in perfect analogy to the PLE case, see also [84].…”

Section: Reminder About Existing Resultsmentioning

confidence: 99%

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Bonforte

Simonov

Stan

2022

Journal de Mathématiques Pures et Appliquées

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Section: Reminder About Existing Resultsmentioning

confidence: 99%

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Bonforte

Simonov

Stan

2022

Journal de Mathématiques Pures et Appliquées

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“…It is worth noting that the operator (−∆ p ) s φ is the nonlocal counterpart of the local doubly nonlinear operator (−∆ p )φ (see, for example [PV93,EU05,CH16]). In addition, the fractional p-Laplacian (−∆ p ) s is a natural generalization of the well-known linear fractional Laplacian (−∆) s .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A doubly nonlinear evolution problem involving the fractional p-Laplacian

Collier¹,

Hauer²

2021

Preprint

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In this article we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional p-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in R d and composed with a continuous, strictly increasing function. We establish well-posedness in L 1 in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain L q − L ∞ regularity estimates for mild solutions, implying decay estimates and extending the property of strong solutions for more initial data. Moreover, we prove local and global Hölder continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.

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“…This evolution problem includes, as particular cases, the heat equation and the p-Laplacian equation, and has been widely studied starting from the sixties by the classical papers of Nash, Moser, Ladyženskaja, Solonnikov, Ural'ceva, J.L. Lions, Aronson, Serrin, Trudinger (see [2,33,34,36,37,53] and the references therein) to nowadays and it is really a difficult task to give a complete bibliography on this subject (see for example [1,3,4,[6][7][8][10][11][12]17,[19][20][21][22][23][24][25][26][29][30][31][32]35,[39][40][41]45,45,51,52,54,56,57] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Regularity and time behavior of the solutions to weak monotone parabolic equations

Porzio

2021

J. Evol. Equ.

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In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

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The smoothing property for a class of doubly nonlinear parabolic equations

Cited by 10 publications

References 20 publications

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

A doubly nonlinear evolution problem involving the fractional p-Laplacian

Regularity and time behavior of the solutions to weak monotone parabolic equations

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