On the behavior of the solutions of degenerate parabolic equations

Ishige, Kazuhiro

doi:10.1017/s0027763000006978

Cited by 33 publications

(32 citation statements)

References 14 publications

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“…Let U be the reflection of U as given by (5.2). Then by Lemma 5.1, U is a nonnegative weak solution to the degenerate parabolic equation (5.3) in (t, x, y) ∈ (0, 1) × B 2 × (−2, 2).For this equation the Harnack inequality of Chiarenza-Serapioni [11] (see also Gutiérrez-Wheeden [16], Ishige [18]) applies. Therefore…”

Section: Proof Let φ Be a Smooth Test Function With Compact Support mentioning

confidence: 92%

“…For the Harnack inequality for uniformly parabolic equations in divergence form see Moser [24], and for degenerate parabolic equations in divergence form see Chiarenza-Serapioni [11], also Gutiérrez-Wheeden [16] and Ishige [18].…”

Section: Corollary 16 (Comparison Principle -Uniqueness)mentioning

confidence: 99%

“…After this transformation is performed, we obtain a function U (2) that solves again a degenerate parabolic equation with bounded measurable coefficients and A 2 -weight for (t, x, y) ∈ (−2, 2) × R n + ∩ B 2 × R and vanishes continuously for (t, x, y) ∈ (−2, 2) × (∂R n + ∩ B 2 ) × (−2, 2). We can apply the boundary Harnack inequality of Ishige [18] (or Salsa [28] in the case s = 1/2) to U (2) and the conclusion follows by transforming back to u. Indeed,…”

Section: Proof Let φ Be a Smooth Test Function With Compact Support mentioning

confidence: 99%

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Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

Stinga¹,

Torrea²

2017

SIAM J. Math. Anal.

108

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Abstract. We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operatorThis nonlocal equation of order s in time and 2s in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for spacetime nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for (∂t−∆) s u(t, x) and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. Hölder and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic Hölder spaces. Our methods involve the parabolic language of semigroups and the Cauchy Integral Theorem, which are original to define the fractional powers of ∂t − ∆. Though we mainly focus in the equation (∂t − ∆) s u = f , applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

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Section: Proof Let φ Be a Smooth Test Function With Compact Support mentioning

confidence: 92%

Section: Corollary 16 (Comparison Principle -Uniqueness)mentioning

confidence: 99%

Section: Proof Let φ Be a Smooth Test Function With Compact Support mentioning

confidence: 99%

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Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

Stinga¹,

Torrea²

2017

SIAM J. Math. Anal.

108

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“…Degenerate parabolic operators: [77] establishes a Harnack inequality; [78] allows for time-dependent weights; [79] establishes bounds for the fundamental solution; [80] allows for terms of lower order; [81] studies a class of hypoelliptic evolution equations.…”

Section: As Above We Consider the Operator DIV A(·)∇u)mentioning

confidence: 99%

Harnack Inequalities: An Introduction

Kaßmann

2007

Boundary Value Problems

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The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.

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“…Let ðx; tÞ 2 D Â ð0; 1Þ; and r 0 be a positive number such that r 0 ominð ffiffi t p ; distðx; @DÞÞ and the ball Bðx; r 0 Þ is included in a coordinate neighborhood of x: Then there exists a positive constant C such that for any r 2 ð0; r 0 and any non-negative solution u of (3.8) there holds the inequality This parabolic Harnack inequality plays a crucial role in studying positive solutions of parabolic and elliptic partial differential equations. As for the proof, see [ArSe,CW,FS,Gr1,GW1,2,Ishige,Moser,Stur3]. (ii) For every y 2 E; the function uðx; tÞ ¼ p E ðx; y; tÞ is a solution of the equation…”

Section: Minimal Fundamental Solutions Of Parabolic Equations and Thementioning

confidence: 99%

Martin Boundaries of Elliptic Skew Products, Semismall Perturbations, and Fundamental Solutions of Parabolic Equations

Murata

2002

Journal of Functional Analysis

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On the behavior of the solutions of degenerate parabolic equations

Abstract: In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness and continuity of the solutions.

Cited by 33 publications

References 14 publications

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

Harnack Inequalities: An Introduction

Martin Boundaries of Elliptic Skew Products, Semismall Perturbations, and Fundamental Solutions of Parabolic Equations

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