A Certain Property of Solutions of Parabolic Equations With Measurable Coefficients

Крылов, Н. В.; Safonov, M. V.

doi:10.1070/im1981v016n01abeh001283

Cited by 437 publications

(404 citation statements)

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“…Fix any c > w * (e, z, u 0 ). From the above asymptotic connectedness property and from the global smoothness of Ω, Harnack inequality applied in Ω (see [17,30,31]) yields the existence of η > 0 such that…”

Section: General Propertiesmentioning

confidence: 99%

The speed of propagation for KPP type problems. II: General domains

Berestycki

Hamel

Nadirashvili

2009

J. Amer. Math. Soc.

131

107

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This paper is devoted to nonlinear propagation phenomena in general unbounded domains of R N \mathbb {R}^N , for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. This article is the second in a series of two and it is the follow-up of the paper The speed of propagation for KPP type problems. I - Periodic framework, by the authors, which dealt which the case of periodic domains. This paper is concerned with general domains, and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data. We study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition. Some a priori bounds are proved for large classes of domains. The case of exterior domains is also discussed in detail. Lastly, some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited; the construction is based on some new heat kernel estimates in such domains.

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Section: General Propertiesmentioning

confidence: 99%

The speed of propagation for KPP type problems. II: General domains

Berestycki

Hamel

Nadirashvili

2009

J. Amer. Math. Soc.

131

107

View full text Add to dashboard Cite

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“…We remark that, while these authors work in Lipschitz cylinders, one can easily see that their proofs can be generalized to the setting of bounded NTA-cylinders. While the works Fabes, Safonov and Yuan completed, for linear uniformly parabolic equations, the line of research considered in this paper, contributions by other researchers are contained in [17], [23], [25], [33], [16], [21], [44]. For the elliptic versions of Theorem 1.1-Theorem 1.3 we refer to [18], [19], [20], and we emphasize that in the elliptic case the assumption λ ∈ A 2 (R n ) on the weight is sufficient for the validity of the corresponding versions of Theorem 1.1-Theorem 1.3.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Boundary estimates for solutions to linear degenerate parabolic equations

Nyström

Persson

Sande

2015

Journal of Differential Equations

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Let Ω ⊂ R n be a bounded NTA-domain and let Ω T = Ω × (0, T ) for some T > 0. We study the boundary behaviour of non-negative solutions to the equationWe assume that A(x, t) = {a ij (x, t)} is measurable, real, symmetric and thatfor some constant β ≥ 1 and for some non-negative and real-valued function λ = λ(x) belonging to the Muckenhoupt class A 1+2/n (R n ). Our main results include the doubling property of the associated parabolic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes, Kenig, Jerison, Serapioni, see [18], [19], [20], to a parabolic setting.

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“…For uniformly elliptic and uniformly parabolic equations such estimates were established in the classical paper [1]; see also [2]. In [3], this result was generalized to the equations with unbounded lower-order coefficients.…”

Section: §1 Introductionmentioning

confidence: 93%

“…The result of [1] shows that the Dirichlet problem for a uniformly elliptic equation (1.1) has approximative solutions for any continuous boundary data. Moreover, solutions of the Dirichlet problems for equations (6.1) converge to the approximative solution uniformly in s Ω.…”

Section: A I Nazarov §5 Parabolic Casementioning

confidence: 99%