Heat Kernels and Spectral Theory

Davies, E. B.

doi:10.1017/cbo9780511566158

Cited by 1,871 publications

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“…To do so, we first state a general upper bound for the heat kernel in domains satisfying the extension property. This upper bound follows from general results of Davies [10] and Grigor'yan [16]. …”

Section: 3mentioning

confidence: 72%

“…Let C 0 > 4 be given. Since Ω satisfies the extension property, it follows from Theorem 2.4.4 by Davies [10] that there exists C 1 > 0 such that 0 ≤ p(t, z, x) ≤ C 1 t −N/2 for all 0 < t ≤ 1 and for all (z, x) ∈ Ω × Ω. The maximum principle then yields p(t, z, x) ≤ C 1 for all t ≥ 1 and (z, x) ∈ Ω × Ω.…”

Section: 3mentioning

confidence: 96%

“…Furthermore, we prove here that the same property turns out to be true for the large class of domains satisfying the extension property. This class of domain is defined now: quoting Davies [10], a nonempty open subset Ω of R N is said to have the extension property if, for all 1 ≤ p ≤ +∞, there exists a bounded…”

Section: 5mentioning

confidence: 99%

See 2 more Smart Citations

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: 3mentioning

confidence: 72%

Section: 3mentioning

confidence: 96%

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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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“…[21], [9], [12], [4], [18], [5]). Since the work of Nash [19] and Aronson [1], many methods have been discovered for deriving Gaussian upper and lower bounds of H(x, y, t), see e.g., [7], [18], [10], [13], [11], [17]. One of the methods was developed by Li-Wang in [17].…”

Section: Introductionmentioning

confidence: 99%

Davies type estimate and the heat kernel bound under the Ricci flow

Zhu

2015

Trans. Amer. Math. Soc.

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Abstract. We prove a Davies type double integral estimate for the heat kernel H(y, t; x, l) under the Ricci flow. As a result, we give an affirmative answer to a question proposed in [8]. Moreover, we apply the Davies type estimate to provide a new proof of the Gaussian upper and lower bounds of H(y, t; x, l) which were first shown in [6]. IntroductionOn a complete Riemannian manifold (M n , g ij ), the heat kernel H(x, y, t), is the smallest positive fundamental solution to the heat equation ∂u ∂t = ∆u.( is the distance between U 1 and U 2 .In this paper, we consider the heat kernel of the time-evolving heat equation under the Ricci flow on a complete manifold M n , i.e., where ∆ t is the Laplacian with respect to a complete solution g ij (t), t ∈ [0, T ) and T < ∞, of the following Ricci flow equationThe existence and uniqueness of the heat kernel H(y, t; x, l) to (1.3) were proved in Our main goal is to give an affirmative answer to the question above. More specifically, we prove Theorem 1.2. Let (M n , g ij (t)) be a complete solution to (1.4) on [0, T ) and T < ∞. Suppose that H(y, t; x, l) is the heat kernel of (1.3), and Rc ≥ −K 1 on [0, T ) for some nonnegative constant K 1 . Then for any open sets, U 1 and U 2 , with compact closure in M, and 0 ≤ l < t < T , we havewhere C 0 is a constant depends only on n.Using Theorem 1.2, we are able to provide a new proof of the Gaussian upper bound and lower bounds of H(y, t; x, l) which was first shown by . For the Gaussian upper bound, we have where constants C 1 , C 2 ,, C 3 and C 4 depend only on n.Next, following a method of Li- Tam H(y, t; x, l) ≥ C 5 e −C 6 e C 7 Λ+C 8 K 1 T exp − 4d 2 t (x, y) (t − l)Vol t (B t (x, t−l 8 ))

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“…It follows from the Beurling-Deny criterion ( [11], Theorem 1.3.3) or ( [28], Corollary 2.17) that the semigroup is submarkovian.…”

Section: Dirichlet Regularity and Degenerate Diffusion 5863mentioning

confidence: 99%

Dirichlet regularity and degenerate diffusion

Arendt¹,

Chovanec²

2010

Trans. Amer. Math. Soc.

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Abstract. Let Ω ⊂ R N be an open and bounded set and let m : Ω → (0, ∞) be measurable and locally bounded. We study a natural realization of the operator m in C 0 (Ω) := u ∈ C(Ω) : u |∂Ω = 0 . If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C 0 (Ω) wheneverdoes not go fast enough to 0 as x → ∂Ω, then Dirichlet regularity is necessary. However, if |m(x)| ≤ c·dist(x, ∂Ω) 2 , then we show that m 0 generates a semigroup on C 0 (Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.

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Heat Kernels and Spectral Theory

Cited by 1,871 publications

References 0 publications

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

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

Davies type estimate and the heat kernel bound under the Ricci flow

Dirichlet regularity and degenerate diffusion

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