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Large-time estimates for the Dirichlet heat equation in exterior domains

José A. Cañizo, Alejandro Gárriz, Fernando Quirós

TL;DR

The paper investigates the large-time behavior of the Dirichlet heat equation in exterior domains with a bounded hole, establishing both uniform and L1 asymptotics with explicit convergence rates that reflect the domain geometry through a harmonic profile φ and mass loss through the boundary.It develops a robust entropy-based framework by a change of variables that converts the exterior-domain problem into a Fokker–Planck type dynamics with transient equilibria F_τ and a moving domain, enabling quantitative decay estimates.A central contribution is the precise kernel asymptotics p_Ω(t,x,y) ∼ φ(x)φ(y)p(t,x,y) in d ≥ 3 and analogous scaled forms in d = 2 and d = 1, including rates and diffusive scaling information, together with L1 and weighted L1 convergence results for general initial data with finite moments.The results yield a unified, dimension-dependent description of mass loss through the boundary and convergence to Gaussian-type profiles corrected by domain geometry, advancing the understanding of diffusion in perforated environments and offering new rates for kernel and solution asymptotics.

Abstract

We give large-time asymptotic estimates, both in uniform and $L^1$ norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of $\mathbb{R}^d$ satisfying certain technical assumptions. We always assume that the initial datum has suitable finite moments (often, finite first moment). All estimates include an explicit rate of approach to the asymptotic profiles at the different scales natural to the problem, in analogy with the Gaussian behaviour of the heat equation in the full space. As a consequence we obtain by an approximation procedure the asymptotic profile, with rates, for the Dirichlet heat kernel in these exterior domains. The estimates on the rates are new even when the domain is the complement of the unit ball in $\mathbb{R}^d$, except for previous results by Uchiyama in dimension 2, which we are able to improve in some scales. We obtain that the heat kernel behaves asymptotically as the heat kernel in the full space, with a factor that takes into account the shape of the domain through a harmonic profile, and a second factor which accounts for the loss of mass through the boundary. The main ideas we use come from entropy methods in PDE and probability, whose application seems to be new in the context of diffusion problems in exterior domains.

Large-time estimates for the Dirichlet heat equation in exterior domains

TL;DR

The paper investigates the large-time behavior of the Dirichlet heat equation in exterior domains with a bounded hole, establishing both uniform and L1 asymptotics with explicit convergence rates that reflect the domain geometry through a harmonic profile φ and mass loss through the boundary.It develops a robust entropy-based framework by a change of variables that converts the exterior-domain problem into a Fokker–Planck type dynamics with transient equilibria F_τ and a moving domain, enabling quantitative decay estimates.A central contribution is the precise kernel asymptotics p_Ω(t,x,y) ∼ φ(x)φ(y)p(t,x,y) in d ≥ 3 and analogous scaled forms in d = 2 and d = 1, including rates and diffusive scaling information, together with L1 and weighted L1 convergence results for general initial data with finite moments.The results yield a unified, dimension-dependent description of mass loss through the boundary and convergence to Gaussian-type profiles corrected by domain geometry, advancing the understanding of diffusion in perforated environments and offering new rates for kernel and solution asymptotics.

Abstract

We give large-time asymptotic estimates, both in uniform and norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of satisfying certain technical assumptions. We always assume that the initial datum has suitable finite moments (often, finite first moment). All estimates include an explicit rate of approach to the asymptotic profiles at the different scales natural to the problem, in analogy with the Gaussian behaviour of the heat equation in the full space. As a consequence we obtain by an approximation procedure the asymptotic profile, with rates, for the Dirichlet heat kernel in these exterior domains. The estimates on the rates are new even when the domain is the complement of the unit ball in , except for previous results by Uchiyama in dimension 2, which we are able to improve in some scales. We obtain that the heat kernel behaves asymptotically as the heat kernel in the full space, with a factor that takes into account the shape of the domain through a harmonic profile, and a second factor which accounts for the loss of mass through the boundary. The main ideas we use come from entropy methods in PDE and probability, whose application seems to be new in the context of diffusion problems in exterior domains.

Paper Structure

This paper contains 37 sections, 48 theorems, 392 equations.

Table of Contents

  1. Introduction and main results
  2. Precedents and informal discussion of results
  3. Known bounds for kernels.
  4. Asymptotic results for kernels.
  5. Asymptotic results for solutions
  6. Main uniform estimates
  7. Some notation.
  8. Strategy and $L^1$ estimates
  9. Organisation of the paper
  10. Change of variables and entropy
  11. The heat equation in the full space
  12. The heat equation in an exterior domain
  13. A useful expression for the remainder term
  14. Some preliminary estimates
  15. Estimates on the harmonic profile $\phi$
  16. ...and 22 more sections

Key Result

Theorem 1.2

In dimension $d=1$ take $\Omega = (x_0,+\infty)$, $x_0\in\mathbb{R}$; in dimension $d \geq 2$, assume $\Omega \subseteq \mathbb{R}^d$ is an exterior domain satisfying eq:hypOmega1 and Hypothesis hyp:logsob. Let $u$ be the standard solution to the heat equation eq:heat-ext in $\Omega$ with nonnegativ Then there exists a constant $C > 0$ which depends only on $d$ and the domain $\Omega$ such that fo

Theorems & Definitions (101)

  • Theorem 1.2: Uniform estimates of solutions
  • Remark 1.3: Initial data
  • Remark 1.4: Optimal rates
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8: Sign-changing solutions
  • Corollary 1.9: Uniform estimates of the heat kernel
  • proof : Proof of Corollary \ref{['cor:heat_kernel_bounds']}
  • Theorem 1.10: Weighted $L^1$ estimates
  • ...and 91 more