Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition

Abstract: We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}<sup>N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial--boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.

• The paper derives an explicit fundamental solution for the half-space heat equation with a dynamic diffusive boundary, yielding a symmetric and strictly positive kernel formulation.
• It provides detailed sharp bounds and decay estimates that ensure optimal L^p-L^q decay rates and precise regularity of the solutions.
• The work rigorously analyzes multiple asymptotic diffusion limits, offering clear insights into bulk, boundary, and tangential scaling regimes.

Explicit Fundamental Solution and Diffusion Limits for the Heat Equation in a Half-Space with a Diffusive Dynamical Boundary Condition

Problem Formulation and Context

This work provides a comprehensive analysis of the linear heat equation in the half-space $\Omega = \mathbb{R}^{N-1} \times (0,\infty)$ subject to a diffusive dynamical boundary condition on the hyperplane boundary $\partial\Omega$. The system is given by: $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$ where $\epsilon, \delta, k > 0$, $\Delta'$ is the Laplace operator tangential to the boundary, and $$(\phi, \psi) \in L^\infty(\Omega) \times L^\infty(\partial\Omega)$$ are initial data. The focus is on the explicit construction, sharp bounds, and asymptotic analysis of the fundamental solution for this system, as well as rigorous examination of various diffusion limits with respect to the parameters $\epsilon$, $\delta$, and $k$.

The motivating physical framework arises in models where the boundary supports tangential diffusion and has a finite storage capacity, mediating exchange with the interior—a generalization relevant in materials science, reaction-diffusion systems, and surface–bulk coupling phenomena. While dynamical boundary problems have been deeply investigated in bounded domains, explicit characterizations in unbounded domains like the half-space were previously unavailable.

Explicit Construction of the Fundamental Solution

A significant achievement of the paper is an explicit formula for the fundamental solution $G(x, y, t)$ for the half-space heat equation with diffusive dynamical boundary condition. The construction exploits the classical heat kernel $\partial\Omega$0, the Dirichlet heat kernel $\partial\Omega$1, and auxiliary kernels, yielding

$\partial\Omega$2

where

$\partial\Omega$3

This representation is symmetric, strictly positive, and sharply characterized in both the interior and up to the boundary. The kernel $\partial\Omega$4 incorporates both normal and tangential diffusion and encodes the intricate bulk-boundary coupling.

A constructive integral representation for bounded classical solutions follows directly via Duhamel-like formulas involving $\partial\Omega$5 and explicit integration over initial and boundary data. The approach is notably robust, relying on fundamental solution machinery—no spectral or abstract semigroup theory is required.

Sharp Bounds and Decay Estimates

The authors provide matching lower and upper bounds for the kernel $\partial\Omega$6 and hence for $\partial\Omega$7. The estimates are regionally stratified according to spatial and temporal configuration and parameter regimes—especially through the regimes $\partial\Omega$8–$\partial\Omega$9 defined according to the scaling of $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$0. This produces, for example,

$$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$1

where $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$2 are parameter-dependent minima and maxima. The scalar factors $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$3 capture the respective kernel behaviors in near-field, far-field, and transitional regimes and are individually computed for all regions. These pointwise bounds imply optimal $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$4-$$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$5 decay rates and sharp regularity properties for solution operators.

A notable technical result is the Täcklind-type uniqueness theorem for classical solutions with controlled exponential growth at infinity, providing a foundation for all representation and limiting arguments.

Diffusion Limits: Asymptotic Analysis

A cornerstone of the analysis is a detailed investigation of multiple parabolic and elliptic limits as the key parameters approach singular or vanishing regimes. The results are both highly nontrivial and sharp in the rate of convergence.

1. Bulk Diffusion Limit ($$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$6): Solutions approach a boundary-layer-dominated regime, converging (with explicit $$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$7 rate) to solutions of a Laplace equation in the half-space with the same diffusive dynamical condition,

$$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$8

The fundamental solution in this limit is provided explicitly.

2. Boundary Diffusion Limit ($$\begin{cases} \epsilon \partial_t u - \Delta u = 0 & \text{in } \Omega \times (0, \infty), \ \delta \partial_t u - k \Delta' u - \partial_{x_N}u = 0 & \text{on } \partial\Omega \times (0, \infty), \ u = \phi & \text{on } \Omega \times \{0\}, \ u = \psi & \text{on } \partial\Omega \times \{0\}, \end{cases}$$9): The system converges (linearly in $\epsilon, \delta, k > 0$0) to the heat equation in the half-space with a (diffusive) Neumann-type condition determined by

$\epsilon, \delta, k > 0$1

and the explicit kernel is again computed.

1. Tangential Diffusion Limit ($\epsilon, \delta, k > 0$2, $\epsilon, \delta, k > 0$3): The tangentially non-diffusive ($\epsilon, \delta, k > 0$4) and strongly diffusive ($\epsilon, \delta, k > 0$5) limits are characterized, including in the latter case the effective reduction to instantaneous boundary homogenization, and explicit rates of convergence (in terms of $\epsilon, \delta, k > 0$6, $\epsilon, \delta, k > 0$7 for $\epsilon, \delta, k > 0$8 data) are achieved.
2. Simultaneous and Composite Limits: The framework supports treating more singular scaling limits (such as the so-called "fast/slow surface–bulk coupling"), leading to new parabolic or elliptic limit systems depending on joint asymptotics of $\epsilon, \delta, k > 0$9.

All these limits are accompanied by explicit error control and identification of all possible sharp optimal rates, with examples demonstrating that these rates are unimprovable.

Well-Posedness, Regularity, and Further Applications

The explicit kernel machinery yields—without recourse to abstract parabolic theory—uniqueness, existence, and regularity for bounded classical solutions with general $\Delta'$0 initial and boundary data. The detailed kernel structure allows for explicit analysis of boundary trace behavior, identification of when initial layers propagate, and precise conditions for strong or weak convergence in function spaces.

Further, the paper derives explicit fundamental solutions for (1) the Laplace equation with diffusive dynamical boundary condition and (2) the heat equation with a diffusive Neumann boundary condition in the half-space, both cases previously inaccessible explicitly.

Implications and Theoretical Significance

The explicit formulas and sharp asymptotic analyses provide a structural understanding of bulk-boundary coupled parabolic systems that was previously unavailable even for canonical geometries such as half-spaces. This directly advances mathematical control over the mechanisms of tangential and normal diffusion interaction, quantitative scaling regimes, and the role of boundary dynamics in diffusion-dominated processes.

Practically, the results supply exact simulation and computation methods for such systems and can serve as benchmarks for numerical schemes, asymptotic expansions, and stochastic representations.

Theoretically, the work establishes a blueprint for systematic study of limiting processes in extended operator settings, provides insight into how nontrivial boundary dynamics modify heat propagation, and lays groundwork for the investigation of nonlinear analogues, random walks with surface diffusion, and singular perturbation problems in unbounded domains.

Potential future developments include:

• Extension to general unbounded (possibly non-flat) domains via flattening and localization,
• Analysis of nonlinear systems and reaction–diffusion–surface coupling,
• Stochastic interpretations (surface–bulk Markov processes),
• Connections to probabilistic trace theorems, spectral theory, and interface homogenization.

Conclusion

This work fills a critical gap in the explicit, sharp analysis of heat flow with diffusive dynamical boundary condition in half-space geometry. The results provide a rigorous, comprehensive kernel-based characterization of the system's dynamics and the rich tapestry of its asymptotic diffusive limits. The techniques and explicit formulas will serve as essential tools for future investigations in parabolic free-boundary, diffusion, and coupled surface-bulk systems.

Reference: "Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition" (2604.00497)