Nonlocal Dynamic Boundary Condition

• Nonlocal dynamic boundary conditions are defined as boundary evolution laws where the boundary trace satisfies its own time-dependent equation coupled with interior PDEs through mechanisms like tangential diffusion and fractional derivatives.
• They encompass various models such as Laplace equations with fractional reactive–diffusive conditions, nonlocal Cahn–Hilliard systems, and nonlocal diffusion on boundary strips, illustrating diverse applications.
• Key analyses focus on operator-theoretic structures, existence and uniqueness proofs, long-time asymptotics, and numerical formulations that address the integration of spatial nonlocality and temporal dynamics.

Searching arXiv for recent and relevant papers on nonlocal dynamic boundary conditions. arXiv search query: "nonlocal dynamic boundary condition Laplace Cahn-Hilliard diffusion" A nonlocal dynamic boundary condition is a boundary evolution law in which the trace of a field on a boundary or interface satisfies its own time-dependent equation, coupled to the interior PDE through normal fluxes, tangential diffusion, boundary integral operators, fractional derivatives, or memory terms. In the recent literature, this designation covers several distinct but structurally related settings: the Laplace equation with a fractional reactive–diffusive boundary law, nonlocal Cahn–Hilliard systems with bulk–surface exchange, nonlocal diffusion posed on a thin boundary strip, semilinear transmission problems on rough interfaces with fractional interfacial operators, and exact or asymptotically compatible nonlocal truncation conditions for Helmholtz and heat-type problems (Capitanelli et al., 2024, Lv et al., 2024, Berna et al., 2019, Gal et al., 2015, Kirby et al., 2020).

1. Canonical formulations

A prototypical analytic model is the Laplace equation in a bounded smooth domain $\Omega \subset \mathbb{R}^n$ with boundary $\Gamma=\partial\Omega$, where the interior remains harmonic,

$\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$

while the boundary trace evolves according to

$$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$

with $k\in \mathbb{R}$, $l>0$, $\alpha\in(0,1]$, and initial datum $u(0,y)=u_0(y)$. Here $D_t^\alpha$ is the Caputo–Dzhrbashyan fractional derivative, so the boundary law is simultaneously dynamic and nonlocal in time (Capitanelli et al., 2024).

A second major class arises in nonlocal Cahn–Hilliard systems. In the bulk one has

$$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$

or, in the singular-potential formulation,

$\Gamma=\partial\Omega$0

On the boundary one introduces a surface order parameter and surface chemical potential through

$\Gamma=\partial\Omega$1

together with the kinetic-rate dependent coupling

$\Gamma=\partial\Omega$2

with $\Gamma=\partial\Omega$3. The parameter $\Gamma=\partial\Omega$4 distinguishes instantaneous exchange when $\Gamma=\partial\Omega$5, finite-rate adsorption/desorption when $\Gamma=\partial\Omega$6, and decoupling in the limit $\Gamma=\partial\Omega$7 (Lv et al., 2024, Lv et al., 12 Sep 2025, Knopf et al., 2020).

A different formulation appears in nonlocal diffusion equations with dynamical boundary conditions, where the “boundary” is represented by a thin strip

$\Gamma=\partial\Omega$8

In the simplest linear model, the interior problem on $\Gamma=\partial\Omega$9 is elliptic in time,

$\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$0

whereas on $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$1 the solution evolves dynamically,

$\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$2

Variants allow full jumps on $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$3, $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$4-Laplacian-type nonlinearities, and singular kernels $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$5 (Berna et al., 2019).

On rough interfaces, Gal and Warma study a semilinear transmission problem in which an interfacial trace $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$6 obeys

$\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$7

Here $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$8 may have Hausdorff dimension $\Delta u(t,x)=0,\qquad x\in \Omega,\ t>0,$9, and $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$0 is a nonlocal operator induced by a symmetric kernel $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$1 on the interface (Gal et al., 2015).

In computational scattering and truncation problems, nonlocal boundary conditions may be exact rather than phenomenological. For the exterior Helmholtz equation truncated by an artificial boundary $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$2, the outgoing condition can be written exactly as

$$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$3

where $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$4 and $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$5 are single- and double-layer potentials built from the free-space Green’s function. The same framework motivates a time-domain analogue involving retarded boundary integrals and therefore a dynamic nonlocal boundary condition with memory (Kirby et al., 2020).

2. Operator-theoretic structure

The operator content of nonlocal dynamic boundary conditions depends on the model, but several recurrent mechanisms appear. In the Laplace problem, the key boundary operators are the Laplace–Beltrami operator $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$6 and the Dirichlet–to–Neumann operator $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$7, defined by harmonic lifting: for $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$8, if $$D_t^\alpha u(t,y)=k\,\partial_n u(t,y)+l\,\Delta_\Gamma u(t,y)+f(t,y),\qquad y\in \Gamma,\ t>0,$$9 in $k\in \mathbb{R}$0 and $k\in \mathbb{R}$1, then

$k\in \mathbb{R}$2

The total boundary operator is

$k\in \mathbb{R}$3

on $k\in \mathbb{R}$4 with domain $k\in \mathbb{R}$5. For $k\in \mathbb{R}$6, $k\in \mathbb{R}$7 is self-adjoint and generates an analytic semigroup $k\in \mathbb{R}$8 (Capitanelli et al., 2024).

In nonlocal Cahn–Hilliard models, nonlocality is carried by the convolution operators $k\in \mathbb{R}$9 in the bulk and $l>0$0 on the surface, with even nonnegative kernels $l>0$1 and $l>0$2 satisfying Sobolev regularity assumptions such as $l>0$3 and $l>0$4 for $l>0$5. The energetic structure uses bulk and surface interaction energies of the form

$l>0$6

and

$l>0$7

supplemented by singular potentials such as the logarithmic potential (Lv et al., 2024, Lv et al., 12 Sep 2025).

For rough interfaces, the nonlocal operator is encoded by a bilinear form

$l>0$8

with

$l>0$9

so that a prototypical example is the fractional Laplacian $\alpha\in(0,1]$0 on $\alpha\in(0,1]$1 (Gal et al., 2015).

In exact truncation for Helmholtz, the nonlocal part is a boundary integral operator acting from the physical boundary $\alpha\in(0,1]$2 to the artificial boundary $\alpha\in(0,1]$3. In asymptotically compatible local-to-nonlocal coupling for diffusion, the nonlocal Robin-type collar operator contains geometry-dependent coefficients $\alpha\in(0,1]$4 and $\alpha\in(0,1]$5 and a penalization term $\alpha\in(0,1]$6, where $\alpha\in(0,1]$7 is the orthogonal projection onto the interface. These formulations make clear that “nonlocal” may refer to integration along the boundary, across the bulk–boundary interface, or over a finite interaction horizon $\alpha\in(0,1]$8 (Kirby et al., 2020, You et al., 2019).

3. Existence, uniqueness, and representation formulas

For the Laplace equation with fractional reactive–diffusive boundary condition, the boundary trace $\alpha\in(0,1]$9 solves the fractional evolution equation

$u(0,y)=u_0(y)$0

and the harmonic lift $u(0,y)=u_0(y)$1 is the unique solution. If $u(0,y)=u_0(y)$2 is an orthonormal eigenbasis of $u(0,y)=u_0(y)$3,

$u(0,y)=u_0(y)$4

and $u(0,y)=u_0(y)$5 denotes the harmonic lift, then

$u(0,y)=u_0(y)$6

where $u(0,y)=u_0(y)$7 is the Mittag–Leffler function. For $u(0,y)=u_0(y)$8 this reduces to the classical semigroup formula $u(0,y)=u_0(y)$9 (Capitanelli et al., 2024).

The same paper gives a probabilistic representation in terms of a boundary process $D_t^\alpha$0 with generator $D_t^\alpha$1 and an inverse subordinator $D_t^\alpha$2 associated with the Laplace exponent $D_t^\alpha$3:

$D_t^\alpha$4

Equivalently, for the purely boundary fractional Cauchy problem,

$D_t^\alpha$5

This identifies the fractional derivative with a random time-change and the boundary operator $D_t^\alpha$6 with a jump–diffusion on $D_t^\alpha$7 (Capitanelli et al., 2024).

For nonlocal diffusion on the boundary strip, Berná and Rossi prove existence and uniqueness in several regimes. For the linear smooth-kernel problem (P), if $D_t^\alpha$8 with $D_t^\alpha$9, there is a unique solution

$$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$0

and for the nonlinear $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$1-Laplacian problem (P2) there is a unique strong solution

$$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$2

For the singular fractional problem (P3), if $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$3, there is a unique strong solution

$$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$4

The proofs reduce the system to an evolution equation on $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$5 with a completely accretive operator and then apply nonlinear semigroup theory (Berna et al., 2019).

For nonlocal Cahn–Hilliard systems with $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$6, Lv and Wu establish existence and uniqueness of global weak solutions by a Yosida approximation of singular potentials and a Faedo–Galerkin scheme. In the formulation with singular graphs $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$7, the energy identity reads

$$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$8

Knopf and Signori prove weak and strong well-posedness for a related model with boundary penalization and a Robin-type coupling between bulk and surface chemical potentials (Lv et al., 2024, Knopf et al., 2020).

4. Long-time behavior, asymptotics, and emergent memory

In nonlocal Cahn–Hilliard dynamics, the kinetic parameter $$\partial_t \phi=\Delta \mu,\qquad \mu=a_\Omega \phi-J*\phi+F'(\phi),$$9 controls both coupling and asymptotic behavior. Lv and Wu prove that for each conserved mean $\Gamma=\partial\Omega$00, the semigroup $\Gamma=\partial\Omega$01 possesses a connected global attractor $\Gamma=\partial\Omega$02, bounded in $\Gamma=\partial\Omega$03, for all $\Gamma=\partial\Omega$04. For $\Gamma=\partial\Omega$05 and under extra analyticity and separation hypotheses, there exists an exponential attractor $\Gamma=\partial\Omega$06, semi-invariant under $\Gamma=\partial\Omega$07, which attracts bounded sets at an exponential rate in spaces such as $\Gamma=\partial\Omega$08 for $\Gamma=\partial\Omega$09 and $\Gamma=\partial\Omega$10. Under real-analyticity assumptions on $\Gamma=\partial\Omega$11 and $\Gamma=\partial\Omega$12, every weak solution converges in $\Gamma=\partial\Omega$13 to a single stationary pair (Lv et al., 12 Sep 2025).

The finite-$\Gamma=\partial\Omega$14 theory is complemented by asymptotic limits. In the well-posedness paper, Lv and Wu show that under the stronger assumption $\Gamma=\partial\Omega$15,

$\Gamma=\partial\Omega$16

and

$\Gamma=\partial\Omega$17

with analogous statements for $\Gamma=\partial\Omega$18. They also establish instantaneous strict separation for $\Gamma=\partial\Omega$19 by a De Giorgi iteration scheme (Lv et al., 2024).

For nonlocal diffusion with dynamical boundary conditions on $\Gamma=\partial\Omega$20, long-time relaxation depends on the kernel and the nonlinearity. In the linear smooth-kernel problems (P) and (P*), mass is conserved and one has exponential convergence to the mean:

$\Gamma=\partial\Omega$21

under the stated geometric hypotheses. In the nonlinear $\Gamma=\partial\Omega$22-Laplacian problem (P2), the convergence remains valid, with exponential decay for $\Gamma=\partial\Omega$23 and algebraic decay for $\Gamma=\partial\Omega$24:

$\Gamma=\partial\Omega$25

For the singular fractional problem (P3), the decay rate is algebraic:

$\Gamma=\partial\Omega$26

for $\Gamma=\partial\Omega$27 (Berna et al., 2019).

A distinct asymptotic mechanism appears in critical-scale homogenisation. Díaz, Gómez-Castro, Shaposhnikova, and Zubova consider a parabolic equation in a perforated domain with microscopic dynamic boundary condition

$\Gamma=\partial\Omega$28

In the homogenised limit the effective problem contains a coupled PDE–ODE system,

$\Gamma=\partial\Omega$29

$\Gamma=\partial\Omega$30

so that the microscopic dynamic boundary produces a nonlocal-in-time memory term in the macroscopic equation. The limit system satisfies a comparison principle (Díaz et al., 2019).

5. Numerical formulations and exact truncation conditions

In exterior Helmholtz problems, the nonlocal boundary condition introduced by Gillman and Barnett is exact and arises directly from Green’s formula. After truncating the exterior domain by an artificial boundary $\Gamma=\partial\Omega$31, the variational form reads: find $\Gamma=\partial\Omega$32 such that

$\Gamma=\partial\Omega$33

with

$\Gamma=\partial\Omega$34

The form is split as $\Gamma=\partial\Omega$35, where $\Gamma=\partial\Omega$36 is the nonlocal contribution generated by the double-layer potential (Kirby et al., 2020).

Under the assumptions that $\Gamma=\partial\Omega$37 is Lipschitz, $\Gamma=\partial\Omega$38, the exact solution belongs to $\Gamma=\partial\Omega$39, and the adjoint problem enjoys $\Gamma=\partial\Omega$40-regularity, the form satisfies a Gårding inequality,

$\Gamma=\partial\Omega$41

and the Galerkin solution is quasi-optimal when $\Gamma=\partial\Omega$42. For $\Gamma=\partial\Omega$43 elements with $\Gamma=\partial\Omega$44,

$\Gamma=\partial\Omega$45

provided $\Gamma=\partial\Omega$46 is sufficiently small. The dense nonlocal operator is applied matrix-free via FMM or GIGAQBX, and the purely local transmission-BC operator $\Gamma=\partial\Omega$47 is used as a preconditioner (Kirby et al., 2020).

The same paper sketches a time-domain extension obtained by inverse Fourier transform. The resulting boundary law contains a retarded operator,

$\Gamma=\partial\Omega$48

which is a dynamic nonlocal boundary condition with memory (Kirby et al., 2020).

For local-to-nonlocal coupling of the heat equation without overlap, You, Yu, and Kamensky derive a nonlocal Robin-type condition in the collar near the interface,

$\Gamma=\partial\Omega$49

They show that the corresponding nonlocal Neumann-type problem converges in $\Gamma=\partial\Omega$50 to the local Neumann solution at rate $\Gamma=\partial\Omega$51, and that in the coupled scheme the overall framework converges to the local limit with an $\Gamma=\partial\Omega$52 rate when $\Gamma=\partial\Omega$53 is fixed and $\Gamma=\partial\Omega$54. The nonlocal solver uses GMLS meshfree discretization, the local solver uses linear finite elements, and the Robin coefficient is selected by minimizing the amplification factor of the fully discrete coupling matrix (You et al., 2019).

6. Relations to local boundary conditions and conceptual distinctions

A nonlocal dynamic boundary condition should be distinguished from a static nonlocal boundary operator. The defining feature is the presence of boundary evolution, typically through $\Gamma=\partial\Omega$55, $\Gamma=\partial\Omega$56, or an analogous time-dependent law on $\Gamma=\partial\Omega$57 or $\Gamma=\partial\Omega$58. The nonlocality may then enter through a spatial operator such as $\Gamma=\partial\Omega$59, $\Gamma=\partial\Omega$60, $\Gamma=\partial\Omega$61, $\Gamma=\partial\Omega$62, a finite-horizon interaction, or a retarded boundary integral (Capitanelli et al., 2024, Lv et al., 2024, Gal et al., 2015, Kirby et al., 2020).

The relation between local and nonlocal theories is particularly explicit in peridynamics. Aksoylu, Beyer, and Celiker construct a nonlocal operator as a bounded function of a classical self-adjoint operator $\Gamma=\partial\Omega$63 with built-in Dirichlet, Neumann, periodic, or antiperiodic boundary conditions:

$\Gamma=\partial\Omega$64

Because $\Gamma=\partial\Omega$65 commutes with $\Gamma=\partial\Omega$66, the original local boundary conditions are automatically inherited. In this framework, a nonlocal governing operator can preserve local boundary conditions while remaining genuinely nonlocal in the interior (Aksoylu et al., 2014).

A common misconception is that nonlocal dynamic boundary conditions are synonymous with fractional-in-time memory. The literature shows a broader taxonomy. In the Laplace problem, memory is explicit through the Caputo derivative and its inverse-subordinator representation. In the homogenisation problem, the microscopic dynamic boundary condition generates macroscopic memory only after passage to the limit. In nonlocal Cahn–Hilliard systems, by contrast, the principal nonlocality is spatial and energetic, while the dynamics remain first order in time. This suggests that “nonlocal” and “dynamic” are independent structural attributes whose combination can occur in several mathematically distinct ways (Capitanelli et al., 2024, Díaz et al., 2019, Lv et al., 12 Sep 2025).

At a broader level, the subject links analytic semigroup theory, spectral and resolvent methods, nonlinear semigroup theory, stochastic time-changes, homogenisation, gradient-flow formulations in $\Gamma=\partial\Omega$67-type metrics, and fast boundary-integral numerics. The resulting models are used to describe heat conduction with boundary memory, surface reactions with anomalous rates, phase separation with bulk–surface exchange, diffusion on boundary strips, transport across rough interfaces, and exact radiation or coupling conditions in wave and diffusion problems (Capitanelli et al., 2024, Lv et al., 2024, Berna et al., 2019, Gal et al., 2015, You et al., 2019).