Papers

We establish local well-posedness for a class of non-linear heat equations on the torus $\mathbb{T}^n$ whose highest order non-linearities have scaling-critical dimension $-1/k, \, k\in \mathbb{N}$, when the initial condition is a random field satisfying a spectral gap inequality involving a suitable Lebesgue norm. The corresponding subcriticality condition matches that of the Gaussian free field in dimension $d<2+2/k$. This extends previous subcritical well-posedness results beyond the Gaussian setting. Moreover, our proof is simpler and avoids diagrammatic arguments.

We study the Cauchy problem $$ v_{tt}-v_{xx}=\abs{v_t+v_x}^{m}\abs{v_t}^{n}, \qquad v(x,0)=\eta\varphi(x),\quad v_t(x,0)=\eta\psi(x), $$ on $\R$, with compactly supported profiles $\varphi\in C_0^2(\R)$, $\psi\in C_0^1(\R)$, exponents $m,n>1$, and amplitude $\eta>0$. We show that the sign of $P_0=\psi+\varphi'$ decides the behaviour of small solutions, whatever the sign of $\psi-\varphi'$. If $P_0$ is positive at some point, the lifespan $T(\eta)$ obeys explicit two-sided bounds of order $\eta^{-(m+n-1)}$ for every $\eta>0$, and $$ \lim_{\eta\to0^+}\eta^{m+n-1}T(\eta)=\frac{2^{n}}{(m+n-1)\,(\max_{\R}P_0)^{m+n-1}} . $$ If $P_0\le0$, the solution is global for every amplitude below an explicit threshold. If moreover $P_0\not\equiv0$, the component $v_t+v_x$ decays at the universal rate $\bigl(2^{n}/((m+n-1)t)\bigr)^{1/(m+n-1)}$, and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if $P_0\equiv0$, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of $v_t+v_x$ is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents $m,n\ge1$ and to sources $f(v_t+v_x)g(v_t)$, for which $T(\eta)$ is asymptotic to an Osgood-type integral.

We consider an inverse scattering problem for the scalar wave equation in which the point emitter, a point-like scatterer, and multiple receivers are all moving. The goal is to reconstruct the trajectory of a moving point-like scatterer from time-dependent measurements of the scattered field. To this end, we establish a rigorous point-interaction model for the moving scatterer in the time domain, proving well-posedness of the forward scattering problem, including existence, uniqueness and continuous dependence of the scattered field on the scatterer trajectory and scattering parameter. By exploiting the retarded-time structure, we introduce distance functions that connect emission, scattering, and observation processes, and show that they satisfy a coupled system of nonlinear ordinary differential equations determined by the measurement data. Based on this formulation, we propose a reconstruction algorithm that combines initial localization with the solution of the derived ODE system. Numerical experiments demonstrate that the method is accurate and robust with respect to noise, with reconstruction errors remaining stable under moderate perturbations.

This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of $\mathbb{R}^d$, $d\ge 1$. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.

In this article, we prove boundary observability estimates for wave equations on bounded domains $\Omega \subseteq \mathbb{R}^n$, with a critically singular potential that diverges as the inverse square distance to $\partial \Omega$. The result is applicable in all dimensions and for wave equations with general time-dependent lower order terms. The key geometric assumption is a convexity condition on $\Omega$. The main tool for our result is a global Carleman estimate that is carefully adapted to the critical singular nature of the potential.

We study the long-time behaviour of solutions to kinetic Fokker-Planck equations with power law confinement potentials and local equilibria with fat tails. Without relying on a priori moment bounds or perturbative regimes, we establish global estimates on weighted norms (main result) using two methods: a coupling of the weighted norms with entropy dissipation (DMS method) and a Foster-Lyapunov condition approach. As a consequence, we establish an entropy - entropy production inequality and the convergence rates to equilibrium in terms of the exponents associated to the growth of the spatial confinement and the tail decay of local equilibria. For sake of simplicity, we assume that stationary states are factorized.

This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.

We study the Neumann problem $u_t-\varepsilon\Delta u=e^u-1-au\ (a>1)$ on a smooth bounded domain $\Omega\subset\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $\xi_a$ is unstable, and solutions starting above $\xi_a$ blow up in finite time. Although finite-time blow-up persists at every diffusivity, we show that sufficiently large diffusion recovers this scalar trichotomy uniformly on every bounded $H^1$ ball, and that blow-up occurs precisely when the spatial mean crosses $\xi_a$. For initial data with $\|u_0\|_{H^1}\le R$ and spatial mean at most $\xi_a-\delta$, let $\varepsilon_{\mathrm{unif}}(R,\delta)$ denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove $\log \varepsilon_{\mathrm{unif}}(R,\delta)=R^2/(8\pi)+O(\log R)$ as $R\to\infty$. The domain-independent coefficient $1/(8\pi)$ arises from the sharp mean-zero Moser–Trudinger inequality. A matching lower bound is obtained from boundary-concentrating Moser profiles via a localized Kaplan argument.

We obtain weighted $L^2$ estimates for solutions to the Dirac equation with weights in the Kerman-Sawyer class. This class has been introduced as a necessary and sufficient condition for the boundedness of fractional integral operators in weighted $L^2$ spaces.

We study an irregular double obstacle problem with Orlicz growth on a bounded nonsmooth domain. The nonlinear operator is assumed to satisfy standard monotonicity and growth conditions, together with a small bounded mean oscillation condition in the spatial variable, while the domain is Reifenberg flat. We establish a global weighted Orlicz estimate for the gradient of the weak solution in terms of the nonhomogeneous datum, the gradients of the two obstacles, and the boundary datum.

We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.

The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an $N$-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [Commun. Math. Sci., 15(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio $\varepsilon$ and the mass ratio $\eta$. Furthermore, under the Vlasov-Navier-Stokes scaling assumption $O(1)\varepsilon^3 \leqslant \eta \leqslant o(1)\varepsilon^2$ as $\varepsilon \rightarrow 0$, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.

This paper studies the $C^1$ existence and dynamical stability of temporal periodic solutions involving strong transonic shocks for the quasi–one–dimensional full compressible Euler equations in diverging nozzles. We reveal the physical structural dissipation mechanism of strong transonic shocks hidden in the Rankine–Hugoniot conditions. This allows us to estimate the resonance at the shock boundary caused by the left acoustic waves in the subsonic downstream region. With this dissipation, we complete the proof by developing a fraction–step linearized iterative method.

In this paper, we firstly study Strichartz estimates $\|e^{it D^\alpha} u_0\|_{L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))}\leq C(d,\alpha,q,r,s)\|u_0\|_{\dot{H}^s}.$ We show some counterexamples for $(q,r) = (2,\infty)$. Then we consider the embedding $X^{s,b}_\alpha \hookrightarrow L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))$, where $\|u\|_{X_\alpha^{s,b}}:=\|\langle\xi\rangle^s\langle \tau-|\xi|^\alpha\rangle^b\hat{u}(\tau,\xi)\|_{L^2_{\tau,\xi}}$. We present the necessary and sufficient conditions for this embedding.

In this article, we first investigate the necessary and sufficient conditions on the ranges of $s,s_1,s_2\in R$, $1\leq p_1,p_2,q\leq \infty$, $\theta_1,\theta_2\geq 0$ for the validity of the fractional Gagliardo–Nirenberg inequalities $$\|D^su\|_{L^q(R^d)}\lesssim \|D^{s_1}u\|_{L^{p_1}(R^d)}^{\theta_1}\|D^{s_2}u\|_{L^{p_2}(R^d)}^{\theta_2}.$$ Secondly, we consider the anisotropic Gagliardo–Nirenberg inequalities $$\|u\|_{L^q(R^d)}\lesssim \|u\|_{L^{p_0}(R^d)}^{\theta_0} \prod_{j=1}^n\|D_{x_j}^{s_j}u\|_{L^{p_j}(R^d)}^{\theta_j}.$$ We show the sharp conditions of these inequalities except “one" case. Finally, we establish the profile decomposition associated with the above inequalities when $p_j = 2$, $0\leq j\leq n$. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.

We study the continuum scaling limit of the Langevin dynamics for three-dimensional U(1) lattice Yang–Mills theory. The model is defined on the discrete 3D torus with a general class of plaquette actions that are suitably normalized, including Wilson, Manton, and Villain actions. Under the weak-coupling scaling and in the DeTurck gauge, we prove that the rescaled logarithmic field converges to the solution of the one-form stochastic heat equation in probability. In particular, the limiting dynamics are universal and do not depend on the higher-order details of the plaquette action.

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose–Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

We study the bifurcation of two-dimensional Darcy-Bénard convection (DBC) in a rectangular domain, a canonical model for thermal convection in porous media with applications in geophysics and engineering. The momentum equation lacks advection and viscous dissipation, being regularized solely by a linear Darcy damping term. As a result, the linearized operator generates a semigroup that is neither analytic nor compact and the nonlinear term fails to be Lipschitz. The system is thus placed outside the scope of the standard center-manifold theorem. To overcome these obstructions, we develop a center-manifold reduction adapted to DBC system. Our main result is a constructive proof of the existence of the center manifold function h and local attractivity of the center manifold-the exponential convergence of small solutions toward it-without relying on analyticity of the full linear semigroup and Lipschitz nonlinearity. We circumvent these difficulties by exploiting the partially dissipative structure: the temperature equation is governed by the Laplacian, which generates an analytic semigroup and provides the smoothing needed to compensate for the lack of regularity in the velocity and the absence of global Lipschitz bounds. Through carefully designed inequalities, we establish both the construction of the center manifold function h and the exponential convergence of nearby solutions. Explicit approximations for the center manifold are derived in two scenarios-one simple eigenvalue and two distinct eigenvalues-yielding reduced systems of ordinary differential equations whose analysis determines the bifurcation type. Numerical simulations are presented to corroborate the theoretical results.

After Corollary 1.3 in Calculus of Variations and Partial Differential Equations 60 (2021), Paper 231, Djitte, Fall, and Weth asked whether, when $1<p<2$, a volume-constrained local minimizing domain for the best fractional Sobolev constant must still be a ball. This paper solves that problem. Let $0<s<1$, $1<p<2$, and let $\Omega\subset\mathbb{R}^N$ be a bounded $C^3$ domain. If $\Omega$ is a local minimizing domain for $\lambda_{s,p}(\Omega)=\inf{[u]_s^2:u\in\mathcal{H}*0^s(\Omega),\ |u|*{L^p(\Omega)}=1}$ under smooth volume-preserving deformations, then $\Omega$ is a ball. The proof first uses the fractional Hadamard formula to reduce shape minimality to the overdetermined boundary condition $u/\delta^s=C_0$. To overcome the moving-plane obstruction caused by the failure of $u^{p-1}$ to be Lipschitz at zero, we establish a weighted singular narrow-domain maximum principle whose absorption factor is exactly the $sp/N$ power of the measure of the negative set. The difficulty at a corner is resolved by a finite boundary expansion: setting $\rho=sp$, the boundary quotient is composed of finitely many constant-coefficient normal powers $\delta^{k\rho}$ and a $C^{1,\varepsilon}$ remainder; when $k\rho=1$, the unique resonant correction is $\delta\log\delta$. This expansion makes all lower-order normal terms on the two sides of an orthogonal corner cancel, thereby yielding the first-order tangential vanishing required by the moving-plane corner lemma. The method does not require the domain to be convex and covers the full range $0<s<1$ and $1<p<2$.

We study a two-species model for tissue growth in which both populations are transported by an external drift and by a velocity potential determined through Brinkman's law. The pressure is generated by a stiff constitutive relation depending on the total density. Our main result establishes the incompressible limit as the stiffness exponent tends to infinity, in arbitrary space dimension and for merely integrable initial data. The limit system consists of the two balance laws coupled to Brinkman's equation, the hard-congestion constraint $0\leq n_\infty\leq 1$, the graph relation $p_\infty(1-n_\infty)=0$, and the corresponding complementarity relation. A key point of the analysis is a new $L^2$-based compactness theory that avoids both uniform $L^\infty$-bounds on the pressure and the kinetic reformulation used in earlier approaches. We first construct weak solutions for bounded data and then remove the boundedness assumption by means of a weighted compactness argument inspired by Bresch–Jabin. We also prove an Aubin–Lions–Simon type lemma based on oscillation control, yielding time continuity of the constructed solutions. Finally, a refined dissipation estimate for the Bresch–Jabin compactness functional gives strong compactness of the pressure in the stiff limit and implies a regularising effect: the limiting pressure is bounded even when the approximating pressures are only integrable.

We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamiltonian with real-analytic dispersion, a Fermi sea confined to a finite momentum range, and nonzero Fermi velocity at every Fermi point, we analyze a particular Abel-regularized transport response (obtained by tracing over the Fermi sea) and prove that its large-time limit equals the Euler characteristic. We derive an explicit finite-time formula, show that quantization requires the large-time limit, and obtain a convergence rate under additional nonstationarity assumptions. We also prove the lattice analog, where the sharp commutator is trace-class and a completely filled band contributes zero. Finite-volume calculations illustrate the prescribed order of the thermodynamic and large-time limits.

Motivated by a conjecture of Burq and Gérard, we investigate quantitative uniqueness principles for functions on $\mathbb T^2$ whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond $L^\infty$.

A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction.

In this paper, we study the dependence of the propagation speed on the diffusion ratio in a Lotka-volterra competition-diffusion system under strong competition. The model is known to admits a unique monotone travelling front connecting two exclusion equilibria and the sign of its speed determines which species invades the territory occupied by the other. We establish smoothness results for the wave speed and the wave profile. Our main result shows that the wave speed is a strictly decreasing function of the diffusion ratio on explicit unbounded regions of parameter space. This monotonicity had been observed numerically but, had not previously been proved. The proof combines the smooth dependence of both the travelling front and its speed on the diffusion ratio, an adjoint identity for the linearized operator and a maximum principle argument. At zero speed, we also prove that there exists a smooth threshold, we obtain an exact formula for its derivative and deduce its monotonicity. These threshold results also provide a complete characterization of the sign of the wave speed.

On the lattice graph $\mathbb{Z}^n$, we establish the existence of sign-preserving solutions to the Tzitzéica equation. We prove the existence of positive solutions and two classes of negative solutions under different assumptions, and derive their decay estimates. The proof is based on a suitable approximation scheme, the monotone convergence theorem, and an exhaustion argument. These results extend those of Hua, Huang, and Wang (Anal. PDE, 2026) by establishing the existence of both positive and negative sign-preserving solutions together with their decay estimates.

In this paper, we establish systematic estimates for the entropy and its density on Riemannian manifolds, focusing on the more challenging cases where the Ricci curvature changes sign or the boundary is nonconvex. For example, the refined second law of thermodynamics states that the entropy in a compact domain in $\mathbb R^n$ is increasing in time, furthermore, it is concave if the domain is convex. The concavity property is equivalent to the property that the Fisher information is decreasing, which also holds for convex domains in a Riemannian manifold with nonnegative Ricci curvature (cf. \cite{NiLei}). In view of the wide application of entropy in mathematics, information theory, physics, etc., there is certain desire in the community to extend the property to broader settings, especially to the case with nonconvex boundary (see e.g. \cite[p. 3]{CFM}), even in the Euclidean case. Here, we prove that the refined second law still holds if the domain is not too far from convex and the negative part of the Ricci curvature is not too large, in an explicit, nonperturbative sense, thus realizing some of the expectations. The proof is based on a new, sharp second order log Poincaré inequality that does not require explicit curvature conditions of the manifold. If the negative part of the Ricci curvature is too large, a counterexample to the concavity is given. Some other related estimates for the entropy density (Hamilton type estimates) are also proven.

We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost Kähler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.

Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint Hölder regularity and uniform convergence of the direction maps in the end-reversing case.

Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.

We study entire solutions of the Allen–Cahn equation $\Delta_g u = W'(u)$ on Cartan–Hadamard manifolds of the form $u = U\circ b$, where $b$ is a Busemann function and $U\in C^2(\mathbb{R})$; we call these Busemann profiles. Our main result is a rigidity principle requiring no homogeneity of the manifold: if $\inf \Delta_g b>0$ and $U$ has finite limits $\ell_\pm$, then $W(\ell_-)\le W(\ell_+)$, with equality exactly when $U$ is constant. In particular no nonconstant Busemann layer joins the two wells of a balanced double-well potential, in any dimension $n\ge 2$; under $K\le-\kappa_2<0$ the hypothesis is automatic by Hessian comparison, and if $W'$ is locally Lipschitz it holds under the mere nonvanishing of $\Delta_g b$. The mechanism is a dissipation identity for the Hamiltonian $\tfrac12(U')^2-W(U)$ along the flow lines of $\nabla b$, in which the mean curvature of the horospheres acts as a wave speed. Conversely, when $\Delta_g b\equiv c$ and $W$ is bistable and unbalanced, strictly increasing Busemann fronts joining the wells exist precisely when $c$ is the Fife–McLeod speed; their interfaces are then horospheres of constant mean curvature. On warped-line and rotational surfaces we test sharpness: monotone warpings admit no equal-level profile and rotational surfaces no radial two-well profile, while an even convex warping carries a nonconstant monotone layer whose nodal set is the central geodesic leaf.