Papers

Inverse problem to determine simultaneously a general space- and time-dependent source and an initial state in a fractional diffusion equation from an a posteriori measurement of the normal derivative of the state on a portion of a boundary of the space domain is considered. Uniqueness for this problem is proved under the assumption that an order of a fractional derivative involved in the equation is irrational. The uniqueness result is generalized to an inverse problem for a coupled system of fractional diffusion equations, where both sources and initial states are unknown and first component of the state is measured on the boundary.

We consider the transmission problem in presence of interfaces with imperfect bonding. The imperfect bonding condition is characterized by the positive resistance along the interface, which causes discontinuity of the potential across the interface while the flux is continuous. If the interface resistance is zero, then the interface is of perfect bonding, where both the potential and the flux of the solution are continuous across the interface. In this paper, we first construct using layer potentials the solution to the transmission problem with imperfect interfaces. We then prove that the solutions converge in various Sobolev spaces to the solution to the transmission problem with perfect interfaces as the interface resistance tends to zero. In particular, it is shown that the gradient of the solution converges in the uniform norm if the boundary is sufficiently regular.

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted weak type estimate for both full and lacunary fractional spherical maximal functions at the boundary of the maximal $L^p-L^q$ bounded regions.

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First, we obtain a new lower bound for the first nontrivial eigenvalue of the $p$-Laplacian, which improves upon existing estimates in certain parameter regimes and for thin domains. Second, we prove an existence theorem for nonlinear boundary value problems of the form \[ -\Delta_p u = \lambda f(u) \quad \text{in } \Omega, \qquad u=0 \quad \text{on } \partial \Omega, \] with $f$ nonnegative, continuous and nondecreasing. A striking consequence is the emergence of a stabilization phenomenon: for every such nonlinearity there exists a threshold $p_0 \colon = p_0(f,\lambda,\Omega)$ such that for all $p \geq p_0$ solutions exist. To our knowledge, this stabilization effect with respect to $p$, that apparently has not been observed before, suggests a connection to the $\infty$-Laplacian.

We study the resonance spectrum of the multiflow induced on a flag manifold by the action, through multiplication by the exponential map, of the Cartan subalgebra of the underlying Lie group. We give a definition of joint resonance for the flow, then prove its discreteness and existence of resonant states. We conclude by explicit characterization of the spectrum in the special cases of Projective spaces and manifolds of full flags.

We study positive solutions of the superlinear Lane–Emden inequality \[ -\Delta u\ge \sigma u^q,\qquad q>1, \] on infinite locally finite weighted graphs and connected domains. When the Dirichlet Green function is finite, the existence of a positive solution is equivalent to \[ G_\Omega\bigl(\sigma g_\Omega(o,\cdot)^q\bigr)(x) \le C g_\Omega(o,x) \] for some pole \(o\in\Omega\). Under Green function estimates, this yields sharp existence criteria and the Serrin-type exponents on \(\mathbb Z^d\) and orthant domains. For nonexistence, the principal method is flow decomposition. Its basic estimate bounds Green energy from below in terms of the relative capacities of intrinsic balls. %It yields annular conductance, capacity-to-infinity, and Nash–Williams cut-resistance criteria. For \(\sigma>0\), set \(\nu=\sigma\mu\). We show that if \(d_\rho\) is a complete \(\nu\)-adapted path metric and \[ \int_1^\infty \frac{r^{2q-1}}{\nu(B_{d_\rho}(o,r))^{q-1}}\,dr=\infty, \] then every nonnegative solution is identically zero. The proof combines a flow decomposition of the acyclic Green current, a pathwise Hardy estimate, and a relative capacity estimate. It requires none of (VD), (PI), (P$_0$), or the (3G) condition.

Using the recent work of Brendle–Wang on the Riemannian positive mass theorem, we prove the spacetime positive mass theorem for asymptotically flat and asymptotically hyperboloidal initial data sets in arbitrary dimensions.

The existence of global weak solutions to the compressible Navier-Stokes equations for the density of endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, is established in a three-dimensional torus for energy-finite initial data. The coupling of the equations arises through the chemotaxis force, which contributes to the momentum balance equation, and the signal production due to the cells in the chemotaxis equation. The equations model the self-assembly of endothelial cells during the early stages of blood vessel formation. The existence result holds for adiabatic pressure exponents $\gamma>4/3$, matching the exponent found in the existence analysis for the degenerate Keller-Segel equations. The proof leverages an approximation via Korteweg and drag terms, the BD entropy inequality, and a construction of weak solutions that are renormalized in the velocity variable.

The aim of this paper is to establish multiple positive normalized solutions $(u,v,\lambda_1,\lambda_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schrödinger system involving Sobolev critical exponent: $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha|u|^{\alpha-2}u|v|^\beta, x\in \mathbb{R}^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta|v|^{\beta-2}v|u|^\alpha, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $\mu_1,\mu_2, \nu, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q<2+\frac{4}{N}, \alpha>1, \beta>1$, and $\alpha+\beta=2^*:=\frac{2N}{N-2}$. For sufficiently small $\nu>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [J. Funct. Anal., 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extend the result of Gou and Jeanjean [Nonlinearity, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis “either $p,q\leq \alpha+\beta-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $\nu$, and the limiting profiles for $\nu\rightarrow 0^+$.

We study the nonlinear generalized heat equation $C(u)u_t=\frac{1}{z^{\nu}}\left(K(u)z^{\nu}u_z\right)_z$, where $C(u)$ and $K(u)$ are temperature-dependent thermal coefficients and $\nu>0$ is a geometric parameter describing the varying cross-section geometry. By applying the classical Lie symmetry method, we derive the determining equations and perform a complete classification of the admitted Lie point symmetries according to the functional dependence between $C(u)$ and $K(u)$. The analysis shows that the symmetry structure splits naturally into two principal cases: $C(u)/K(u)$ non-constant and $C(u)/K(u)=\beta$ constant. In the first case, only the basic symmetries are admitted for arbitrary coefficients, whereas additional generators appear under special compatibility relations. In the second case, the equation can be transformed to a linear radial heat equation by the substitution $v=\int K(u)du$, yielding an extended symmetry algebra. For each case, we construct the infinitesimal generators, commutator tables, one-parameter transformation groups, and corresponding invariant reductions. Invariant and similarity solutions are obtained and then specialized to several physically relevant subclasses, including power-law, exponential-type, and linear constitutive coefficients. The results provide a unified symmetry-based model for the analysis of generalized nonlinear heat equations in non-Cartesian geometries.

We construct transported PDEs on self-similar fractal domains from reference equations posed on the unit interval, and derive explicit self-similar interacting particle systems that approximate the resulting dynamics. The construction combines a measure-preserving isometry between $L^2$-spaces on $[0,1]$ and on the fractal \cite{Med2026}, a nonlocal-to-local approximation of differential operators \cite{PauTre2025}, and a Galerkin discretization on the canonical self-similar partitions. This yields a two-parameter approximation scheme whose error separates a nonlocal consistency term from a Galerkin network term. We work out the transport, Burgers, and heat equations, discuss the relation with intrinsic operators on fractals, and outline extensions to local charts and to pullbacks of nonlocal equations on fractal domains. Moreover, the reverse mapping transforms a nonlocal evolution equation on a fractal domains into the evolution equation on the unit interval, where the methods of classical numerical analysis can be applied. This suggests a promising direction for the development of numerical methods for nonlocal models on fractals, including fractional heat equation, fractal scattering, and related models.

Via a new inequality à la Gagliardo–Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-\Delta)^s u + \frac{\mu}{|y|^{2s}} u + \lambda u = f(u), \quad \mathbb{R}^N \ni x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = \rho \end{cases} \end{equation*} in the mass-critical and mass-subcritical regimes, where $s>0$, $N \ge K \ge 2$, $\mu \in \mathbb{R}$ belongs to a specific range, $\rho>0$ is given a priori, and $\lambda \in \mathbb{R}$ is unknown. Additionally, we obtain similar results for the problem above with $\mu=0$ and $N \ge 1$ as well as a related curl-curl equation. Finally, we provide a thorough insight into the threshold for $\rho$ that divides the scenarios of negative and zero least energy.

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions.

The article is devoted to the solvability of a system of integro-differential equations in the case of the difference of the standard Laplacian and the bi-Laplacian in the diffusion terms. The proof of the existence of solutions is based on a fixed point technique. We use the solvability conditions for the elliptic operators without the Fredholm property in unbounded domains.

We study the fast reaction limit for a two-component reaction-diffusion system with asymmetric reaction terms, where only one component diffuses. For nonnegative and mutually segregated initial data, we prove that the initial interface vanishes instantaneously. More precisely, the diffusive component converges uniformly to the solution of the heat equation, while the non-diffusive component vanishes away from the initial time. The proof is based on explicit barriers and a comparison argument, and applies under both Dirichlet and Neumann boundary conditions.

We study local regularity properties of solutions to stationary anisotropic magnetic Schrödinger equations in $\mathbb{R}^d$, $d \ge 2$, arising from singular magnetic potentials concentrated along manifolds of general codimension $2 \le n \le d$. The magnetic interaction is modeled through a covariant gradient of the form \[ \nabla_m u = (iM\nabla + A)u, \] where $M^T M$ is a uniformly elliptic matrix encoding anisotropy and $A$ is a magnetic potential with critical Hardy-type scaling along the $n$-codimensional singular set $\Sigma_0$; that is, $A\sim \mathrm{dist}(\cdot,\Sigma_0)^{-1}$. We establish local Hölder $C^{0,\alpha}$ and Schauder $C^{1,\alpha}$ estimates for weak solutions via a blow-up analysis adapted to the magnetic structure. The regularity is deeply influenced by the combined effect of anisotropy and the singular magnetic potential, which determines the spectrum of the limiting spherical Laplace-Beltrami operator arising in the blow-up at the singular set. Our model is motivated by the study of magnetic potentials generated by shrinking solenoids onto an axis $\Sigma_0$, in the three-dimensional setting $d=3$, $n=2$, leading to Aharonov-Bohm-type (AB) models. In this framework, we show that the geometry of the solenoidal loops plays a crucial role: in particular, any deviation from planar cross-sections orthogonal to $\Sigma_0$ induces a twofold effect. On the one hand, it breaks the ideal AB configuration, in the sense that the magnetic field outside the solenoid is no longer vanishing. On the other hand, it yields an unexpected regularizing mechanism on the wave functions, through a positive shift in the eigenvalues of the asymptotic spectral problem. This purely three-dimensional effect is consistent with our $C^{1,\alpha}$ regularity.

We study the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians on the flat torus. Starting from the discrete Lax–Oleinik equation, we introduce natural discrete analogues of these sets and analyze their convergence, as the time step tends to zero, in the sense of Kuratowski. Our results show that the semi-discrete variational framework captures not only the ergodic constant, but also the minimizing invariant geometry of the continuous dynamics. In full generality, we prove upper Kuratowski limit inclusions for both the Aubry and Mather sets. For the Aubry set, we establish full convergence under a hyperbolicity assumption on the continuous Aubry set. For the Mather set, we prove full convergence under a genericity assumption ensuring that the Lagrangian admits finitely many ergodic Mather measures. This provides a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets in the Tonelli setting, and clarifies how discrete variational models recover the central geometric objects of weak KAM and Aubry–Mather theory.

We establish the approximate controllability in $L^2$ for the nonlinear Benjamin-Ono equation on torus via two-dimensional control input. Our proof is based on adaptations of geometric control approach introduced by Agrachev and Sarychev. As an application of this control result, we study long-time dynamics of a randomly forced equation. It is proved that the trajectories are unbounded in Sobolev norms almost surely, when the random force is nondegenerate and statistically periodic in time.

We study a distributed optimal control problem for a nonisothermal Caginalp-type phase-field model that describes tumour growth under thermal therapy. The PDE system couples a possibly viscous Cahn-Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. Chemotaxis and active transport effects are taken into account, and hyperthermia appears as a control variable. We introduce a suitable tracking-type cost functional and show the existence of optimal controls. Then, we analyse the differentiability of the control-to-state operator and establish necessary first-order conditions expressed through a variational inequality involving the adjoint state variables.

We present QED, an open-source multi-agent system that turns human-provided research questions into complete mathematical proofs without further human guidance. Its pipeline is designed to overcome common failures of single-query proof generation by separating planning, proving, and verification: a decomposition agent structures the proof search, prover agents generate candidate arguments, and verifier agents check correctness. In collaboration with domain experts, we evaluated QED on 18 research-level projects of varying difficulty. QED produced five original works across algebraic geometry, fluid PDEs, probability, and inverse problems. Expert assessments regard these works as solid specialized research contributions, with three comparable in difficulty and scope to work commonly published in established specialist mathematics venues. QED is released at https://github.com/proofQED/QED.

Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. 1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional curvature has upper bound $\frac{1}{2(k-1)}$, we prove that the Ricci shrinker is isometric to a finite quotient of $\mathbb{R}^{n-k}\times \mathbb{S}^k$. 2. If $M$ has constant scalar curvature $R=\frac{n-2}{2}$, and each level set of $f$ has vanishing Weyl curvature, we prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^{n-2}$. This can be seen a generalization of Cheng-Zhou's four dimensional result \cite{Cheng-Zhou} to high dimension, since the level set of the potential function $f$ has vanishing Weyl curvature automatically when $n=4$.

We present a survey on multiplicity results for the Allen–Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through $\Gamma$-convergence and isoperimetric theory. In the scalar case, the Allen–Cahn functional converges to perimeter, giving rise to minimal and constant-mean-curvature hypersurfaces, while vectorial Allen–Cahn systems lead to multi-phase isoperimetric clusters. The main methodological tool discussed is the photography method, a variational-topological approach based on localized approximate solutions and barycenter maps, which enables one to encode the topology of the ambient manifold into multiplicity results. We compare problems posed on closed manifolds with those on manifolds with boundary, describing the distinct geometric effects induced by Neumann and Dirichlet boundary conditions. The survey highlights both the effectiveness and the limitations of this framework, particularly in the vectorial case, where the lack of a full classification of isoperimetric clusters creates fundamental analytical challenges.

We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.

We prove that the branching set of a solution to a two-dimensional two-phase Bernoulli problem with constant coefficients is locally finite. We do this via a Weierstrass representation formula, which allows to transform the problem into a new geometric two-phase problem for capillary minimal surfaces. We also apply this method to the obstacle problem establishing a connection between the directional derivatives of solutions to the obstacle problem and the linear thin two-membrane problem.

In this paper, we consider the semilinear damped wave equation with Hartree-type nonlinearity $\mathcal{I}_\gamma\left(|u|^{p_1}\right)|u|^{p_2}$, where $0<\gamma<n$, with initial data possessing additional negative regularity in $\dot{B}_{2, \infty}^{-\beta}$, $\beta \in [0,n/2)$. We first establish decay estimates for the corresponding linear equation in homogeneous Besov spaces $\dot{B}_{q,r}^{s}$. A key point is that the low-frequency part of the data is measured only in $\dot{B}_{m,\infty}^{-\beta}$, while the heat-like behavior yields the required $\ell^r$ summability under the condition $s+\beta+n(1/m-1/q)>0$. This formulation is particularly suited to the nonlinear analysis. We then establish the existence of a unique global mild solution for sufficiently small initial data whenever $$ p_1+p_2\geq 1+\frac{4+2\gamma}{n+2\beta}, $$ under the remaining admissibility conditions stated in the existence theorem. In particular, the critical case is covered whenever the critical line satisfies these conditions. Conversely, for initial data satisfying an explicit positive lower bound, the test-function method rules out global weak solutions when $$ 2<p_1+p_2<1+\frac{4+2\gamma}{n+2\beta}. $$ Thus, whenever the critical line is admissible and the subcritical interval is nonempty, $1+(4+2\gamma)/(n+2\beta)$ gives a Fujita-type threshold for the sum $p_1+p_2$.

In 1965 Kraichnan pointed out that a sufficiently strong background magnetic field, i.e. the case of small Alfvén number, will reduce the nonlinear interaction and inhibit the formation of strong gradients in the magnetohydrodynamic (abbr. MHD) system with $\mu=\nu\geqslant 0$, where ${\mu}$ and $\nu $ are the coefficients of kinematic viscosity and resistivity resp.. This means that the MHD system with ${\mu}=\nu\geqslant 0$ admits global-in-time large perturbation solutions with small Alfvén numbers. The existence of such large perturbation solutions was first mathematically verified in Hölder spaces by Bardos–Sulem–Sulem for the case ${\mu}=\nu= 0$ in 1988, and in Sobolev spaces by Cai–Cui–Jiang–Liu for the case ${\mu}=\nu> 0$ recently. In this paper, we further found a similar result for the general case “${\mu}>0$ and $\nu>0$", and provide a rigorous proof by developing a new approach, which includes a key bilinear estimate for dealing with the nonlinear interaction terms. Moreover both additional results for the vanishing behavior of the nonlinear interaction and the small Alfvén number limit of solutions are also established.

We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss–Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of \(\nabla u\), rather than by a sign condition on \(u\). The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up.

Since the pioneering work of Korteweg (1901) and the subsequent refinement of capillary fluid models by Dunn and Serrin (1985), the global existence of strong solutions to the multi-dimensional compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data has stood as a formidable open problem in fluid mechanics. This challenge was recently addressed by [Gu-Huang-Meng-Zhou, arXiv:2603.11762], who established the global existence of strong solutions for arbitrarily large initial data on the periodic domain $\mathbb{T}^N$ ($N=2,3$), provided that the viscosity coefficients satisfy a BD-type algebraic relation ($\mu(\rho) = \nu \rho^\alpha, \lambda(\rho) = 2\nu(\alpha-1)\rho^\alpha$) and the Korteweg stress tensor complies with a generalized Bohm identity ($\kappa(\rho) = \varepsilon^2 \alpha^2 \rho^{2\alpha-3}$). However, the existence of global strong solutions for the Cauchy problem under these conditions has remained an open question. In this paper, we resolve this problem by proving the global existence of strong solutions for the Cauchy problem ($\mathbb{R}^N$, $N=2,3$) with arbitrarily large initial data and non-vacuum far-field density. By employing a refined truncation analysis combined with an original modified Nash-Moser type iteration scheme, we overcome the difficulties arising from the lack of integrability for the density in the whole space. This result extends the large-data theory of compressible Navier-Stokes-Korteweg equations from bounded torus $\mathbb{T}^N$ to unbounded whole space $\mathbb{R}^N$, thus applicable to more general physical settings.

In this paper, we are concerned with differential inequalities with $(p,q)$-Laplacian operator on Riemannian manifolds. Using a test function argument, we establish Liouville-type theorems under the manifold's geometry and the potential's behavior at infinity.

Gaussian basis functions provide an efficient and flexible alternative to spline activations in KANs. In this work, we introduce the partition-of-unity Gaussian KAN (PU-GKAN), a Shepard-type normalized Gaussian KAN in which the Gaussian basis values on each edge are divided by their local sum over fixed centers. This produces a partition-of-unity feature map with trainable coefficients, while preserving the standard edge-based KAN structure. The normalized construction gives exact constant reproduction at the edge level and admits an explicit finite-feature kernel interpretation. We formulate both the standard Gaussian KAN (GKAN) and PU-GKAN from a finite-feature and additive-kernel viewpoint, making the induced layer kernels and empirical feature matrices explicit. Using the first-layer feature matrix as the reference object, we adopt a practical scale-selection interval for \(\epsilon\), with the lower endpoint determined by adjacent-center overlap and the upper endpoint determined by a conservative conditioning threshold. Numerical experiments show that PU-GKAN reduces sensitivity to \(\epsilon\), improves validation accuracy for most smooth and moderately non-smooth targets, and gives more stable training behavior. The benefit persists across sample-size and center-number sweeps, higher-dimensional architectures, Matérn RBF bases, and physics-informed examples involving Helmholtz and wave equations. These results indicate that Shepard-type partition-of-unity normalization is a simple and effective stabilization mechanism for RBF-based KANs.