Papers

For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 \rho = \boldsymbol{v} \cdot \nabla \rho$ (where $\rho \in C^\infty_c(\mathbb{R}^d)$) is essentially skew-adjoint on $L^2(\mathbb{R}^d)$; 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field $\boldsymbol{v}\colon \mathbb{R}^3 \to \mathbb{R}^3$ for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.

We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.

Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $G:U\to H$ is a linear bounded operator and $B:[0,T]\times H\to H_\beta$ is a locally $\theta$-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable $\theta\in(0,1)$. Here, $H_\beta$ is a Hilbert space which contains $H$ with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space $H$ into itself. The coefficient $\beta\geq0$ measures the order of such a singularity and the case $\beta=0$ corresponds to a drift term with values in $H$. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler–Bernoulli beam equation up to dimension $3$, even in the hyperbolic case.

Colding established a sharp gradient estimate for the Green function on manifolds with non-negative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely $\fint (1-|\nabla b|^2)\,dV$, controls the volume deficit in an optimal quantitative manner; here $b$ is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with $\ric\ge(n-1)g$ and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.

We study inverse problems for a time-harmonic one-photon model describing the interaction of a single photon with a medium of stationary two-level atoms. After time-harmonic reduction, the unknown compactly supported atomic density appears as a frequency-dependent potential in a scattering equation for the half Laplacian. We prove high-frequency uniqueness results for three types of intensity data: source-driven measurements, renormalized far-field intensity measurements, and phaseless far-field measurements obtained from coherent superpositions of incident plane waves. In each case, the corresponding data, given at all sufficiently large frequencies, determine the atomic density uniquely; the source-driven result requires a geometric visibility condition on the source and observation sets.

A critical quasilinear chemotaxis model with indirect signal production in space dimensions higher than two is studied under radial symmetry. Exploiting the variational structure of the system, the existence of a sharp mass threshold M\textsubscr{c} > 0 is established: all solutions with mass below M\textsubscr{c} are global and bounded, while those with mass above M\textsubscr{c} and sufficiently negative energy are global but blow up in infinite time. Moreover, the value of the critical mass M\textsubscr{c} is related to the best constant of a variant of a Hardy-Littlewood-Sobolev inequality.

We establish a non-Archimedean Caffarelli-Silvestre extension theory for the Vladimirov-Taibleson operator $D^s$ for $s>0$ and, more generally, for hierarchical Laplacians $L_C$. For every $f\in\mathcal{S}(\mathbb{Q}_p^n)$, we equip the Bruhat-tits tree $\mathcal{T}_{p^n}$ with the weight $w_{v_{k-1}v_k}=p^{k(s-n)}$ under the horocyclic coordinates. We prove that the resulting Dirichlet problem admits a unique bounded weighted-harmonic solution that is continuous on the end compactification. Its boundary traces converge to $f$ uniformly and in $L^2$, while the associated deformed normal derivative converges pointwise and in $L^2$ to $D^s f$. We derive explicit Fourier and Poisson representations of the extension and establish an energy identity between its weighted tree energy and the quadratic form of $D^s$. Motivated by $p$-adic AdS/CFT, we also give an equivalent massive formulation on the unweighted tree, based on a renormalized trace and a scale-corrected normal derivative. Finally, under natural local-finiteness and scaling assumptions, we extend the construction to hierarchical Laplacians $L_C$ on ultrametric spaces. We characterize the canonical edge conductances by requiring the cancellation Green operator on the ultrametric tree $\mathcal{T}_X$ to coincide with $L_C^{-1}$ on $\mathcal{S}_0(X)$. In the corresponding canonical flux class, the extension is unique, its Dirichlet-to-Neumann map is $L_C$, and it satisfies the associated energy identity. The Vladimirov-Taibleson construction is recovered as the homogeneous special case.

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $\eta\in(0,\infty)$ and ResNets $\Psi_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $\eta d^{\eta}\varepsilon^{-\eta}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $\xi>0$, the explicit bound $C_\xi d^{4+\xi}\varepsilon^{-(3+\xi)}$ on the number of parameters.

We prove the local Hadamard well-posedness of the higher-order nonlinear Schrödinger equation with negative dispersion coefficient on the half-line under nonzero Dirichlet and Neumann boundary data. The paper uncovers a striking feature of this model: unlike the positive-dispersion case studied in (Alk{\i}n, Mantzavinos, Ozsari, 2024), well-posedness requires two boundary conditions rather than one. This reflects a structural limitation in the underlying global relation, which allows elimination of at most one unknown boundary trace. Using the unified transform of Fokas, we derive an explicit solution formula for the associated forced linear problem and establish sharp Sobolev and Strichartz estimates on the half-line. These linear estimates, together with corresponding Cauchy problem bounds, yield local well-posedness in both high- and low-regularity Sobolev spaces, as well as local Lipschitz continuity of the solution map.

We consider the Schrödinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schrödinger equation with Gaussian fluctuations. We propose a stochastic Schrödinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh–Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatorname{Spin}(7)$ analogue of Theorem 10.1 of Chodosh–Gianocca: a complete two-sided minimal immersion $M^7\to\mathbb{R}^8$ of index one and finite total curvature is a higher-dimensional catenoid.

We establish the existence and uniqueness theorem for the exterior Dirichlet problem for the special Lagrangian equation with prescribed asymptotic behavior at infinity, in both the viscosity setting for all the phases and classical setting for the critical and supercritical phases. These results generalize previous work by the second author by removing restrictive assumptions on the asymptotic matrix and improving the decay rate to the order $2-n$. We also solve the interior Dirichlet problem for the critical special Lagrangian equation and, as applications, all the above-mentioned corresponding problems for the three dimensional quadratic Hessian equation without any admissibility condition.

We consider a vegetation-rainfall-bushfire interaction model consisting of two partial differential equations describing the evolutions of bushfire intensity and water availability, together with an ordinary differential equation governing the vegetation density. The main feature of the model is that the ODE contains a nonlocal coefficient multiplying the vegetation density, which makes the mathematical analysis highly nontrivial. In this paper, we regard this nonlocal coefficient as the generator of a deformation of the metric structure of the underlying real Hilbert space. This interpretation enables us to formulate the ODE as a twisted gradient flow on state-dependent Hilbert spaces and to establish the existence of strong solutions to the associated initial-boundary value problem.

In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in all three dimensional regimes. If $2\leq k<\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\partial\Omega$. If $k=\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$; if $k>\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$. In each case, for a positive constant $c$, $|Du|=c>0$ on $\partial\Omega$ forces $\Omega$ to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter $q=(n-k)/k$ and the scale function \[ F_q(t)=\frac{t^{1-q}-1}{1-q},\qquad F_1(t)=\log t. \] After normalization, $U=F_q(v)$ and $M_q[v]=vD^2v-qDv\otimes Dv$ satisfies $\sigma_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $|Dv|\le b$;a boundary viscosity contact then yields $H_k\ge qbH_{k-1}$. The integral mechanism that closes the argument changes at the critical exponent: Rellich–Pohozaev identities are used for $q>1$, a scale-invariant Wronskian–Newton current for $q=1$, and a renormalized Newton–Jacobi mass for $q<1$. When $q\leq 1,$ we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.

Let $(M,g,J)$ be a closed Kähler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler–Lagrange equations associated with the Beckner–Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and Kähler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of $(M,g)$.

J. Danskin proved in 1966 that the envelope of a family of continuous functions $\mathrm F : \mathbb R^n \times \mathrm K \longrightarrow \mathbb R$, parameterised by the points of a compact $\mathrm K \subseteq \mathbb R^m$, given by \[ f(u) := \max_{k \in \mathrm K} \mathrm F(u,k), \ \ \ \ f \, : \, \mathbb R^n \longrightarrow \mathbb R, \] is semi-differentiable on $\mathbb R^n$, if $\mathrm F$ is sufficiently regular. This result is of utmost importance in numerous applications. Extensions have been proved towards almost every direction, but none permits to replace “max over $\mathrm K$" with “essential sup over $\mathrm K$". We establish the semi-differentiability of generalised $\mathrm L^\infty$ envelopes when $ \mathrm F$ is defined on the product of a Banach space with a measure space. As an application, we obtain a new foundational regularity result in the Calculus of Variations in $\mathrm L^\infty$, asserting that supremal functionals are semi-differentiable everywhere, with an explicit formula for their semi-derivative, despite generally being non-differentiable. Moreover, we establish a variational characterisation of (absolute) minimisers of general level-convex $\mathrm L^\infty$ functionals via their semi-differentials, revealing the genuine $\mathrm L^\infty$ counterpart of the Euler-Lagrange equations. Conventional Aronsson equations and related divergence PDE involving measures, deducible from $\mathrm L^p$ approximations as $p\to \infty$, in general are not sufficient for minimality in $\mathrm L^\infty$. Our results dislodge the trademark necessity for extrinsic $\mathrm L^p$-approximations as $p\to \infty$ to derive and study PDEs, reinterpreting the existing $\mathrm L^\infty$ theory. Most crucially, they provide a new powerful intrinsic $\mathrm L^\infty$ framework, evocative of the straightforward variational methods for integral functionals.

Energy-based analysis of the three-dimensional incompressible Navier–Stokes equations with spatially localized dissipation is developed at the level of kinetic energy and Leray–Hopf weak solutions. A localized coercivity property linking viscosity and damping to global $L^2$ control is introduced and related to observability and spectral inequalities for the associated Stokes operator. Under this condition, exponential energy decay is established without regularity or linearization assumptions, and time-dependent damping is treated within a nonautonomous framework. A finite-dimensional spectral quantity is proposed as a computable proxy for coercivity, yielding explicit decay estimates for reduced-order models.

Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical–residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.

We prove the existence of multi-peak solutions for singularly perturbed elliptic problems in the so-called zero-mass case by applying the classical Lyapunov-Schmidt reduction method. The goal is to construct solutions concentrating around several distinct interior points of the domain, identified as nondegenerate critical points of a potential.

We consider the following chemotaxis-growth system, featuring density-suppressed motility and a $\theta$-logistic growth term that incorporates a strong Allee effect: \begin{equation*} \left\{ \begin{aligned} u_t &= \Delta(\gamma(v)u) + \mu u(1-u^\theta)(u-A), & x \in \Omega, \quad t > 0, v_t &= \Delta v - v+ w^\beta, & x \in \Omega,\quad t > 0, w_t &= -\delta w +u, & x \in \Omega, \quad t > 0, \end{aligned} \right. \end{equation*} subject to homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^d (d\geq 2)$ with smooth boundary. Here $\mu\in \mathbb{R} , \delta,\beta>0, \theta\geq1$, and the positive motility function $\gamma(v) \in C^3([0,\infty))$ fulfills $\gamma'(v) \leq 0$ for all $v\geq 0$. The main objective of this paper is to establish the global existence and boundedness of classical solutions to the proposed problem. Our analysis combines the Schauder fixed point theorem for proving local-in-time existence with the extensibility criterion, fundamental energy estimates, $L^p-$bounds, the Gagliardo-Nirenberg interpolation inequality, Young's inequality, semigroup estimates, and a Moser-type iteration scheme to derive uniform-in-time $L^{\infty}-$bounds, thereby ensuring global bounded classical solutions. More precisely, we establish that the chemotaxis-growth system admits a unique globally bounded classical solution whenever $\beta < \frac{2(\theta+2)}{d}$. Furthermore, if the logarithmic derivative of the motility function is assumed to be uniformly bounded on $[0,\infty)$, that is, $\frac{\gamma'(v)}{\gamma(v)} \in L^{\infty}([0,\infty))$, then the above restriction on $\beta$ can be weakened to $\beta < \frac{d(\theta+1)+2(\theta+2)}{2d}$, provided that $\theta > \frac{4-d}{d-2}$.

We study open sets of nearly plane symmetric data for the $3D$ compressible Euler equations with dynamic entropy and non-trivial vorticity. In our prior work, we proved that the solutions develop a gradient singularity along a singular boundary, which is a hypersurface that emanates from a co-dimension 2, spacelike submanifold called the crease. In the present paper, we prove that a null hypersurface, called a Cauchy horizon, emanates from the crease, and propagates in a direction transverse to the singular boundary. The solution remains smooth up to the Cauchy horizon, even though it “feels” the influence of the gradient singularity at the crease. The union of the Cauchy horizon, the crease, and the singular boundary, make up a connected portion of the boundary of a maximal globally hyperbolic development (MGHD) of the data. Roughly, an MGHD is the “largest” way that smooth data can evolve into a classical solution. Known examples show that the only way one can guarantee uniqueness of an MGHD is by proving that certain structural properties hold along its entire boundary. We prove that a local version of the needed properties are satisfied along the Cauchy horizon and singular boundary. Our work provides the first description of the formation, structure, and stability of an $O(1)$-size portion of the Cauchy horizon in $3D$ without symmetry assumptions. The presence of dynamic entropy and vorticity stretching, both of which are absent in simpler settings such as $2D$ isentropic Euler, introduces significant difficulties that we resolve with novel techniques. Our approach relies on new foliations of spacetime dynamically adapted to the shape of the crease, and a PDE framework based on double-null foliations and the energy identities we derived in earlier work. Collectively, our techniques allow us to handle a challenging new solution regime where the crease lacks strict convexity.

We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. We consider both the nonresonant and resonant cases. Our approach is based on the Conley index theory applied to study invariant sets of the associated parabolic semiflow. Using properties of the Schrodinger semigroup with Kato-class potentials, which follow from its Feynman-Kac representation, we derive a priori estimates for bounded solutions in the $L^\infty$ and $L^2$ norms, as well as regularity bounds in Sobolev spaces. As a consequence, we obtain conditions ensuring the existence of connecting orbits between stationary solutions, which in turn yield the existence of nontrivial standing waves.

We establish convergence rates of solutions to traveling waves for a reaction diffusion model with nonlinear advection. The convergence rate is proven using the shape defect function. We introduce a novel energy methods-based approach that uses a weighted Nash inequality to obtain $L^\infty$-estimates on the shape defect function. In determining the convergence rates, we also establish the front location for the model.

For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax–Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of $\phi$, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity $R_\phi$, which measures the $C^{1,1}$ deviation of $\phi$, and prove the quantitative relation $R_\phi(x)\sim 1/\tau_{\phi,H}(x)$ with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton–Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the $C^{1,1}$ support, and the degree of regularity.

We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{\alpha_k}, $$ where $c_k, \alpha_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $\alpha_k$. For linear combinations of square roots of positive quadratic polynomials on $\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $\Delta u + \lambda u = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\mathbb{R}^3\setminus\{(x,0,0)\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.

Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from partial incompressible flow observations using continuous data assimilation (CDA). We propose a simple and efficient parameter recovery algorithm and also a nonlinear solver for CDA-NSE. Together, this creates a highly efficient technique for recovering an unknown viscosity from partial solution data. Our analysis establishes the well-posedness of steady CDA-NSE, quadratic convergence of the parameter recovery algorithm, and quadratic convergence of a CDA-Picard + CDA-Newton nonlinear solver. Numerical experiments illustrate that the methods are very effective in restoring parameters quickly, even with poor initial guesses.

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation $$ \vartheta_{tt}+\gamma \vartheta_t-\Delta\vartheta+|\vartheta|^{p-2}\vartheta=0 $$ on a smooth bounded domain $\varOmega\subset\mathbb{R}^d$, where the evolution is projected onto the tangent space of the Hilbert manifold $$\mathcal{M} = \left\{ \vartheta\in L^2(\varOmega):\|\vartheta\|_{L^2(\varOmega)}=1 \right\}, $$ which is the unit sphere in $L^2(\varOmega)$. We assume that $$p\in[2,\infty)\ \text{ for }\ d=1,2, \ \text{ while }\ 2\leq p\leq \frac{2(d-1)}{d-2} \ \text{ for }\ d\geq3.$$ By employing the Faedo–Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold $\mathcal{M}$ is invariant under the flow, and hence the $L^2$-constraint is preserved throughout the evolution. Using the Lusternik–Schnirelmann theory, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking Webb's theorem and Barbalat's lemma. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in $H_0^1(\varOmega)$ to the unique positive ground-state solution.

In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_{0}, u_{0t})=(u, u_{t})|_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_{t})$ in $ W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.

We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace–Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya–Szegő conjecture, which asserts that the regular $n$-gon minimizes the first Laplace–Dirichlet eigenvalue among $n$-gons of equal area. The proof uses a Rayleigh–Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh–Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.