Papers

We study finite-time acoustic wave tracking using effective interior Dirac-type actuators generated by clusters of subwavelength resonant bubbles driven from exterior transducers. Starting from a time-domain point-interaction approximation, the full scattering problem is reduced to a finite-dimensional delayed system of interacting retarded monopoles. After localization to a bounded observation domain, we formulate the tracking problem on a prescribed finite-dimensional Dirichlet spectral subspace. At the ideal level, a full-rank spatial coupling condition yields exact tracking of arbitrary sufficiently smooth compatible trajectories in this subspace by interior point sources. These ideal actuators are realized using small, high-contrast bubbles, whose Laplace-domain transfer operator encodes the Minnaert resonances and collective attenuation. The principal mode of each cluster is isolated by the Perron–Frobenius structure and the intra-cluster spectral gap. Clusters tuned to the same band are then treated as a collective resonant block. The selected Dirichlet bands are matched with the corresponding Minnaert bands. For prescribed slowly time-modulated resonant pressure profiles, with sufficiently flat initialization, we construct causal exterior controls and obtain a quantitative tracking estimate on every fixed time interval $[0,T]$. In contrast, on frequency ranges uniformly separated from the Minnaert resonances, the actuator gain is $O(\varepsilon)$ and the corresponding realization cost is asymptotically unbounded as $\varepsilon \ll 1$, $\varepsilon$ being the maximum radius of the bubbles. Thus the Minnaert resonances convert accessible exterior excitations into robust effective interior controls.

We consider the one dimensional space-periodic Vlasov-Poisson equations and construct, close to symmetric flat velocity strips, small amplitude traveling quasi-periodic electron-layers, namely strip-shaped patches of electrons in the phase space. These solutions are found for most values of the strip area. The proof uses a Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. Thanks to a suitable linear transformation of the unknowns, we reveal a connection between the electron patch problem and the classical, physically relevant, electronic Euler-Poisson system with cubic pressure law. As a direct, non trivial consequence, we obtain small-amplitude quasi-periodic traveling waves for this model as well. To the best of our knowledge, this work provides the first rigorous example of construction of quasi-periodic solutions for kinetic models.

We study the zero-dispersion limit for a class of Korteweg–de Vries (KdV)-type initial-boundary value problems on the half-line, with Dirichlet boundary conditions assigned at \(x=0\). We focus on the outflow regime, where the solution of the limiting scalar conservation law does not attain the boundary condition imposed on the dispersive problem. We construct a boundary layer profile, depending on the fast variable, which is uniquely determined, through the associated stationary third-order boundary layer equation, by the mismatch between the boundary conditions, and by the exponential decay at infinity in the fast variable. Our main result shows that, under suitable regularity and compatibility assumptions on the data, the dispersive solution is well approximated by a WKB expansion given by the sum of the smooth solution of the conservation law and the boundary layer profile. In particular, we establish stability of the boundary layer profile by proving quantitative estimates for the remainder term in a weighted energy norm, and show that it converges to $0$ in $H^1$, uniformly in time and up to the lifespan of the smooth solution of the conservation law. The proof is based on the analysis of a linearized energy functional and does not rely on complete integrability or inverse scattering techniques. It applies to general fluxes and requires no smallness assumption on the amplitude of the boundary layer. To the best of our knowledge, this is the first stability result for boundary layers of KdV-type equation on the half line.

We study the pointwise convergence of solutions to the free Schrödinger equation with initial data in the Bessel potential spaces $L_s^p(\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1 \leq p < \infty$, and demonstrate via counterexamples that these indices are sharp for all $1 \leq p \leq 2$ in one dimension, as well as for $p=1$ or $p$ large enough in higher dimensions. The proofs rely on the high-dimensional stationary phase method.

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu |u|^{q-2}u +(I_{\alpha}*|u|^{p^*_{\alpha}})|u|^{p^*_{\alpha}-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & \tau. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p<N$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (\max\{0,N-2p\}, N)$, $p^*_{\alpha}=\frac{p}{2}\left(\frac{N+\alpha}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.

We investigate singular integral operators with variable Calderón–Zygmund kernels and their commutators with $VMO$ functions on Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in $L^\Phi$ under the standard $\Delta_2$ and $\nabla_2$ conditions on the Young function. The analysis combines decomposition techniques with weak-type estimates to derive the main operator bounds. As an application, these results provide a functional-analytic framework for establishing a priori estimates and proving the interior regularity of solutions to higher-order elliptic equations and systems with discontinuous coefficients.

We study the Laplace equation posed in the unbounded rectangular domain $\Pi = I \times (0,\infty)$ with $I= (0,2\pi)$, and subject to nonlocal boundary conditions on $\partial \Pi$ in the trace sense. The analysis is carried out in the Bochner-Sobolev space $W^2_{p,1}(\Pi;X)$, associated with the Bochner space $L^{p,1}(\Pi;X)$, with $ p \in (1,\infty)$ and $X$ is a suitable Banach space. To solve the problem, we employ a generalized spectral method. In particular, we introduce the notion of $\otimes$-basis generated by tensor products and extend the classical scheme known from the scalar case to the present setting. Moreover, we prove that the system of root functions of the corresponding nonlocal spectral problem forms a $\otimes$-basis in $L^p(I;X)$.

We derive the focusing $\Phi^6_1$ measure on the torus $\mathbb{T}$ as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. The main difficulty in the focusing setting is to relate the classical mass cutoff to the quantum particle-number cutoff. Our result reaches the optimal mass threshold for the classical field identified by Oh, Sosoe, and Tolomeo (2022), and thereby extends the earlier work of Rout and Sohinger (2025). At the critical threshold, the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold, the same convergence holds with a polynomial dependence on the temperature. Moreover, we establish a quantum-level phase transition at the same mass threshold. The proof develops the variational framework of Lewin, Nam, and Rougerie (2015) in the focusing setting and relies on two new ingredients: a non-factorized trial state construction and a delicate tail estimate for the interacting lower symbol. These allow us to control the localization and relative entropy errors caused by the particle-number cutoff, as well as the contribution of the focusing exponential weight.

In this paper, we study the eigenvalue problem \[\left\{\begin{array}{cl}-\hbox{div}\left(a(x)\frac{Du}{|Du|}\right)=\Lambda\, b(x)\frac{u}{|u|} & \text{in }\Omega\\u=0 & \text{on }\partial\Omega,\end{array}\right.\] where $a(x)$ and $b(x)$ are suitable nonnegative functions. We prove that the first eigenvalue coincides with the weighted Cheeger constant. To see this identity, we analyze the behavior of the first Dirichlet eigenvalue of the weighted $p$-Laplacian operator as $p$ goes to $1$. In the case that the weight $a(x)$ is Lipschitz-continuous, we show that the limit of eigenvalues of the weighted $p$-Laplacian exists, and it is the weighted Cheeger constant. In addition, we check that the sequence of normalized $p$-eigenfunctions converges to the normalized eigenfunction of our limiting problem, which turns out to be bounded. For more general weights, we identify the first eigenvalue of the weighted 1-Laplacian operator with the weighted Cheeger constant and prove that the associated eigenfunction is bounded.

We study sharp weighted Sobolev-type inequalities of the form\[ \int_{0}^{1}|u(x)|\rho(x) \,\mathrm{d}x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \,\mathrm{d}x \Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \]where $\rho$ is a non-negative weight. We characterize the minimizers and identify the optimal constant $\Lambda(k,\rho)$ by showing that every minimizer has a constant sign and therefore solves a nonlinear eigenvalue problem of polyharmonic type. This yields an explicit characterization of extremizers for a broad class of weights. Moreover, we even provide with a an explicit computation of the optimal constant in terms of the weight function. The new weighted estimates turn to be very useful and, among other applications, allow us to recover several previous sharp estimates and Hardy type inequalities on finite intervals.

This paper establishes a rigorous functional analytic framework for weighted Weyl-Sonine fractional operators on semi-infinite intervals. While the classical Phillips functional calculus relies strictly on completely monotonic Bernstein functions to subordinate linear semigroups, we demonstrate that this structural restriction can be entirely bypassed. By exploiting the intrinsic topological decay induced by an admissible amnesia weight ($\omega$), we prove that the abstract integral representation converges absolutely for general classes of tempered Sonine kernels, thereby unifying the treatment of both diffusive relaxation and oscillatory non-local dynamics within a single framework. Employing the theory of strongly continuous semigroups, we characterize these operators as the infinitesimal generators of a deformed translation semigroup that simultaneously incorporates subjective aging scales ($\psi$) and history-attenuating amnesia effects. This similarity-based approach not only provides a sound abstract foundation for the hypersingular Marchaud representation of these operators but also guarantees their closedness and the analytical well-posedness of the associated abstract fractional Cauchy problems in general Banach spaces. Furthermore, we introduce the associated generalized fractional integral and prove the fundamental inversion theorem, demonstrating the complete analytical symmetry of the theory. Finally, we establish pointwise decay estimates for the solutions via weighted Sobolev embeddings.

Adopting the powerful methods introduced in \cite{li2021carnotcaratheodory, LZ2025}, we investigate the asymptotic behaviour at infinity for the heat kernel associated with the Grushin operator $\Delta_G = \Delta_x + |x|^2 \Delta_u$ on $ \mathbb{R}_x ^n \times \mathbb{R}_u ^{n'} $ for all $ n , \, n' \ge 1 $. We further establish sharp bounds for its spatial derivatives. As a by-product, precise estimates and small-time asymptotic behaviour for the Grushin heat kernel will be provided, as explicit as one can possibly hope for.

This paper develops a trace-regular variational framework for time-harmonic Maxwell scattering problems involving pointwise nonlinear boundary and interface responses. We investigate three canonical classes of models: nonlinear impedance, nonlinear perfect electric conductor, and nonlinear transmission conditions. Since the standard Maxwell tangential trace belongs to a space of negative order, the nonlinearities are formulated in refined functional spaces where the tangential electric field admits an $L^2(\Gamma)$-trace. Under the assumption of a sufficiently small Lipschitz constant for the nonlinear response, we establish the well-posedness of the direct problems via fixed-point arguments leveraging the mapping properties of the associated linear Maxwell operators. Within this framework, we perform a rigorous sensitivity analysis of the electromagnetic fields with respect to perturbations of the scattering interface. By employing the covariant Piola transform, we prove the continuity and Fréchet differentiability of the pulled-back solutions with respect to domain variations. The material derivative is characterized as the unique solution to a corresponding $\mathbb{R}$-linearized Maxwell system, and the shape derivative is shown to satisfy explicit boundary or interface conditions for each of the three nonlinear models. We further demonstrate that the resulting sensitivity expressions possess the Hadamard structure, depending exclusively on the normal component of the boundary deformation. The resulting derivative characterizations provide a mathematical basis for subsequent adjoint-based sensitivity analysis, shape optimization, and gradient-driven inverse reconstruction.

We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_{\theta}^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on \(\mathbb{T}_{\theta}^{n}\), we give local well-posedness, the blow-up alternative, persistence of higher regularity, and instantaneous smoothing in the Sobolev algebra scale \(H^k_\theta\), \(k>n/2\).

In this paper, we construct Poincaré-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal $\mathbb{S}^1$-bundle over $T^2$, with one Poincaré-Einstein end and one end asymptotic to a real or complex hyperbolic cusp. (2) Infinite families of Einstein metrics on $(0,\infty)\times P$, where $P\to \Sigma_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $\Sigma_{\mathtt{g}}$ of genus $\mathtt{g}\geq 2$, with one Poincaré-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $\Sigma_{\mathtt{g}}$. Universal covers of (1) and (2) provide new complete negative Einstein metrics on $\mathbb{R}^4$. These Einstein metrics also exhibit interesting degeneration phenomena. With this construction, we give a negative answer to a question of Anderson concerning cusp formation for Poincaré-Einstein 4-manifolds.

We present an epidemiological model for vector-borne diseases that includes within-host viral load and antibody dynamics using structured transport equations. By incorporating the internal dynamics into the infected and recovered host compartments, the formulation introduces nonlinearities and nonlocalities. We establish analytical properties, including well-posedness and mass conservation, and characterize its characteristic curves. Furthermore, we derive a simplified Uniform Host Response (UHR) model featuring delay-type terms. For both the full and UHR frameworks, the basic reproduction number is determined and shown to serve as a threshold for the existence of an endemic equilibrium, and is related to the linear stability of the disease-free state. Finally, numerical experiments, parameterized specifically for Dengue fever, demonstrate how within-host mechanisms influence population-level epidemiological outcomes.

We study the energy-critical inhomogeneous Hartree equation in space dimensions three and higher. Previous local well-posedness results left open the parameter regime where the inhomogeneity exponent is small and the Riesz potential exponent is either small or large. We establish local well-posedness for a new range of parameters, thereby substantially filling the remaining open parameter regime. In particular, our result completely resolves the remaining gap in dimensions $5$ and $6$.

We prove the soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space $H^s\left(\mathbb{R}\right)$ for $s>-\frac12$. This extends soliton resolution to the sharp regularity, whereas previous results required weighted $H^{1,1}\left(\mathbb{R}\right)$ initial data. We first prove the soliton resolution conjecture in the sharp well-posedness space for the integrable equation. In particular, our work solves three important open problems, including the problem of embedded singular continuous spectrum, the extension of Gérard's explicit formula, and the inverse spectral problem for the Lax operator. A central ingredient of our work is a new modular operator-theoretic framework combining the closed quadratic-form theory of rough Lax operators, weighted limiting absorption, and distorted Fourier analysis, which yields a unified treatment of distributional spectral theory, continuous-spectrum scattering, and radiative asymptotics at the sharp well-posedness threshold. Another important ingredient is a Fourier-side regularity and stability result for the eigenfunctions of the rough Lax operator, established through a zero-frequency identity, resolvent commutator estimates, and an isolating-contour argument. This yields well-defined soliton centers that are stable under rough approximation. Our framework provides a broadly applicable framework for proving soliton resolution conjecture for integrable dispersive equations in low-regularity spaces.

This paper presents a backstepping approach for the boundary control of first-order hyperbolic equations with spatially varying coefficients posed on domains of arbitrary dimension. The method is based on a change of variables induced by the characteristic flow of the time-invariant transport operator, transforming the original multidimensional system into a continuum of decoupled one-dimensional hyperbolic equations evolving along individual characteristic curves. A backstepping controller is then designed for each equation in the decomposition, and the resulting control laws are reassembled in the original coordinates to achieve finite-time stabilization of the full system. The framework relies on the existence of characteristic curves foliating the spatial domain, with uniformly bounded transit times (non-trapping).

We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the first eigenvalue over a class of such domains under perimeter and volume constraints, and under an additional convexity assumption when $N \geq 3$. This result generalizes to a wider class, and extends to higher dimensions, the inequality of Hersch [20], whose approach was substantially based on a construction of the so-called effectless cut by Weinberger [35], so that we call it the Hersch-Weinberger inequality. Our method is based on the analysis of the gradient flow of the first eigenfunction and several approximation procedures, without relying on the effectless cut itself. The effectless cut being a complicated object related to the attractor of the gradient flow, we describe its most fundamental topological properties. In particular, we show that it does not necessarily have to be a hypersurface.

In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem \begin{equation}\label{crit_pro_abstract}\tag{$\mathcal{P}_{2^*}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{2^*-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $N\geq3$, $s\in(0,1)$, $\lambda>0$, and $2\leq p<2^*=\frac{2N}{N-2}$. We establish a compactness result for the following class of subcritical/critical problems \begin{equation}\label{sub_pro_abstract}\tag{$\mathcal{P}_{p_n}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{p_n-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega, \end{equation} where $p_n \in (p,2^* ]$ and $p_n\to 2^*$. Specifically, for $p \in (2+\frac{4s}{N-2},2^*)$ when $N>6-4s$, and for $p \in (2^*-1,2^*)$ when $N\leq6-4s$, we prove that any bounded sequence of solutions $\{u_n\}$ to \eqref{sub_pro_abstract} is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqref{crit_pro_abstract}. To the best of our knowledge, this is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, proving the compactness result requires several delicate new and novel estimates, which we believe will be of independent interest for further studies of related problems. As an application of this compactness result, under the same ranges of $N$ and $p$, we prove that \eqref{crit_pro_abstract} admits infinitely many sign-changing solutions.

(A) It is known that among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,\theta}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,\theta}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. Note that, for the initial velocity, there is neither a size restriction on other components, nor a parity assumption. There are four key ingredients in the proof. (1) Three new good unknowns are introduced, and a self-closed energy estimate for them is derived. (2) An elliptic estimate for pressure is established to control boundary terms arising from the boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of $rv_\theta$. (4) A new anisotropic Hardy's inequality is derived for weighted mean-zero functions to overcome the lack of parity of $\boldsymbol{v}$. (B) Based on (A), we introduce and prove the so-called controlled regularity for the above problem, i.e. for suitable initial data without any smallness assumption, there exists an external force supported away from the axis of symmetry such that the corresponding problem admits a global strong solution. This seems to add a little weight to the regularity scenario for ASNS, since the force is supported away from the axis which is the only place regularity may break down. We also prove that if there exists a solution that blows up in finite time, an unstable blow-up solution must exist.

We show that SU(1,1) NLFT can diverge pointwise for square-summable coefficients. As a consequence, we prove that the classical pointwise asymptotics of polynomials orthogonal on the unit circle can fail for measures in the Szegö class. We also discuss some special cases when the pointwise convergence holds.

In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$ for given any $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya–Prodi–Serrin condition at the endpoint $(2,\infty)$ and the solutions live on the borderline of the Beale–Kato–Majda criterion. To the best of our knowledge, this is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts, we also obtain non-uniqueness for the Navier–Stokes equations in $L^2_t L^p$ with $1\le p<\infty$, and for the MHD system with large $\mathrm{BMO}^{-1}$ initial data.

We study hypercontractivity for underdamped Langevin dynamics with a convex confining potential whose spatial Gibbs marginal satisfies a logarithmic Sobolev inequality (LSI). Unlike in the overdamped case, the noise acts only on the velocity variable, so the usual LSI-based argument does not apply. Nevertheless, under an additional tame Hessian-growth assumption on the potential, we prove that, when the spatial LSI constant is $\rho$ and the friction parameter is of order $\sqrt{\rho}$, the semigroup satisfies a Gross-type $L^p$-to-$L^q$ estimate in which the integrability exponent grows exponentially on the kinetic time scale $t \sim \rho^{-1/2}$. The central ingredient, of independent interest, is a space-time LSI, valid for controlled kinetic paths of finite action, which quantifies how dissipation in the velocity variable is transferred to the position variable. We derive it from a controlled version of the hypocoercive entropy-decay estimate, and we then restore the Gross hypercontractivity mechanism through a duality argument built on a non-reversible forward/backward interpolation of the underdamped Langevin semigroup. As a corollary, Rényi divergences of every order decay exponentially at the sharp hypocoercive rate $\mathcal{O}(\sqrt{\rho})$.

This paper establishes a sharp, expanded wave-breaking criterion for a class of nonlinear nonlocal Whitham-type equations, significantly generalizing the classical threshold introduced by Seliger. While the system of inequalities governing the spatial extrema of the fluid deviation has been extensively studied, establishing a sharp condition has remained a challenge. This difficulty is primarily due to the non-cooperative nature of the system, which precludes the application of standard comparison principles. We rigorously overcome this analytical obstacle by identifying the exact nonlinear threshold that confines the solutions of the inequality system and analyzing its time evolution.

This paper investigates a multilayered Helmholtz model in $\mathbb{R}^d$ ($d \ge 2$) characterized by concentric layers of materials with alternating positive and negative refractive indices. To overcome the loss of coercivity induced by the sign-changing material parameters, we construct a bespoke $\mathbb T$-coercivity operator to restore the coercive structure of the problem. Furthermore, to address the inherent lack of compactness on unbounded domains, we integrate a complex-wavenumber Dirichlet-to-Neumann (DtN) operator into this framework. By combining this variational synthesis with sharp a priori estimates, we rigorously establish the limiting absorption principle and prove the well-posedness of the corresponding transmission problem in appropriate function spaces. Crucially, we quantify the dependence of uniqueness on the domain geometry by explicitly analyzing the optimal trace constant, thereby providing a rigorous mathematical criterion for the design of multi-layer metamaterials.

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -\Delta v-\frac{1}{2} y \cdot \nabla v=\alpha v+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, \alpha=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case, and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions.

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Hardy term \begin{equation*} \begin{cases}u_t-\Delta u-\mu \frac{u}{|x|^2}=|u|^{2^*-2} u & \text { in } \mathbb{R}^N \times(0, T), u(x, 0)=u_0(x) & \text { in } \mathbb{R}^N , \end{cases} \end{equation*} into the following parabolic equation \begin{equation*} \begin{cases} v_s-\Delta v-\frac{1}{2} y \cdot \nabla v=\beta v+\frac{\mu v}{|y|^2}+|v|^{2^*-2} v &\text { in } \mathbb{R}^N \times(0, S), \left.v\right|_{s=0}=v_0 & \text { in } \mathbb{R}^N, \end{cases} \end{equation*} where $N \geqslant 3$, $\mu\in [0,(N-2)^2 /8]$ and $2^{\ast}=2N /(N-2)$. For this equation, we establish a weighted Hardy inequality. Furthermore, by virtue of the modified potential well method and Palais-Smale sequence analysis, we investigate the long-time behavior and finite-time blow-up properties of solutions to the parabolic equation.

We prove the existence of time-periodic solutions for a driven–damped $\phi^4$ wave equation posed on a bounded spatial domain with periodic boundary conditions. The result is obtained at the level of the full partial differential equation. The analysis is based on a Lyapunov–Schmidt decomposition in a time-periodic function space. A key structural feature is the presence of linear damping, which ensures uniform invertibility of the associated linear operator on the complement of the kernel and removes the need for nonresonance conditions. After introducing a truncation to control the cubic nonlinearity, we solve the complement equation via a contraction mapping argument and reduce the problem to a scalar kernel equation. The latter is treated using a mean/zero-mean decomposition and a fixed-point argument. A period-averaged energy estimate then allows one to remove the truncation and obtain a time-periodic weak solution under explicit smallness conditions on the forcing parameters