Papers

We prove a global-in-time uniform bound for solutions to thermo-diffusive systems of reaction-diffusion equations with $L^\infty$ initial conditions.

We construct nanopteron solutions to the differential equation \[ \epsilon^2u^{(4)}+u''-u+u^2 =0, \] which is a singular perturbation of the traveling wave problem $u''-u+u^2=0$ for the Korteweg–de Vries (KdV) equation. These nanopteron solutions are the superposition of the sech$^2$-type KdV traveling wave profile, a small localized error, and a periodic "ripple." The ripple is parametrized by its amplitude and its phase shift. In the process of constructing the nanopteron, we preselect one of these parameters and then solve for the other, and we provide detailed commentary on how the amplitude and phase shift solution processes differ. Our work revisits several prior constructions of nanopteron solutions for this singularly perturbed KdV equation in light of modern developments of nanopterons for lattice differential equations. In the context of these contemporary methods, the simple structure of the singularly perturbed KdV equation allows it to serve as an accessible template for nanopteron constructions in more complicated problems.

In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for convex solutions, as shown by counterexamples due to Lu \cite{LuGeneral}. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient equations for $2\leq k\leq n-1$ in arbitrary dimensions.

We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear or fully nonlinear pattern-forming systems close to the first instability. We present an approximation theory in uniform local Sobolev spaces which is applied to a quasilinear Swift-Hohenberg model, to a quasilinear version of Bénard's problem with velocity dependent viscosity, and to abstract quasilinear reaction-diffusion-advection systems.

Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically requiring smoothness of the underlying domain; see the recent survey by Niebel & Valentini. In this work, we prove a new trace estimate for kinetic Sobolev spaces over the sphere for $\rho$-convex domains which, in general, may have a non-smooth boundary. Similar to the work of Cessenat, we use characteristics to obtain trace estimates in weighted trace spaces with explicit constants. For completeness, we also present a density result for the corresponding function spaces on Lipschitz domains.

This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.

We identify a previously unnoticed gradient-flow structure of the Fokker–Planck equation arising in the spontaneous particle aggregation model. For an exponential response function, the equation is a generalized Wasserstein gradient flow of the attractive McKean–Vlasov free energy with a nonlocal mobility. We show that, within a natural class of local entropies and symmetric interaction energies, this structure essentially singles out the exponential response. We discuss consequences for stationary states, global minimizers, the single-cluster structure in one dimension, and formally outline convergence to equilibrium using a Wasserstein–\mboxŁojasiewicz inequality.

We study the long range behavior of solutions to $i\partial_tu=H_\alpha u+\lambda|u|u$ on $\mathbb R^2$, where $H_\alpha$ is the Friedrichs realization of the Aharonov-Bohm Hamiltonian with a single pole. The logarithmic phase of the long range ansatz may push a profile out of the domain of $H_{\alpha}$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac 12$; at half flux $\alpha=\frac 12$, no nonzero trace survives. However, every profile in the full domain of $H_{\alpha}$ with small $L^\infty$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+\nu_\alpha$, $\nu_\alpha=\min\{\alpha,1-\alpha\}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $\alpha\neq \frac 12$, we can construct profiles with an error of size $t^{-1/2-\nu_\alpha}\log t$, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in $L^2$; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with $1<b<2$.

We establish time-decay estimates for the two-dimensional fourth-order Schrödinger operator $H=\Delta^2+V$ with a real-valued decaying potential $V$, covering all possible zero-energy threshold obstructions. When zero is a regular point or a first-kind resonance, we prove \[ \left\| H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H) \right\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{2+\alpha}{4}}, \qquad -2<\alpha\leq2, \] which matches with the free sharp decay rate throughout the full range of $\alpha$. For a second-kind resonance, the decay rate is $|t|^{-(2+\alpha)/4}(\log(2+|t|))^2$ for every $-2<\alpha\leq2$, with only a logarithmic loss. For the stronger threshold singularities, we show that the large-time behavior is governed by the presence of a \(d\)-wave resonance. If zero is a third-kind resonance, or an eigenvalue accompanied by a \(d\)-wave resonance, we obtain the sharp decay $(\log|t|)^{-1}$ for $\alpha=0$ and $|t|^{-\alpha/4}(\log|t|)^{-2}$ for $0<\alpha\leq2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<\alpha\leq2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<\alpha\leq2$ and $s>0$: \[ \left\| \omega^{-s} H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H)\omega^{-s} \right\|_{L^1\to L^\infty} \lesssim \frac{1} {|t|^{\frac{2+\alpha}{4}}(\log|t|)^s}, \qquad |t|\geq2, \] where $\omega(x)=\log(2+|x|)$. By contrast, zero is a second-kind resonance for the free operator $\Delta^2$, and the free evolution admits no such logarithmic gain. Thus, in the regular and first-kind cases, the potential changes the zero-energy spectral structure of the free operator, and this change is accompanied by improved weighted decay.

We present a potential theoretic approach reducing the analysis of the asymptotic shape of free surfaces to the analysis of a precise ordinary differential equation resulting from the reduction process. Although the approach relies mainly on the principal part of the PDE operator to allow for a representation formula and is thus not restricted to problems of elliptic type, we present it at the clean-cut example of three-dimensional axially symmetric steady incompressible cavity flows, which are Neumann-type Bernoulli free boundary problems and for which frequency formulas are unknown and, if they do exist, insufficient to yield the very precise asymptotic behavior we prove here. In 1946 Norman Levinson derived by a power-law ansatz with a slowly varying correction a precise formula for the asymptotic shape of such cavities. However his result requires very strong assumptions such that it has remained an open problem for 80 years whether the cavity solutions we know to exist by a result by Garabedian-Lewy-Schiffer [12] actually share this asymptotic behavior, or whether at least one solution possessing the Levinson asymptotics exists. Here we answer both questions affirmatively, and we obtain complete rigidity at infinity of the Levinson solution in the class of axially symmetric solutions, that is, any solution satisfying mild and natural assumptions at the fixed boundary and infinity converges asymptotically to the Levinson profile $(\log r)^{-1/4}\sqrt{r}$.

We study an optimal control problem for a quasilinear parabolic equation modeling nonlinear heat conduction during heat treatment of metals. Under general assumptions, we prove existence of an optimal control and characterize the associated optimal control-state pair in a practically relevant framework. The main difficulty is the quadratic gradient term, which we handle through energy estimates and differentiability properties of the control-to-state map.

This work studies surface-localized transmission eigenfunctions for time-harmonic electromagnetic scattering. Prior results on this phenomenon are mostly qualitative, proving existence without quantifying distribution. Here we provide a quantitative analysis for the Maxwell transmission eigenvalue problem, covering both TE and TM modes. We first establish a Weyl asymptotic law for the full counting function, with cubic growth with respect to the radius $R$. We then prove Weyl-type upper and lower bounds for the counting functions restricted to surface-localized modes, and demonstrate that they share the same growth order three. To our knowledge, this is the first quantitative result of its kind, demonstrating that surface-localized eigenfunctions form a non-negligible fraction of the high-frequency spectrum.

We study a Serrin-type overdetermined problem for the fractional Laplacian in a bounded open set with an isolated non-removable interior singularity. For positive weak solutions of (-Delta)^s u=f(u) in Omega\{0}, u=0 in R^N\Omega, and (partial_eta)^s u=c on partial Omega, with 0<s<1, N>2s and c<0, we isolate a first-order tangential cancellation of the boundary quotient u/delta^s that matches the first-order cancellation needed to close the fractional corner argument and is weaker than requiring u/delta^s in C^1 in a full boundary neighborhood. Under this condition, Omega is a ball centered at the singular point and u is radial and strictly decreasing in the radial variable. No pointwise blow-up rate at the pole is assumed. If Omega is of class C^{2,alpha}, the cancellation is automatic when s>1/2, and for every s in (0,1) when f is constant near zero. In particular, in the torsion case f=1, u(x)=tau_R(x)+kG_R(x,0), where tau_R and G_R are respectively the fractional torsion function and the Green function of the ball.

We provide explicit examples showing that the existing assumptions in literature on the force are sharp for the stated properties of strong solutions to the 2D Navier-Stokes equations. If they fail to be satisfied, finite time blow-up may occur in different forms.

We prove comparison inequalities for Dirichlet-to-Neumann maps corresponding to different non-positive Helmholtz parameters. For convex domains our bounds are sharp, and the resulting eigenvalue inequalities partially confirm an earlier conjecture, which we show does not hold in full generality. We further obtain geometry-dependent versions for arbitrary sufficiently regular domains, together with extensions to compact Riemannian manifolds with boundary. We also discuss analogous questions for metric graphs.

We study the classical anisotropic Calderón problem associated to the elliptic equation $\text{div}(\sigma\nabla u)=0$, where the complex admittivity $\sigma$ is of the form $\sigma=A(\cdot,a(\cdot))$ in a domain $\Omega\subset\mathbb{R}^n$, $n\ge3$. We establish boundary stability estimates for $\sigma$ and its derivatives of arbitrary order from a local Neumann-to-Dirichlet map. Our results extend those of Comm. Partial Differential Equations, 34 (2009) from the real-valued conductivity setting to complex anisotropic admittivities.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

Combining classical group theory and partial differential equation theory, this paper investigates the symmetry group $\operatorname{Sym}(u)$ of the unique solution $u$ to the second-order linear elliptic boundary value problem on an $n$-dimensional bounded domain $\Omega$ $-\sum_{i,j=1}^{n} a_{ij}(x)u_{x_ix_j} + \sum_{i=1}^{n} b_i(x)u_{x_i} + c(x)u = f(x), x\in \Omega$, $u(x) = h(x), x\in \partial \Omega$, The following symmetry groups are defined and characterized respectively: the symmetry group $\operatorname{Sym}(A)$ of the second-order coefficient matrix function $A(x)=(a_{ij}(x))_{n\times n}$; the symmetry group $\operatorname{Sym}(b)$ of the first-order coefficient column vector function $b(x)=(b_{1}(x),b_{2}(x),\cdots,b_{n}(x))^{T}$; the symmetry group $\operatorname{Sym}(c)$ of the zero-order coefficient function $c(x)$; the symmetry group $\operatorname{Sym}(f)$ of the internal source function $f(x)$; and the symmetry group $\operatorname{Sym}(h)$ of the boundary source function $h(x)$. This paper rigorously proves that the common symmetry group $\operatorname{Sym}(A)\cap\operatorname{Sym}(b) \cap\operatorname{Sym}(c) \cap\operatorname{Sym}(f) \cap\operatorname{Sym}(h)$ is a subgroup of $\operatorname{Sym}(u)$. In addition, if the common symmetry group contains several mirror symmetry elements, the original second-order linear elliptic boundary value problem on the entire domain $\Omega$ can be reduced to the corresponding boundary value problem on a certain subdomain. It is strictly proven in this paper that the new boundary condition imposed on the boundary of the subdomain is the homogeneous generalized Neumann boundary condition. The linear finite element method is used to numerically solve the second-order linear elliptic boundary value problem on the subdomain, thereby achieving domain reduction and significantly reducing the computational cost.

The main result of this paper concerns the stability of Monge solutions to eikonal equations in metric spaces. Although the subslope used in the standard definition of Monge solutions is, in general, not stable under uniform convergence, we overcome this difficulty by introducing an equivalent nonlocal characterization based on sphere-wise subslope conditions. This approach complements existing methods based on metric viscosity solutions and enables us to establish stability directly under uniform convergence and Gromov-Hausdorff perturbations of the domain and metric, without the use of test functions. The method extends to inhomogeneous equations via the optical metric and to a broader class of Hamilton-Jacobi equations. Our stability argument further enables us to develop Perron's method for Monge solutions in general complete length spaces. This yields a new Perron-type existence result for Hamilton-Jacobi equations in metric spaces, without relying on test functions.

We extend the quantum modulated energy developed in [17] in order to de- rive the magnetized pressureless Euler-Poisson equation as a semiclassical and mean field semiclassical limit from the magnetized Schrödinger-Poisson and von-Neumann equations, respectively. Local well-posedness of the underlying monokinetic PDE is also addressed. In both limits, the magnetic field is external and may be spatially non-uniform. Our results fall in the broader scope of semiclassical and quantum mean field limits for magnetized quantum dynamics.

We study the focusing quadratic nonlinear Schrödinger system \[ \begin{cases} i\partial_t u+\Delta u=-2v\overline{u}, \\ i\partial_t v+\kappa\Delta v=-u^2, \end{cases} \qquad (t,x)\in I\times\mathbb R^4, \] where $\kappa>0$. In the non-mass-resonant case $\kappa\neq \frac12$, previous works of Inui–Kishimoto–Nishimura and Dinh–Forcella showed that radial solutions with negative energy must either blow up in finite time or exist globally while their $H^1$-norm grows without bound. In this paper, we prove that every radial $H^1\times H^1$ solution with negative energy blows up in finite time, both forward and backward in time. No finite-variance assumption is required. The main ingredient is a localized virial argument based on the bounded exponential weight \[ \nabla\phi_R(x)=2x e^{-|x|^2/R^2}. \] A radial weighted interpolation estimate allows us to control the nonlinear error terms by the corresponding weighted kinetic term, up to an $O(R^{-2})$ error depending only on the conserved mass. Moreover, the localized virial quantity itself can be bounded directly in terms of the same weighted kinetic defect. Combining these estimates yields a superlinear Riccati-type differential inequality, which cannot persist for all time and therefore forces finite-time blow-up.

This paper investigates the global solvability, boundedness, and uniqueness of weak solutions to the chemotaxis system \begin{equation*} \begin{cases} u_t = \Delta u^m - \chi \nabla \cdot (u \nabla v) + u(a - b u), & \text{in } (0,\infty)\times\mathbb{R}^N, \\ \tau v_t = \Delta v - \lambda v + \mu u, & \text{in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where \(m>1\), \(\tau\in\{0,1\}\), \(\lambda,\mu,a,b>0\), and \(\chi\in\mathbb{R}\). For every \(m>1\), we establish the existence of weak solutions for initial data that are not necessarily integrable, although such solutions need not be bounded in general. We then show that globally bounded weak solutions exist in the parabolic-parabolic case \((\tau=1)\) when \(m>\frac{2N}{N+2}\), and also when \(1<m\le \frac{2N}{N+2}\) provided that the logistic damping coefficient \(b\) is sufficiently large. In the parabolic-elliptic case \((\tau=0)\), we prove the existence of globally bounded weak solutions when \(m>2-\frac{2}{N}\), and also when \(1<m\le 2-\frac{2}{N}\) provided that \(b\) is sufficiently large. Finally, for \(1<m\le 3\), we prove uniqueness of weak solutions that are Hölder continuous up to the initial time.

Let $N\ge1$, $p\in[1,\infty)$, $\gamma\in(0,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $\lambda\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{\lambda,p,\gamma}(u;\Omega) :=\lambda\iint_{\Omega\times\Omega} \mathbf 1_{\left\{(x,y)\in\Omega\times\Omega:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+\gamma}}\geq\lambda\right\}} |x-y|^{\gamma-N}\,dx\,dy. \end{align*} In this article, we prove that, as $\lambda\to\infty$, the family $G_{\lambda,p,\gamma}$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\mathrm{cell}}(u;\Omega):= \begin{cases} C_{N,p,\gamma}^{\mathrm{cell}}\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^{\mathrm{cell}}|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}, \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$ and characterized by a cell formula. This gives an affirmative answer to the problem posed by Brezis [Open Problem~9.3, Rend. Lincei Mat. Appl. 2023].

Consider the Landau equation with a Coulomb potential linearized around traveling Maxwellians. We prove that the microscopic part of the linear solution decays with an inverse polynomial rate, while the macroscopic part remains bounded but in general does not decay. In particular, the long-time dynamics are different from the linearization around either (1) traveling Maxwellians for moderately soft or hard potentials or (2) global (non-traveling) Maxwellians (for any potentials).

For a diffeomorphism $\Phi$ of a domain $\overline{\Omega}$ onto itself, which is the identity on $\partial\Omega$, we prove that local regularity of the push-forward $\Phi^*[I]$ implies local regularity of $\partial\Omega$, provided a certain non-degeneracy condition is satisfied. To be precise, if $\nu$ is a normal to $\partial\Omega$ at a point $P$, then the condition $(\Phi^*[I](P)-I)\nu \neq 0$ and the assumption that $\Phi^*[I]$ is of class $C^{k+1,\alpha}$ near $P$ imply that $\partial\Omega$ is also of class $C^{k+1,\alpha}$ near $P$. This result naturally complements recent regularity results for non-scattering inhomogeneities.

We construct infinite-time Sobolev cascades for the cubic nonlinear half-wave equation on the real line. For $\frac12<s\leq1$ and $0\leq\kappa<2s(2s-1)/(4s-1)$, spatial gluing of permanently saturated two-soliton packets gives a dense set of strictly mass-subthreshold data for which $\|u(t)\|_{H^s}/(1+t)^\kappa\to\infty$ in the focusing case. A second construction replicates finite-burst Szegő packets without rescaling. It yields a dense $G_\delta$ subset of $H^\infty(\mathbb R)$ whose solutions perform logarithmic cascades in every $H^s$ with $s>\frac12$ in the focusing case and in the defocusing case.

Nonlinear reaction-diffusion equations arise in a very wide range of applications. Here we address a specific multidimensional class in which the source term is non-autonomous in time, this choice being motivated both by its relevance to a broad class of (autonomous) `near-linear' reaction-diffusion equations and by the unusually rich nature of the Lie symmetries that results (including for arbitrary time-dependence, and with four even more exceptional cases arising). The class of non-autonomous equation analysed is derived by an asymptotic approach, with the Lie symmetry classification and a further analysis subsequently carried out. Both Lie and non-Lie reductions to ordinary differential equations are also noted and relevant exact solutions are constructed.

Finite-player nonzero-sum optimal stopping games typically lead to coupled equilibrium systems whose complexity grows rapidly with the number of players. We introduce an independently randomized formulation in which each stopping rule is represented by an adapted, nondecreasing cumulative stopping process. The canonical embedding preserves pure-profile payoffs, and a pure profile is a Nash equilibrium of the original game if and only if its embedding is a Nash equilibrium of the randomized game. We adopt the $\alpha$-potential approach to construct an $\alpha_N$-potential function, with the error $\alpha_N=O(N^{-1})$ under weak-interaction. We also identify an exact-potential subclass with a closed-form threshold equilibrium. For local stopped-status interactions, randomized payoffs admit a local stopped-mass representation, and potential maximization can be formulated as a multidimensional singular-control problem with local gradient constraints and a nonlocal condition for finite jumps. Under suitable regularity assumptions, we study the associated Hamilton-Jacobi-Bellman quasi-variational inequality and its regularity properties. For unknown model coefficients, we propose a bounded-intensity Potential-CT-DDPG learning algorithm. Numerical experiments closely match the analytical benchmark and yield estimated best-response improvements consistent with $N^{-1}$ scaling.

Let $A$ be $m$-accretive in a real Banach space $X$. Given closed sets $K(t)\subset X$, we develop an existence theory for mild solutions of the constrained problem \[ u'(t)+Au(t)\ni f(t,u(t)), \qquad u(t)\in K_A(t):=K(t)\cap\overline{D(A)}, \] where the forcing is Carathéodory on the effective tube $K_A$. We also assume joint measurability of $f$ on the moving graph or a Carathéodory extension to a fixed cylinder. Under a linear-growth hypothesis and the corresponding regular- and exceptional-time subtangential conditions, we reduce the problem to a bounded effective tube with a common integrable bound. % A first central result constructs a closed separable resolvent-invariant subspace preserving distances to the moving sets and the subtangential conditions, permitting use of the Scorza–Dragoni property for the reduced forcing. A second central ingredient is a two-level approximation. Mild solutions $u_\varepsilon$ driven by $w_\varepsilon$ are accompanied by auxiliary paths $v_\varepsilon$ and current-time selectors $x_\varepsilon$ satisfying $x_\varepsilon(t)\in K_A(t)$ and $w_\varepsilon(t)=f(t,x_\varepsilon(t))$ for almost every $t$ in a closed regularity set of almost full measure. For every $t$ there is also $\sigma_\varepsilon(t)\in[(t-\varepsilon)^+,t]$ with $v_\varepsilon(\sigma_\varepsilon(t))\in K_A(\sigma_\varepsilon(t))$. The current-time selectors identify the limiting forcing, while the lagged nodes guarantee viability. % This yields well-posedness under time-dependent constraints for such forcings $f$ when they are locally Lipschitz in the state variable. Further viability results follow under various compactness conditions. Application of the abstract viability theory yields comparison principles, nonautonomous Lyapunov pairs, periodic solutions, and time-dependent bounds for abstract reaction–diffusion systems.

In this article, we consider an energy-critical quintic wave equation on a bounded domain $\Omega\subset\mathbb{R}^3$ with nonlinear and nonlocal Kelvin–Voigt damping of the form $-\|\nabla u_t\|_{L^2(\Omega)}^\alpha\Delta u_t$, where $\alpha\in\mathbb{R}_+=[0,\infty)$. Under suitable hypotheses on the quintic source term, we establish the well-posedness of the problem and investigate its long-time dynamics in the natural energy space $\mathcal H=H_0^1(\Omega)\times L^2(\Omega)$. For every $\alpha\in\mathbb{R}_+$, we show that the associated dynamical system $(\mathcal H,S^\alpha(t))$ is gradient and dissipative, and we prove a stabilization estimate that yields asymptotic smoothness and, consequently, the existence of a compact global attractor $\mathcal A_\alpha$. The same estimate provides an upper bound for the Kolmogorov $\varepsilon$-entropy of $\mathcal A_\alpha$ and, in the limiting case $\alpha=0$, reduces to a quasi-stability inequality, which implies that $\mathcal A_0$ has finite fractal dimension. Furthermore, we prove that the family $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ is uniformly bounded in the higher-regularity space $\mathcal H_1=(H^2(\Omega)\cap H_0^1(\Omega))\times H_0^1(\Omega)$. Finally, we establish the upper semicontinuity of $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ at $\alpha=0$, showing that the attractors associated with the nonlinear and nonlocal Kelvin–Voigt damping converge to the global attractor of the limiting problem with classical linear Kelvin–Voigt damping.