In this paper, we study the advective unstable Cahn–Hilliard equation on T2 with shear flow:{ut+Av1(y)∂xu+εΔ2u=Δ(au3+bu2)onT2;uperiodicon∂T2, where u0∈H02(T2), A,ε>0, a<0, and b∈R. The condition a<0 puts the model in an unstable phase-field regime: the nonlinear chemical potential may amplify, rather than restore, concentration fluctuations, as in spinodal decomposition. The shear term Av1(y)∂xu models imposed stirring along the shear direction; through mixing, it enhances dissipation and counteracts the growth driven by the unstable cubic term Δ(au3). Assuming that the shear profile has finitely many critical points and that linearly growing modes occur only in the shear direction, we prove that the L2-energy converges exponentially to zero, provided that |a| and ‖∫Tu0(x,⋅)dx‖Ly2 are sufficiently small and that the flow amplitude A is sufficiently large.
Motivated by the convergence theory of gradient systems, this paper studies the asymptotic behavior of quasi-gradient systems beyond the classical symmetric gradient setting. We consider both first- and second-order dynamics. For the first-order system, the coefficient matrix is assumed to satisfy −(A+AT)≻0. For the second-order system, we derive an additional explicit admissibility condition involving the coefficient matrix, inertia, damping, and a bound on the Hessian of the potential. Under these respective conditions and either of the following two sets of assumptions, (i) the potential is real-analytic and the trajectory is bounded, or (ii) the potential is real-analytic and periodic, we prove convergence to an equilibrium together with explicit convergence-rate estimates. We also provide a non-diagonal example with numerical illustrations for both the first- and second-order theories and apply the framework to several Kuramoto-type models.
In this paper, we study the following mixed-order conformally invariant system with exponentially increasing and convolution-type nonlinearities in Rn:(⁎){(−Δ)su=epv in Rn,(−Δ)n2v=(|⋅|−σ⁎u2n−σn−2s)u2n−σn−2s in Rn, where n≥2, s∈(0,n2), σ∈(0,n), p∈(0,+∞), u≥0 in Rn and v is allowed to change sign. System (⁎) can be viewed as a coupling of two conformally invariant models: the critical Hartree equation associated with (−Δ)s and the critical-order Liouville equation involving (−Δ)n2. It combines the Liouville-type exponential nonlinearity with a nonlocal Hartree-type critical density.Motivated by this mixed conformal structure, we establish a complete classification of classical solutions to system (⁎). More precisely, under the finite total curvature condition ∫Rn(|⋅|−σ⁎u2n−σn−2s)u2n−σn−2s<+∞, together with the mild growth assumption that u(x)=O(|x|τ) at infinity for some finite τ>0, and the additional assumption that v(x)=o(|x|2) at infinity when n≥3, we obtain a complete classification of classical solutions to (⁎) without imposing any sign condition on v. The main ingredients of the proof consist of establishing super poly-harmonic properties, performing a sharp asymptotic analysis at infinity, and applying the method of moving spheres to the associated integral system. As an application, we obtain a nonexistence result for classical solutions to system (⁎) in the supercritical-order cases (i.e., s>n2).Our argument provides a unified framework for the classification of solutions to fractional and higher order conformally invariant systems in all critical dimensions, and we believe that these techniques can be applied to other related problems.
We establish mixed-norm regularity estimates for the parabolic equationut+Δνu=f,u(0,⋅)=0, where Δν denotes the multivariate Bessel operator on (0,∞)n with ν=(ν1,…,νn)∈(−1,∞)n. Let γν=max1≤j≤nmax{0,−12−νj}. We prove the regularity of the equation on Lp([0,∞))Lq(R+n) for all 1<p<∞ and 11−γν<q<1γν. When νj≥−12 for all j, the estimate holds for the full range 1<p,q<∞. The results reveal that temporal regularity remains uniform even when spatial regularity is restricted when ν∈(−1,−∞)n∖[−1/2,∞)n. The proof relies on kernel difference estimates for Bessel operators and localized operator bounds, extending maximal regularity theory beyond the Gaussian setting.
In this paper, we turn to the following Cauchy problem for semilinear models with power nonlinearity and integrable decaying speed of propagation:(1){ϕtt−a(t)2Δϕ+m1a(t)A(t)ϕt+m22(a(t)A(t))2ϕ=|ϕ|p,(t,x)∈(0,∞)×Rn,(ϕ(0,x),ϕt(0,x))=(f(x),g(x)). Here A(t)=∫t∞a(τ)dτ, m1 and m2 are positive constants. The model (1) generalizes the de Sitter model (a(t)=e−t). We are interested to study the interplay of m1 and m2 on the global (in time) well-posedness of small data Sobolev solutions. First of all we propose a classification of models (1). Then our considerations are devoted to two of the models (mass dominant and dissipation dominant in the case of non-effective dissipation). We will see that a large m2 produces a loss of critical exponent pcrit>1.
This paper is concerned with the time-domain scattering of acoustic waves by a bounded elastic obstacle embedded in the lower layer of a two-layer compressible fluid separated by an unbounded rough interface. To deal with the unbounded geometry, we impose exact transparent boundary conditions on suitable artificial boundaries and thereby reformulate the original acoustic–elastic coupling problem as an equivalent initial–boundary value problem in a strip-like domain. In the Laplace domain, we combine these transparent boundary conditions with integral representations to derive a system of boundary integral equations posed on the rough interface, the artificial boundaries, and the fluid–solid interface. The analysis of this system is carried out in a single-trace variational framework, in which the transmission conditions across all physical interfaces are enforced at the level of the trial and test spaces. This leads to a bounded and coercive sesquilinear form for all Laplace parameters in a suitable right half-plane. We then obtain existence and uniqueness of the weak solution to the Laplace-domain problem. In particular, our results provide a systematic Laplace-domain boundary integral analysis of time-domain acoustic–elastic scattering in such a two-layer unbounded configuration. Finally, by the theory of vector-valued distributions and the inversion of the Laplace transform, we transfer these results back to the time domain and establish the well-posedness of the original acoustic–elastic scattering problem in the two-layer unbounded structure.
The nonstationary non-Newtonian flow, with strain rate dependent viscosity in a thin tube structure, with no slip boundary condition, is considered. Applying the weak Banach fixed point theorem we prove the existence and uniqueness of a solution, its high order regularity, and a priori estimates.
This paper investigates the linearization problem of quasi-periodic reversible systems with small perturbations. By the normal form theory, KAM iteration method, and technique of preserving the reversibility of transformations, we establish the analytic linearization result for such systems under appropriate non-resonance conditions and the formal linearizability assumption. Specifically, if the system is formally linearizable and the frequency parameters satisfy the non-resonance conditions, there exists a convergent quasi-periodic reversible transformation that reduces the original system to its linear part. As an application, we study the linearization and stability of a class of nonlinear reversible oscillators by the main result.
This paper investigates the time decay rates of solutions to the isentropic compressible MHD system on T2 when the bulk viscosity coefficient is sufficiently large. It is shown that if the initial velocity and magnetic field belong to H1(T2) and the initial density is bounded and nonnegative, then the solution converges exponentially to a constant state, with the decay rate explicitly characterized in terms of the viscosity coefficients. Arbitrary vacuum regions are allowed, and no compatibility condition is imposed on the initial data.
We investigate the three-dimensional incompressible inhomogeneous Navier–Stokes equations with distinct viscous coefficients in the horizontal and vertical directions. Our aim is to establish the global well-posedness of this anisotropic system when the initial density is a small perturbation of a positive constant and the vertical viscosity is sufficiently large compared with the horizontal one. The main analytical difficulty lies in obtaining the L1(R+;Lip(R3)) estimate for the convection velocity in the density transport equation. A key ingredient of our analysis is to derive the time-decay properties for the global solution of a so-called 2.5-dimensional inhomogeneous Navier–Stokes system, which may be viewed as a singular perturbation of a two-dimensional system in three-dimensional space. Owing to the strong anisotropy of the problem, the analysis is carried out within the framework of anisotropic Besov spaces based on Littlewood–Paley theory.
This paper investigates an initial-boundary value problem for the p-system with spatiotemporal damping on a half-line. By employing the time-weighted energy method, we establish that under suitable smallness assumptions on the initial data, the solutions to the p-system with Neumann boundary conditions converge over time to a modified diffusion wave. Furthermore, we derive the asymptotic behavior and convergence rates of the solutions in the L2-sense.
This paper studies the global asymptotic stabilization of solutions to a viscous balance law system derived from a Keller-Segel-type chemotaxis model with logarithmic sensitivity. The system is considered under time-dependent Dirichlet boundary conditions that are not necessarily equal–a significant generalization of previous works. By developing novel entropy-based energy methods and refined compensation schemes, the global existence and asymptotic stability of strong solutions for several parameter regimes, including both parabolic-parabolic and parabolic-hyperbolic cases, are established. The results also cover the global stability of a spatially heterogeneous steady state, demonstrating how nonlinear chemical kinetics can compensate for boundary asymmetry.
In this paper, we introduce a natural infinite-dimensional extension of the Rüssmann nondegeneracy condition and investigate the long-time weak convergence of statistical ensembles for a class of integrable Hamiltonian systems under both finite- and infinite-dimensional Rüssmann nondegeneracy conditions. Combining Fourier analysis in the angle variables with oscillatory integral decay estimates for finite-type phases, we prove that the ensemble average converges as t→∞ to an angular-averaged equilibrium determined by the initial marginal distribution in the action variables. The results of this work yield verifiable equilibration criteria even when the classical twist condition fails, elucidates a relaxation mechanism driven by higher-order frequency geometry, and provides analytic tools for statistical-mechanical analysis and quantitative convergence questions in integrable and near-integrable systems, including related Hamiltonian PDEs.
We study the emergent dynamics of the Justh–Krishnaprasad (JK) model for self-propelled particles with unit speed. The JK model encodes heading alignment through position-dependent interactions and interpolates between Cucker–Smale's flocking model and Kuramoto's synchronization model. While previous works have focused mainly on the emergence and stability of multi-cluster flocking states from well-separated initial data, we provide a condition under which flocking does not emerge from spatially mixed initial configurations, where no a priori cluster partition is prescribed. We also construct Lyapunov-type functionals for general unit-speed trajectories and derive affine asymptotic expansions and lower bounds that detect long-range separation. We then show that each particle admits a finite limit of the projection 〈xi(t),vi(t)〉−t, and that if this limit is sufficiently large compared to the initial projection onto the mean velocity, then asymptotic flocking cannot occur.
This paper focuses on the elliptic Schrödinger System on Riemannian manifolds, either without a boundary or with a smooth boundary under the constraint of the Neumann boundary condition. We improve the range of the index p in part of Quittner-Souplet's results [27]. We establish several Liouville theorems for the elliptic Schrödinger System on such Riemannian manifolds without a boundary or with a smooth boundary subject to the Neumann boundary condition. Additionally, we prove a series of results for positive solutions of the elliptic Schrödinger System, including gradient estimates and Harnack inequalities. Moreover, we have obtained an expansion of the coefficient λi in terms of the solution of the elliptic Schrödinger System.
We study the Hardy–Sobolev equation(−Δ)α/2u=|η|aup in the product half-space Rn−k×R+k, with zero exterior condition. For a>0 and p≥n+α+2an−α, assuming |η|aup−1∈Ln/α(R+n), and when p>n+α+2an−α, u∈Ln(p−1)/α(R+n), we prove that no positive solution exists. For −α<a≤0 at the critical exponent, the same conclusion holds under the local condition |η|aup−1∈Llocn/α(R+n‾) and a slow-decay assumption. The critical proof combines boundary Kelvin transforms with moving planes: the bounded branch gives fast decay, while the singular branch yields tangential symmetry and dimension reduction through a Kelvin isometry. In the supercritical case we use moving planes directly. This extends previous half-space Liouville theorems from the unweighted case and the case k=1 to all 1≤k≤n−1.
In this paper, we investigate the effects of the spatial period in determining the spreading speed of age-structured reaction-diffusion equations in spatially periodic media. We establish the asymptotic behavior of the spreading speed when the spatial period tends to zero and infinity, respectively. By comparing the spreading properties between the age-structured models and the classic Fisher-KPP reaction-diffusion equations, we show that the introduction of the age structure leads to more complicated spreading dynamics. Specifically, in rapidly oscillating media, there does not exist a definitive relationship between their spreading speeds. In contrast, in slowly oscillating media, the two spreading speeds coincide and can be characterized by a family of periodic Hamilton-Jacobi equations, with the zero order term determined by an age-structured equation parameterized by the spatial variable. Finally, we present an explicit formula for the limiting spreading speed in patchy environments, via constructing the viscosity solutions of the associated Hamilton-Jacobi equation.
We analyze a class of one-dimensional mean-field games with quadratic Hamiltonians and congestion effects. Using a forward-forward parabolic reformulation, we establish global well-posedness and show that nonnegative mobility costs enforce convergence to the homogeneous equilibrium. When the mobility cost becomes negative, this equilibrium loses stability, and global bifurcation theory reveals the emergence of infinitely many nonconstant stationary branches. Along the principal branch, we further derive refined asymptotic expansions that capture the formation of monotone boundary spikes in the strong interaction regime. Numerical simulations illustrate these transitions, highlighting the interplay between homogeneous states, global bifurcation, and sharply localized aggregation profiles.
In the study of meromorphic integrability of analytic differential systems, both the differential Galois group and the resonant set serve as significant tools for detecting existence of the meromorphic first integrals.In this paper, we establish a relationship between these two concepts for differential systems near a periodic orbit via dimension of the differential Galois group as a Lie group. Then we apply this result to two cases. One is to provide a unified and concise proof of two previously published results concerning the number of functionally independent meromorphic first integrals near the periodic orbit. Another is to obtain a smaller upper bound on the dimension of the Lie algebra of the differential Galois group associated with the periodic orbit.
