We study a nonlinear Schrödinger system with three-wave interaction: { - Δ u_1 = f_1(u_1) + α u_2u_3 in ℝ^N, - Δ u_2 = f_2(u_2) + α u_3u_1 in ℝ^N, - Δ u_3 = f_3(u_3) + α u_1u_2 in ℝ^N, u⃗=(u_1,u_2,u_3)∈ (H_rad^1(ℝ^N))^3, . where 3≤ N≤ 5 , α∈ℝ and each nonlinearity f_i(ξ ) satisfies the Berestycki-Lions conditions. Let S_i denote the set of all least energy solutions of the scalar equation -Δ u = f_i(u) in H_rad^1(ℝ^N) . A solution of the systems is called vector if all its components are nontrivial. We establish the existence of two distinct families of vector solutions {u⃗_α} with different asymptotic behaviors as α→ 0 . One family satisfies dist(u⃗_α,S_1× S_2× S_3) → 0 , while another satisfies dist(u⃗_α,S_1× S_2×{0}) → 0 . By contrast, we prove that no family of vector solutions satisfies dist(u⃗_α,S_1×{0}×{0}) → 0 . Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.
Lie symmetries, conservation laws, reductions, and new exact solutions of the extended form of the (2+1) -dimensional generalized Zakharov–Kuznetsov (ZK) equation are studied. This generalized ZK model includes three arbitrary exponents m, n and k in the nonlinear terms. The Lie algebra for the generalized ZK model is four dimensional. Conservation laws are derived using the multiplier method and five cases arise under different restrictions on the parameters m, n and k of the model. Lie symmetries associated with the conserved vectors are determined for each case. Furthermore, several reductions and new exact solutions of ZK equation are established using the generalized double reduction theory. These solutions advance the analysis of nonlinear dynamics in physical systems including plasmas, shallow water flows, and other dispersive media.
In this paper, we study the 3D dissipative fluid–dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the Navier–Stokes/Poisson–Nernst–Planck system. We prove that if the partial derivatives of two velocity components (∂ _1 u_1, ∂ _2 u_2) satisfy ∫ _0^T‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ ^8/5-4α_Ḃ_∞ ,∞^-α/1 + ln (e + ‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ _Ḃ_∞ , ∞^-α ) dt< ∞ for 0< α < 1, then the local solution can be smoothly extended past the time t = T . Particularly, a regularity criterion is further established for the critical case α =0 . These results represent further improvements of previous studies by Zhao et al. (2016; 2019; 2025) and Wu (2019). Moreover, we extend the results by Zhang (2008) and Dong et al. (2010; 2011) for the incompressible Navier–Stokes equations.
We investigate the phenomenon of Landau damping within the context of the Vlasov-Riesz system. Employing a nonlinear analysis technique developed by Ionescu, Pausader, Wang, and Widmayer, our primary objective is to extend the understanding of Landau damping to kinetic models beyond the traditional Vlasov-Poisson system. Our analysis is grounded in the use of the Gevrey- γ norm, and we demonstrate that the critical Gevrey index γ =-4/3α +5/3 is dependent on the parameter α from the Riesz interaction, defined as U=( -Δ) ^-α(ρ -1) . This study not only broadens the applicative scope of Landau damping but also deepens the theoretical understanding of its mechanisms in more complex kinetic frameworks.
In this work, we provide an analytical proof of the existence of attracting invariant tori supporting quasiperiodic dynamics in a family of three-dimensional dissipative systems modeling self-oscillatory electronic devices. This family was originally proposed by Kuznetsov, Kuznetsov, and Stankevich, who observed numerically such quasiperiodic dynamics and evidences for the existence of invariant tori. Our results establish the existence of normally hyperbolic attracting tori using recent advances in averaging theory. We also show that the dynamics on these tori are always either periodic or quasiperiodic, and that quasiperiodic motion occurs for a set of parameter values with positive Lebesgue measure. These findings not only confirm and extend the prior numerical observations, but also demonstrate that quasiperiodicity arises with positive probability when parameters are randomly chosen. Moreover, our study highlights the effectiveness of averaging theory as a powerful analytical tool for detecting attractors in applied dynamical systems.
This paper investigates a class of stochastic nonlinear fractional partial differential equations of complex Ginzburg-Landau (CGL) type posed on the multidimensional positive orthant. The model incorporates both interior multiplicative noise and Robin-type stochastic boundary forcing, acting independently along each coordinate hyperplane. The governing equation includes a vectorial Caputo-type fractional Laplacian of order β∈( 3/2,2) , a nonlinear term of the cubic type and non-Gaussian noise introduced via the modulus of Brownian motions convolved with deterministic kernels. We focus on the formulation and analysis of mild solutions under minimal smoothness assumptions. The main analytical challenges stem from the nonlocal nature of fractional diffusion, the loss of martingale structure due to non-centered boundary noise, and the intricate coupling induced by Robin-type conditions in multiple spatial dimensions. We develop a novel framework combining infinite-dimensional Itô calculus, Laplace-transform methods, and weighted fractional Sobolev estimates. Our contributions include the construction of a well-posed mild solution framework, the derivation of probabilistic a priori bounds, and second moment estimates. We also characterize the long-time behavior of solutions, identifying decay rates and stochastic regularization phenomena driven by the fractional Laplacian. The novelty of this work lies in the synthesis of fractional diffusion, nonlinear complex dynamics, and non-Gaussian boundary noise—a setting that remains largely unexplored. Unlike previous studies restricted to Gaussian interior noise, our approach captures realistic dynamics involving delayed boundary responses and spatially distributed stochastic inputs. These results offer new insights into the behavior of boundary-driven fractional SPDEs and provide a foundation for future work in stochastic modeling of anomalous transport and interface phenomena.
In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: {[ u_t = ∇· (D(u)∇ u) - χ∇· (u∇ v) + ξ _1∇· (u^m∇ w) , x∈Ω , t> 0,; v_t = Δ v + ξ _2∇· (v∇ w) - v + u, x∈Ω , t> 0,; 0 = Δ w - w + u, x∈Ω , t> 0,; ∂ u/∂ν = ∂ v/∂ν = ∂ w/∂ν = 0, x∈∂Ω , t > 0,; u(x,0) = u_0(x), v(x,0) = v_0(x), x∈Ω , ]. in a bounded smooth domain Ω⊂ℝ^n (n≤ 3) , where the parameter χ , ξ _1,ξ _2 > 0 , D(u) is supposed to satisfy the following property D(u) ≥ (u + 1)^α with α > 0. Assume that ξ _1≥λ _1^*χ ^2 , where the parameter λ _1^* = λ _1^*(u_0, v_0, Ω ) > 0 ; then the system admits a global classical solution (u, v, w) via subtle energy estimates. Moreover, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u̅_0, u̅_0, u̅_0) provided the initial data u_0 is sufficiently small, where u̅_0 = ∫ _Ω u_0/|Ω | .
We consider the nonlinear Schrödinger equations with nonlinear Neumann boundary conditions on the right half-line ℝ_+= ( 0,∞) . Our purpose is to study the maximal existence time of strong solutions to the nonlinear Schrödinger equations under certain compatibility boundary conditions.
The problem of estimating the maximal number H(m) of limit cycles that planar polynomial vector fields of degree m can exhibit has long been a central question in the qualitative theory of planar dynamical systems. A natural extension to the three-dimensional space is to study the maximum number N(m) of limit tori that can occur in spatial polynomial vector fields of degree m. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number N_h(m) , if finite, increases strictly with m. More precisely, we prove that N_h(m+1) ⩾ N_h(m) + 1 . Our proof relies on two central results established in this paper. The first is that the normal hyperbolicity of compact invariant manifolds is preserved under time reparametrizations. Despite the fundamental nature of this statement, a complete proof has, surprisingly, not previously appeared in the literature, except under rather restrictive assumptions on the flow restricted to the invariant manifold. The second result concerns the torus bifurcation phenomenon near Hopf–Zero equilibria in spatial vector fields. While the conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of torus bifurcation, assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach not only circumvents intricate computations involving higher-order normal forms but also ensures the normal hyperbolicity of the bifurcated torus.
This paper deals with initial-boundary value problems for a thin quasilinear plate equation. It is shown the global existence of weak and strong solutions, that the energy of the weak solution has an exponential decay rate, and the strong solution is uniformly stable (i.e., the solution’s behaviour changes continuously with the data) and consequently it is unique.
Our particular interest is in the periodic traveling waves, which we construct and study in detail. The question of the spectral stability of spatially periodic waves for the Boussinesq and symmetric regularized long-wave equations with fifth-order nonlinearity are studied. The simultaneous consideration of the two equations is justified, because they have many similarities. However, they have certain differences as well. The main result of this paper is a theorem which establishes spectral stability (instability) of the found periodic waves for the studied equations.
In this paper, we establish the existence of a weak solution for the steady RANS model, where the Navier-Stokes equations are coupled with the equation governing the turbulent kinetic energy (TKE). This coupling of the two equations occurs both through the eddy viscosity and eddy diffusion coefficients and also through a Dirichlet boundary condition on a portion of the boundary leading to energy production. Additionally, the right-hand side in the TKE equation contains a term ν (k) |∇u|^2 that is only integrable. The proof of the existence of a weak solution is obtained by regularization of the original problem, relaxing the terms that are not sufficiently regular. Additionally, due to the non-homogeneous Dirichlet boundary condition on a portion of the boundary for the energy variable k, we introduce a change of variable for k using the lifting of its value on the boundary of the domain. This implies a new variational formulation with regularization of terms arising from this lifting.
It is well-known that the gradient of velocity, when considered in its scaling-invariant Lebesgue spaces, ensures the regularity of weak solutions to the three-dimensional incompressible Navier-Stokes equations, as established by Beirão da Veiga [2, 3]. Inspired by the recent work [43] of the second author, and utilizing the concept of effective viscous flux, we derive certain Beirão da Veiga-type blow-up criteria for the isentropic compressible Navier-Stokes equations allowing vacuum, under the assumption of a bounded supernorm of the density. This formulation complements and extends the Ladyzhenskaya-Prodi-Serrin-type blow-up criteria to compressible viscous fluids. Notably, the velocity gradient can be replaced by the deformation tensor, and these criteria remain applicable to isentropic compressible magnetohydrodynamic equations.
The present work deals with a Keller-Segel-Navier-Stokes system in two-dimensional domains, which involves a cell density, an attractive chemical signal consumed by the cells, and a repulsive one produced by the cells. Potential consumption and production rates jointly with a generalized logistic law for the cells are considered, under non-flux boundary conditions for cell and chemical variables and a Dirichlet boundary condition for the velocity field. We establish the existence of global classical solutions for the system under some constraints related to the rates of attraction, consumption, and logistic competition with chemotactic sensitivities.
We study existence and global behavior of solutions to a wave equation with exponential growth source term and locally distributed nonlinear dissipation posed in a bounded domain Ω⊂ℝ^2 . The results of this paper are the local wellposedness in the energy space, uniform decay of the energy as t→∞ in the defocusing case ( +f(u) ), and the dichotomy into global existence/uniform decay and blow-up (for a=1 ) in the focusing case ( -f(u) ) for those solutions with energy less than d of the ground state, where d is the level of the Mountain Pass Theorem. We give a proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof is that the decay rate is independent of the nonlinearity.
This paper considers the following elliptic problem with logarithmic nonlinearity and Hardy-Littlewood-Sobolev critical exponent { -Δ u=μ |u|^p-2uln |u|^2+ (∫ _Ω|u(y)|^2_α ^*/|x-y|^αdy )|u|^2_α ^*-2u, in Ω , u=0, on ∂Ω , . where Ω is a bounded domain of ℝ^N (N≥ 3) with smooth boundary. μ >0 , 0<α
We study the existence of a positive solution for a class of nonlinear Schrödinger equations -Δ u+V(x)u=f(u), u ∈𝒟^1,2(ℝ^N), N≥ 3. Here the potential V is symmetric under a group action G ⊂ O(N) and decay to zero at infinity, and the nonlinearity f, under very mild hypotheses, is asymptotically linear or superlinear and subcritical at infinity, not satisfying any monotonicity condition.
We study a generalized stochastic heat equation driven by space-time white noise and involving a spatially heterogeneous operator. The model describes heat conduction in a medium composed of two different materials. We focus on the temporal power variations of the solution and analyze two types of power variations: the quartic variation and a weighted quadratic variation. For both, we establish Central Limit Theorem and we derive convergence rates under the Wasserstein distance. These theoretical results are then applied to statistical inference for a parametric version of the model with unknown drift parameter. Based on power variations, we construct consistent estimators and study their asymptotic distributions. Our work combines probabilistic analysis with statistical methodology in the context of SPDEs with spatial heterogeneity.
In this paper we study existence, uniqueness, nonexistence and asymptotic behavior of solutions to the following class of elliptic problems with strongly singular nonlinearities { - Δ u -1/2( x·∇ u) =λ u + h(x)u^-α in ℝ^N, u >0 in ℝ^N, u(x)→ 0 as |x|→∞ , . where N≥ 3, λ∈ℝ and α >1 is a real parameter and h: ℝ^N→ℝ is a measurable function. With respect to singular problems, the novelty of this paper is that the condition h ∈ L^1 is not imposed, unlike in most related works. By employing an auxiliary set ℳ (which contains the Nehari manifold 𝒩 ) together with the fibering map, we prove that the functional associated with the problem admits a minimizer u on 𝒩 . The properties of u and 𝒩 are then exploited to control the singular term and show that u is solution of the problem.
