This paper investigates the well-posedness of a class of one-dimensional Boussinesq-type systems modeling nonlinear dispersive water waves over varying bottom topographies and offers the first rigorous analytical framework for the initial-value formulation of the Madsen and Sørensen Boussinesq-type equations. The equations under consideration depend on a real parameter B, which modulates the strength of the dispersive effect, and on the static water depth h_0 . In the case of flat bathymetry, we provide a detailed analysis of the associated key operator, formulate the quasilinear system, and construct a suitable symmetrizer to define an appropriate energy norm. Local well-posedness in the "hyperbolic" space is established under three distinct assumptions on the pair (B, h_0) . The analysis is subsequently extended to the variable bathymetry setting in the critical case B = -1/6 , which is the only value for which the model admits a mathematically tractable structure allowing for a rigorous treatment. For this setting, we derive a symmetrization and prove local well-posedness results under natural conditions.
In this paper we study the dynamics on the invariant spheres for two classes of 3-dimensional polynomial differential systems of the form ẋ=x+Q(x), x=(x,y,z)∈ℝ^3. First, when Q(x) is linear we characterize these differential systems having invariant the unit sphere centered at the origin of coordinates and describe their phase portraits depending on the parameters of the system. Second, when Q(x) is a cubic homogeneous polynomial we classify these differential systems having invariant the unit sphere centered at the origin of coordinates and we describe the dynamics on this sphere depending on the parameters of the system.
This paper establishes the theoretical result concerning the stability of non-zero point of pullback random attractors in a regular space. To be precise, we prove the upper semicontinuity of pullback random attractors for non-autonomous random dynamical systems with delay parameter ρ as ρ tends to ρ _0 ( 0<ρ <ρ _0 ). Since the attractors of the original system and the corresponding limiting system belong to two different spaces, the main difficulty lies in how to place the attractors of these two systems in the same space. In the application part, we consider stochastic retarded magnetohydrodynamics (MHD) equations with non-autonomous forcing term. Since the solutions of MHD equations lack higher regularity for initial data belonging to a space of limited smoothness, we prove the pullback asymptotic compactness of solution operators for such equations via the spectrum decomposition technique, and then obtain the existence and uniqueness of regular pullback random attractors.
In this paper we consider a nonisotropic harmonic oscillator with Coriolis coupling, perturbed by a generalized cubic Hénon-Heiles potential. This model depends on three parameters which play an important role in the symmetry breaking of the classical Hénon-Heiles system. The main goal is to describe the equilibrium solutions of the system and their linear stability character. We also analyze the nonlinear stability of the equilibrium point at the origin, showing how the symmetry breaking, which manifests itself through the parameters of the problem, affects the stability properties through resonances.
We introduce a class of substitutions called homogeneous substitutions, with the Morse substitution as a typical example. We prove that the subshift generated by a homogeneous substitution is τ -uniquely ergodic for any τ∈ [0,1) . As an application, we show that in a subshift of finite type, the ergodic measure generated by a homogeneous substitution can be approximated sublinearly by periodic measures.
For the following two-species chemotaxis system with two signal productions: 0.1 {[ [ u_t=Δ u-χ∇·( u∇ v) , x ∈Ω , t>0,; 0=Δ v-μ _w+w, x ∈Ω , t>0,; w_t=Δ w-ξ∇·( w∇ z) , x ∈Ω , t>0,; 0=Δ z-μ _u+u, x ∈Ω , t>0, ] ]. under homogeneous Neumann boundary conditions in a smooth, bounded domain Ω⊂ℝ^2 , where μ _u=-∫ _Ω u(· ,0) and μ _w=-∫ _Ω w(· ,0) . We establish new criteria distinguishing between finite-time blow-up and boundedness for radially symmetric solutions, that are
We consider a transmission problem for a string composed by three different types of components: a viscoelastic material of memory type, an elastic material (without dissipation) and a Kelvin-Voigt viscoelastic material. The dissipative mechanisms can act on different parts of the string or they can even intersect. The rate of decay will depend on the position of the components in the following sense: if there exists at least one elastic part of the string that connects with the viscoelastic part of Kelvin-Voigt type but not with the memory part, then there is no exponential stability. In fact, the solutions decay polynomially with the rate t^-2 , which is optimal. In every other situation, in which both viscoelastic dampings are present, there exists exponential stability of the solution.
In this work, we analyze the asymptotic behavior of solutions to a generalized semilinear damped wave equation with time-dependent coefficients, subject to homogeneous Dirichlet boundary conditions on a smooth bounded domain Ω⊂ℝ^N , N ≥ 3 . The equation involves functions α _ϵ (t) , β _ϵ (t) , μ _ϵ (t) , and η _ϵ (t) depending continuously on time and on a parameter ϵ∈ [0,1] , and includes a nonlinear term f(u) satisfying suitable growth, regularity, and dissipativity assumptions. By means of a time-rescaling argument, we establish the existence of pullback exponential attractors and pullback attractors whose sections possess uniformly bounded finite fractal dimension. Furthermore, we prove the continuity of the family of pullback exponential attractors and the upper semicontinuity of the family of pullback attractors with respect to the parameter ϵ . These results provide a unified framework for analyzing the robustness and regularity of nonautonomous dissipative dynamics under parameter perturbations.
The existence of rotating wave solutions of a general reaction-diffusion system with nonlocal interaction on a circle are proved using a new bifurcation approach with two parameters, and a normal form for the rotating wave bifurcations is also developed to compute the direction of the bifurcation diagram. Theoretical results are applied to two model systems for the existence of rotating waves which are further verified by numerical simulations.
In this work we consider the class of partially integrable 3-dimensional piecewise smooth vector fields Y=(X^+,X^-) with separation set Σ ={(x,y,z)∈ℝ^3: z=0} and first integral H(x,y,z)=x^2+y^2+z^2 that leaves invariant any sphere centered at the origin, 𝕊_ρ ^2={(x,y,z)∈ℝ^3: x^2+y^2+z^2=ρ ^2} . We denote this class by 𝒳 and by 𝒳_n when X^± are polynomial vector fields of degree n. Our main goal is to study piecewise smooth vector fields in the class 𝒳_1 presenting three periodic annuli on the invariant sphere 𝕊_1^2 proving that there exists a mixed simultaneous configuration with at least five limit cycles bifurcating simultaneously of them, considering polynomial perturbations inside the class 𝒳_2 .
We investigate expansive homeomorphisms of a compact metric space X via the commutative C^* -algebra C(X) of continuous complex-valued functions, interpreted as observables of the dynamical system. We introduce the concept of expansive observables, namely functions in C(X) whose level sets separate distinct orbits. We show that the collection of expansive observables forms an F_σ -subalgebra of C(X) . Moreover, for equicontinuous or minimal distal homeomorphisms, we provide a complete characterization, proving that in this setting the only expansive observables are locally constant functions. We also establish that topologically conjugate homeomorphisms have identical algebras of expansive observables. Within this framework, we prove that the set of periodic points intersects only countably many level sets of any expansive observable. Finally, we show that no homeomorphism of the circle or of the unit interval admits a dense set of expansive observables. As an application, we obtain C^* -algebraic proofs of some classical results in the theory of expansive dynamical systems.
By the use of the Poincaré–Birkhoff theorem, we generalize several results on the existence and multiplicity of periodic solutions for scalar second order differential equations with a periodic nonlinearity, the paradigm of which is the pendulum equation. As a particular case, we extend a recent result by Amster, Cid, and Mawhin on a relativistic model of the pendulum.
Recently, in Glogić et al. (Non-uniqueness of mild solutions to supercritical heat equations. arXiv:2501.17032 (2025)), it has been shown that the focusing power nonlinearity heat equation NLH ∂ _t u -Δ u = |u|^p-1u, p>1, in dimensions d ≥ 3 has non-unique local solutions in L^q(ℝ^d) for q < d(p-1)/2 provided that p < p_JL , where p_JL denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in (NLH) is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of (NLH) with additive noise. Second, we discuss how randomizing the initial conditions of (NLH) affects its well-posedness.
We provide a stochastic characterization of the classical shadowing property of dynamical systems. Then, we reformulate stochastic stability for differentiable systems as well as topological dynamical systems. We also investigate the stability of the set of invariant measures for homeomorphisms on compact metric spaces under C^0 -perturbations. We explore alternative distances, such as the random distance, and introduce the concept of random-stable homeomorphism. These results reveal that while every topologically stable homeomorphism of a compact metric space is also random-stable, the converse is not always true. We analyze the recurrent set and the chain recurrent set of a random-stable homeomorphism on compact manifolds.
The present paper is devoted to the investigation of the long term behavior of a class of higher-dimensional singular diffusion processes that get absorbed by the extinction set in finite time with probability one. Our primary focus is on the analysis of quasi-stationary distributions (QSDs), which describe the long term behavior of the system conditioned on not being absorbed. Under natural Lyapunov conditions, we construct a QSD and prove the sharp exponential convergence to this QSD for compactly supported initial distributions. Under stronger Lyapunov conditions ensuring that the diffusion process comes down from infinity, we show the uniqueness of a QSD and the exponential convergence to the QSD for all initial distributions. Our results can be seen as the higher-dimensional generalization of Cattiaux et al (Ann. Prob. 2009) as well as the complement to Hening and Nguyen (Ann. Appl. Prob. 2018) which looks at the long term behavior of higher-dimensional diffusions that can only become extinct asymptotically. As applications, we show how our results can be applied to many ecological models, among which cooperative, competitive, and predator-prey Lotka-Volterra systems. The cornerstone of our approach revolves around a uniformly elliptic operator that we relate through a two-step transform to the Fokker-Planck operator associated with the diffusion process. This operator only has singular coefficients in its zeroth-order terms and can be handled more easily than the Fokker-Planck operator, which is defined on an unbounded domain and exhibits degeneracy in the extinction set. For this operator, we establish the discreteness of its spectrum, its principal spectral theory, the stochastic representation of the semigroup generated by it, and the global regularity for the associated parabolic equation. These results extend beyond the study of QSDs and are of independent interest, especially in the context of spectral theory for degenerate elliptic operators on unbounded domains. As direct consequences, we show that the extinction rate associated with the QSD and the sharp exponential convergence rate are respectively given by the absolute value of the principal eigenvalue and the spectral gap, between the principal eigenvalue and the rest of the spectrum, of this operator. Such characterizations of the QSD and exponential convergence rate were previously unknown in the context of irreversible singular diffusion processes.
In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the x–variable evolves independently via ẋ = Ax+f(t,x) and influences the y–variable through ẏ = By+g(x,y) . By first analyzing the long–time dynamics of the nonautonomous x–equation and then examining the resulting y–dynamics for each asymptotic state of x, we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on f, and a deeper level of description, when the term f(t, x) is asymptotically autonomous. Finally, we present a description when the term f(t, x) is a small nonautonomous perturbation of an autonomous term.
In this paper we investigate a predator-prey reaction-diffusion system incorporating prey-taxis and nonlinear boundary conditions. The model describes the directed movement of predators toward higher prey densities, while nonlinear boundary conditions model the net flux of populations across the habitat boundary as a nonlinear function of their density. Using principal eigenvalue analysis and bifurcation theory, we derive criteria for the existence and stability of semi-trivial steady states. Furthermore, we prove the existence of local and global bifurcation of positive steady-state solutions from the prey-only equilibrium. Our results reveal that predator persistence is jointly determined by growth rates, predation efficiency, diffusion rates, and boundary-driven mortality. In particular, we identify critical thresholds for boundary loss rates, predator growth rates, and predation rates that govern the transition between extinction and coexistence, highlighting the ecological significance of dispersal strategies and habitat boundaries in shaping predator-prey dynamics.
The problem from the title will be considered separately for explicit differential inclusions with upper-Carathéodory right-hand sides and implicit differential inclusions with Marchaud right-hand sides. In the first case, several earlier results for differential equations will be generalized to a multivalued or discontinuous setting. In the second case, which is a main object of our study, quite rare multiplicity results for implicit boundary value problems will be extended.
We study the long-time behavior of large solutions to the dispersion generalized Benjamin-Ono equation. By means of virial identities, we identify spatial regions around the origin, growing unbounded in time, not containing the soliton region in which every solution belonging in a suitable Sobolev space, necessarily decays to zero along some sequence of times. A similar result is also obtained for solutions of Benjamin equation, a model for gravity-capillary surface waves of the solitary type on deep water.
