We investigate all-time existence of smooth solutions for the non-isentropic slightly compressible Navier-Stokes equations with Dirichlet boundary conditions in 2D exterior domains. By virtue of the decay property of smooth solutions to the limiting system, we establish all time existence of strong solutions for the corresponding compressible system, provided that the Mach number is sufficiently small. Additionally, as the Mach number approaches zero, solutions of the compressible system uniformly converge to that of the incompressible system for all time. In particular, to derive higher-order estimates of density function near the boundary, we utilize the global geometric tools developed by Christodoulou and Lindblad [Commun. Pure Appl. Math. 53 (2000) 1536–1602].
In the present paper, we consider a general Bazykin-type model that describes the interaction of two populations. We provide sufficient conditions for the existence of a positive periodic solution where each population is localized independently within a distinct conical annular region. We illustrate how our conditions can be verified for certain particular cases and support these results with numerical simulations.
In this paper, we investigate the impact of dormancy on the survival of a population in a periodic environment. We extend the classical periodic chemostat model by including a quiescent compartment with nutrient-dependent dormancy and activation rates, together with a maturation delay between nutrient consumption and reproduction. We first analyze the autonomous version of the system and conclude that, consistently with previous results, dormancy does not improve survival in a constant environment. We then introduce a periodic nutrient inflow and, using a skew-product formulation together with the Krein-Rutman theorem and persistence theory, show that uniform persistence holds if and only if the principal Floquet exponent of the washout periodic orbit is positive. Using Horn’s fixed point theorem, we also prove the existence of a non-trivial periodic solution under the same conditions. Next, we characterize optimal switching strategies and show that Heaviside bang-bang dormancy switching rates maximize persistence. Finally, we prove that, under some very mild conditions, there exists a switching strategy ensuring persistence of the species. Our results show that, while dormancy is neutral in constant environments, it can critically improve survival in periodically fluctuating environments when optimally regulated.
This paper presents a thermodynamically consistent framework for the numerical treatment of finite-strain contact problems involving frictional interaction and inelastic material transformations, with particular reference to plastic or superelastic materials. The formulation combines large-deformation continuum mechanics with dissipative constitutive behavior, including viscoelasticity, and unilateral contact constraints. We devise a time-integration scheme based on the midpoint rule, combined with a semi-smooth Newton strategy for the spatial discretization of frictional contact and inelastic transformation, in such a way that continuous energy balance is reproduced at the discrete level. We carry out a theoretical analysis of the discrete energy balance based on the numerical scheme, and verify the obtained estimates by means of a numerical simulation of two nearly-rigid bars compressing a hyper-visco-elasto-plastic ball. The results demonstrate that the proposed approach accurately captures complex nonlinear phenomena while maintaining robust energy behavior over long-term simulations. The study highlights the importance of discrete energy consistency for reliable and physically meaningful simulations of strongly coupled contact and inelastic processes at finite strain.
This paper studies the following attraction-repulsion chemotaxis system involving nonlinear indirect repulsion-signal mechanism{ut=Δu−ξ∇·(u∇v)+χ∇·(u∇z)+u(a−buk),x∈Ω,t>0,0=Δv−v+uγ1,x∈Ω,t>0,zt=Δz−z+wγ2,x∈Ω,t>0,0=Δw−w+uγ3,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=∂z∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),z(x,0)=z0(x),x∈Ω,where Ω⊂Rn(n≥1) is a smoothly bounded domain and ξ, χ, a, b, k, γ1, γ2, γ3 > 0. If ξ and χ are small enough, it is shown that the global classical solution (u, v, z, w) exponentially converges to ((ab)1k,(ab)γ1k,(ab)γ2γ3k,(ab)γ3k) in L∞(Ω) as t → ∞. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.
In this paper, we focus on a Dirichlet problem involving a multi-phase operator with variable exponents and a reaction in which we have the combined effects of a strongly singular term and of a concave term. Then, under very general assumptions on the exponents and the coefficient functions, we produce a non negative solution for the problem under consideration. We point out that we establish such result thanks to variational tools, such as the Ekeland’s variational principle.
For a simple tumor model with a periodic supply of external nutrients σ=ϕ(t) on the boundary, a unique radially symmetric T-periodic positive solution (σ*(r, t), p*(r, t), R*(t)) was established by [1, 2]. We denote by μ the tumor aggressiveness constant. This T-periodic solution is stable for any μ > 0 with respect to all radially symmetric perturbations [1], [2]; and for some threshold μ*, if 0 < μ < μ* then this T-periodic solution is linearly stable under non-radially symmetric perturbations, whereas if μ > μ* then this T-periodic solution is linearly unstable [2, 3]. In this paper, we establish a sequence of non-radially symmetric T-periodic solutions bifurcating from the radially symmetric T-periodic solution.
Inspired by Jang’s work (Arch. Ration. Mech. Anal. 194 (2009), 531–584), we consider the diffusive limit of the Vlasov-Maxwell-Landau system for Coulomb potentials inside a periodic box. More precisely, we establish the global-in-time validity of diffusive expansions for solutions around Maxwellian equilibrium to the rescaled Vlasov-Maxwell-Landau system by using the nonlinear weighted energy method, provided the initial perturbation is smooth enough. This leads to the mathematical derivation of dissipative hydrodynamic equations, which are collectively called the Vlasov-Navier-Stokes-Fourier system. We also obtain uniform estimates for the first-order and higher-order remainders, with the time-decay rate (1+tk)−k.
In this work, we investigate the global regularity properties of the two-dimensional incompressible inviscid Boussinesq system. While the classical Beale-Kato-Majda criterion requires control of the full gradient ∥∇u∥L∞ over time, recent results by Fanelli (to appear Journal of the European Mathematical Society, 2026) establish a geometric continuation criterion for 2D non-homogeneous incompressible Euler equations showing that it suffices to control the velocity gradient along a single dynamically determined direction, namely tangent to the level sets of the scalar temperature field. Let X=∇⊥θ denote the vector field orthogonal to the temperature gradient. Then, if∫0T∥∂X(t)u(t)∥L∞dt<+∞,the smooth solution (u, θ) can be extended beyond time T. The proof exploits the transport structure of X together with the vorticity formulation, thereby avoiding explicit dependence on the pressure term. As a corollary, singularities cannot form in regions where the scalar field is constant, recovering the classical global well-posedness of the 2D Euler equations. This result provides a refined understanding of the directional nature of potential singularities in active scalar systems.
We consider the planar Lane--Emden equation with a positive Robin parameter on a disk. The positive radial solutions are first parametrized by the logarithmic slope of a single normalized Lane--Emden profile; this yields exactly one positive radial solution for every Robin parameter. The zero-eigenvalue condition in the first angular sector is then reduced to the vanishing of an explicit scalar function \( F_p \). For every \( p\geq12 \), a phase--plane estimate proves that \( F_p \) is negative at a point where the logarithmic slope equals \( 1/2 \), whereas \( F_p \) is positive near both endpoints of its interval of definition. We select two sign-changing zeros and prove that both are simple. A mode-by-mode spectral analysis shows that, in a reflection-invariant space, the linearized kernel is one-dimensional at either zero and that the corresponding eigenvalue crosses transversally. The Crandall--Rabinowitz theorem therefore produces two local branches of positive nonradial solutions. Consequently, for \( p\geq12 \), uniqueness among all positive solutions fails for Robin parameters converging to two distinguished values, even though the positive radial solution remains unique for every parameter.
We consider the system of non-autonomous nonlinear Schrödinger equations with homogeneous Dirichlet boundary conditions in a domain [0, L] and time-dependent forcing that models the motion of large-scale Rossby waves. We study the local and global well-posedness and establish the existence and stability of the family of pullback exponential attractors associated with the problem. As a consequence of the techniques employed, we also derive the existence of a pullback attractor whose sections have finite fractal dimension uniformly bounded in time. Finally, we prove the continuity of the family of pullback exponential attractors and the upper semicontinuity of the pullback attractors.
This paper investigates the existence and non-existence of traveling wave solutions in a within-host HIV epidemic model that incorporates spatial diffusion, distributed delay and drug therapy. Precisely speaking, we prove that if the basic reproduction number, denoted by c* > 0, is larger than 1, then there exists a minimal wave speed c* > 0 so that this system admits a non-trivial traveling wave solution for any wave speed c >= c*. Conversely, no such nontrivial traveling waves exist for any 0 < c < c*. If R-0 <= 1, there are no nontrivial traveling waves for this system. Furthermore, the profile of a traveling wave solution is presented via numerical simulations, in addition, we also observe that the upper bound of the distributed delay exerts a significant accelerating effect on a minimal wave speed c* by a parameter sensitivity analysis.
In this paper, we investigate the following incompressible repulsive Keller-Segel-Navier-Stokes system with logarithmic sensitivity { n(t)+u-del n = Delta n+x del-(n/c del c), Chi is an element of Omega, t > 0, c(t )+ u-del c = Delta c-c +n, Chi is an element of Omega, t > 0, u(t )+ (u.del)u = Delta u + del p + n del Phi, Chi is an element of Omega, t > 0, del.u = 0, Chi is an element of Omega, t > 0 in a bounded convex domain Omega subset of R & sup2; with smooth boundary under the no-flux boundary conditions for n, c and the Dirichlet boundary condition for u. We showed that for all x > 0 this system admits a globally defined classical solution for all general regular initial data.
This paper presents a rigorous convergent dimension reduction of a three-dimensional (3D) thermo-viscoelastic thin film of Kelvin-Voigt type into an effective two-dimensional (2D) model within an explicit variational framework. The resulting model accounts for scale-dependent interfacial phenomena. By employing a thickness-dependent scaling and uniform a priori estimates in Sobolev spaces, 3D displacement and temperature fields are shown to converge to limiting 2D fields. This transition leads to a generalized weak Reynolds equation with effective coefficients that embody the thermomechanical coupling at the substrate-film interface. The analysis demonstrates that the reduced 2D problem is well-defined, admits a unique weak solution, and provides an accurate mathematical derivation of the thin-film model under mechanically free and thermally coupled boundary conditions.
The paper provides a well-posedness analysis for a family of stationary Navier-Stokes-type variational-hemivariational inequalities, motivated by applications in fluid mechanics. The family contains various mixed variational equations, mixed variational inequalities and mixed hemivariational inequalities found in the literature as special cases. The main features of the paper are that the existence and uniqueness of both the velocity field and pressure field are established, and the results are proved in an accessible fashion without the need of knowledge of abstract surjectivity results for pseudomonotone operators as required in many references on variationalhemivariational inequalities. The results are applied to the study of a variational-hemivariational inequality of the Navier-Stokes equations for incompressible fluid flows subject to slip conditions of frictional type, both monotone and non-monotone.
This paper investigates limit cycle bifurcations near a cuspidal loop L0 in a class of piecewisesmooth near-Hamiltonian systems. In earlier work, Wei (Nonlinear Analysis: Real World Applications, 2017) derived asymptotic expansions of the Melnikov functions near L0. However, due to computational complexity, the analysis of limit cycles was restricted to cusps of order at most two. In the present paper, we first show that the coefficients of terms of the same order in the expansions of the two Melnikov functions differ only by a constant multiple. Using this observation, we extend Wei's results and establish a general condition for the existence of limit cycles near L0 when the cusp has arbitrary order. Moreover, for a cubic piecewise Hamiltonian system with polynomial perturbations of degree n (n >= 1), we prove that, for suitable parameter values, the system admits at least 5n-4-3[ n ] limit cycles near L0. 2
This study establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space. Additionally, we demonstrate the existence of solutions for this class of problems using the fibering method. Our approach relies on a novel weighted Sobolev embedding developed for the upper half-space.
