Journal of Differential Equations最新文献

We study the Liouville problem for the steady p-Stokes system in the half-space. We prove that a bounded weak solution of the p-Stokes system with $p>1$ vanishes in two dimensions. For the three dimensional case, the same result is concluded, provided that $p>\frac{5}{3}$.

We consider non-isentropic Euler-Maxwell equations with relaxation times (small physical parameters) arising in the models of magnetized plasma and semiconductors. For smooth periodic initial data sufficiently close to constant steady-states, we prove the uniformly global existence of smooth solutions with respect to the parameter, and the solutions converge global-in-time to the solutions of the energy-transport equations in a slow time scaling as the relaxation time goes to zero. We also establish error estimates between the smooth periodic solutions of the non-isentropic Euler-Maxwell equations and those of energy-transport equations. The proof is based on stream function techniques and the classical energy method but with some new developments.

In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ in $C\left(\right[0,T];R^n)$. Subsequently, we demonstrate that any accumulation point of ${\left\{X^{\epsilon }\right\}}_{0<\epsilon ⩽1}$ can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.

In this paper, we study the spectrum $\sigma \left(L\right)$ of the Lamé operator$L=\frac{d^2}{d{x}^2}-12\wp (x+z_0;\tau )\text{in}{L}^2(R,C),$ where $\wp (z;\tau )$ is the Weierstrass elliptic function with periods 1 and τ, and $z_0\in C$ is chosen such that L has no singularities on $R$. We prove that a point $\lambda \in \sigma \left(L\right)$ is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if λ is a zero of the following cubic polynomial:$\frac{4}{15}{\lambda }^3+\frac{8}{5}{\eta }_1{\lambda }^2-3g_2\lambda +9g_3-6{\eta }_1{g}_2=0.$ We also study the deformation of the spectrum as $\tau =\frac{1}{2}+ib$ with $b>0$ varying. We discover 10 different types of graphs for the spectrum as b varies around the double zeros of the spectral polynomial.

In this paper, we prove the existence of periodic solutions with any prescribed minimal period $T>0$ for even second order Hamiltonian systems and convex first order Hamiltonian systems under the weak Nehari condition instead of Ambrosetti-Rabinowitz's. To this end, we shall develop the method of Nehari manifold to directly deal with a frequently occurring problem where the Nehari set is not a manifold.

Community composition in aquatic environments is influenced by habitat conditions, such as location and size. We propose a system of reaction-diffusion-advection equations for a predator-prey model with variable predator habitat in open advective environments. We investigate the effects of the location and length of the predator's habitat on its invasion. Firstly, we show that the closer the predator's habitat is to the downstream, the easier the predator can invade when its habitat length is fixed. Secondly, we find that increment of the predator's habitat length facilitates its invasion when the upstream boundary of its habitat is fixed. However, increment of the predator's habitat length disadvantages its invasion when the downstream boundary of its habitat is fixed. Thirdly, we obtain the uniqueness of positive steady state when two species reside in different domains. Finally, we numerically analyze how the advection rates affect the populations persistence and spatial distributions of the populations. These findings may have important biological implications and applications on the invasion of predators in open advective environments.

In this paper, we are concerned with the uniform regularity and zero dissipation limit of solutions to the initial boundary value problem of 3D incompressible magnetic Bénard equations in the half space, where the velocity field satisfies the no-slip boundary conditions, the magnetic field satisfies the perfect conducting boundary conditions, and the temperature satisfies either the zero Neumann or zero Dirichlet boundary condition. With the assumption that the magnetic field is transverse to the boundary, we establish the uniform regularity energy estimates of solutions as both viscosity and magnetic diffusion coefficients go to zero, which means there is no strong boundary layer under the no-slip boundary condition even the energy equation is included. Then the zero dissipation limit of solutions for this problem can be regarded as a direct consequence of these uniform regularity estimates by some compactness arguments.

In this paper, we explore a Leslie-type predator-prey model with simplified Holling IV functional response and Allee effects in prey. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and saddle-node types nilpotent bifurcations of codimension four and a degenerate Hopf bifurcation of codimension up to four as the parameters vary. Our results indicate that Allee effects can induce richer dynamics and bifurcations, especially high sensitivities on both parameters and initial densities for coexistence and oscillations of both populations. Moreover, strong Allee effects ($M>0$) (or ‘transitional Allee effects’ ($M=0$) with large predation rates) can cause the coextinction of both populations with some positive initial densities, while weak Allee effects ($M<0$) (or transitional Allee effects with small predation rates) make both populations with positive initial densities persist. Finally, numerical simulations present some illustrations scarce in two-population models, such as the coexistence of three limit cycles and three positive equilibria.

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then M is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes C. Butler's result [5] in dimension two. Using completely different techniques, we also prove an extension of [5] to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible.

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. We justify rigorously the validity of the hydrodynamic limit from the Boltzmann equation of soft potentials to the compressible Euler equations by the Hilbert expansion with multi-scales. Specifically, the Boltzmann solutions are expanded into three parts: interior part, viscous boundary layer and Knudsen boundary layer. Due to the weak effect of collision frequency of soft potentials, new difficulty arises when tackling the existence of Knudsen layer solutions with space decay rate, which has been overcome under some constraint conditions and losing velocity weight arguments.