Journal of Differential Equations最新文献

We study the blow-up phenomena for some integrable Camassa-Holm type equations on the line. For the two-component Camassa-Holm system, we give a sufficient condition on the initial data that leads to a blow-up. For the Degasperis-Procesi equation, we establish a local-in-space blow-up criterion which improves considerably the early criterion based on the sign-changing momentum. Besides, we obtain some new blow-up criteria for the Novikov equation and the modified Camassa-Holm equation.

In this paper, we establish and study a stochastic mosquito population suppression model incorporating the release of Wolbachia-infected males and time switching, where stochastic noises are given by independent standard Brownian motions. By combining the actual mosquito control strategy in Guangzhou, we assume that the waiting release period T between two consecutive releases of Wolbachia-infected males is less than the sexually active lifespan $\bar{T}$ of them. The existence and uniqueness of global positive solutions and stochastically ultimate boundedness for the stochastic model are obtained. Some sufficient conditions for the extinction and the existence of stochastic non-trivial periodic solutions are established. Furthermore, we assume that the release function is a general periodic function and some stochastic dynamical behaviors are obtained. Numerical examples are presented to illustrate the theoretical results.

This paper is concerned with the convergence of invariant measures in the Wasserstein sense for the stochastic Schrödinger delay lattice systems as delay parameter ρ or parameter β approaches zero. Through pth-order moment estimates of solutions to systems, as well as the Hölder continuity estimates of solutions with respect to time, we obtain the convergence of solutions about initial data and the above parameters. Then together with high-order moment estimates of invariant measures, we prove that the unique invariant measure of such delay lattice system converges to the invariant measure of limiting system in the Wasserstein sense as delay parameter ρ or parameter β approaches zero, and the corresponding convergence rate is also obtained.

The Ladyzhenskaya-Prodi-Serrin type $L^{s,r}$ criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global $L^{s,r}$ norm is usually large and hence hard to control. Replacing the global $L^{s,r}$ norm with some kind of local norm is interesting. In this article, we introduce a local $L^{s,r}$ space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local $L^{s,r}$ type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local $L^{s,r}$ type norms is necessary and sufficient to affirmatively answer the millennium problem.

New results on the behaviour of the fast motion in slow-fast systems of ODEs with dependence on the fast time are given in terms of tracking of nonautonomous attractors. Under quite general assumptions, including the uniform ultimate boundedness of the solutions of the layer problems, inflated pullback attractors are considered. In general, one cannot disregard the inflated version of the pullback attractor, but it is possible under the continuity of the fiber projection map of the attractor. The problem of the limit of the solutions of the slow-fast system at each fixed positive value of the slow time is also treated and in this formulation the critical set is given by the union of the fibers of the pullback attractors. The results can be seen as extensions of the classical Tikhonov theorem to the nonautonomous setting.

In this paper, we study the Cauchy problem of the inhomogeneous Landau equation with hard potentials under the perturbation framework to global equilibrium. We prove that the solution to the Cauchy problem enjoys the analytic Gelfand-Shilov regularizing effect with a Sobolev initial datum for positive time.

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path $(B^H,W)$ with $H\in (1/3,1/2)$ as $\epsilon \to 0$. Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we embed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.

In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verification of this condition is facilitated through a combination of analytic techniques and rigorous numerical computations. Moreover, an additional condition is derived, establishing that the localized pattern serves as the limit of a family of periodic solutions (in space) as the period tends to infinity. The integration of analytical tools and meticulous numerical analysis ensures a comprehensive validation of this condition. To illustrate the efficacy of the proposed methodology, we present computer-assisted proofs for the existence of three distinct unbounded branches of periodic solutions in the planar Swift-Hohenberg equation, all converging towards a localized planar pattern, whose existence is also proven constructively. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at [1].

This paper considers the Keller-Segel model coupled to stochastic Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of oxygen and bacteria densities evolving within a stochastically forced 2D incompressible viscous flow. Our main goal is to investigate the existence and uniqueness of global solutions (strong in the probabilistic sense and weak in the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with proper regularization and cut-off operators in $H^s\left(R^2\right)$ is introduced, and the existence of approximate solution is proved by some a priori uniform bounds and a careful analysis on the approximation scheme. Under appropriate assumptions, two types of stochastic entropy-energy inequalities that seem to be new in their forms are derived, which together with the Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the sequence of approximate solutions converges to a global martingale weak solution. In addition, when $\chi (\cdot )\equiv \text{const.}>0$, we prove that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem that the KS-SNS system admits a unique global pathwise weak solution.