In this paper, we consider a Boussinesq equation containing weak backward diffusion, delay in the convection term, dissipation and Marangoni effect. By applying geometric singular perturbation theory, a locally invariant manifold diffeomorphic to the critical manifold is established. For Boussinesq equation with delay and weak backward diffusion, the monotonicity of ratio of Abelian integrals is analyzed by utilizing the Picard–Fuchs equation. The conditions on existence of a unique periodic wave and solitary waves are obtained as well as the bound of wave speed. For Boussinesq equation with weak backward diffusion, dissipation and Marangoni effect, the corresponding Melnikov function containing three generic elements is given. The parametric conditions on existence of a unique and two periodic waves are derived respectively. Furthermore, the existence of a unique solitary wave is proved under some parametric conditions.
We show that the hyperbolicity of a linear delay-difference equation with infinite delay, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.
We consider the four-component chemotaxis-Navier–Stokes system in : Utilizing the Fourier localization technique alongside the inherent structure of the equations, we achieve global well-posedness for a class of rough initial data in the context of the 2D incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation for .
In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.
We study the low Mach number limit of the compressible magnetohydrodynamic equations in a bounded domain with ill-prepared initial data. The velocity field satisfies the Navier-slip boundary conditions and the magnetic field satisfies the perfectly conducting boundary conditions. By performing energy estimate in the conormal Sobolev space and proving the maximum principle to the equations satisfied by , we overcome the difficulties caused by the simultaneous occurrence of fast oscillation and boundary layer. As a consequence, the uniform existence and the convergence of solutions are obtained.
In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter , where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.
In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.
The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.
This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.
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