This paper focuses on a system of the two-dimensional magnetohydrodynamic equations with horizontal fractional dissipation in the domain Ω=T×R with T=[0,1] being a periodic box. We study the stability problem on perturbations near any fixed magnetic field A=(A1,A2) where A1,A2∈R. Global stability for this system was proven by Feng et al. [1] in the Sobolev setting H3(Ω); however, it seems that one can not establish global stability for the lower regularity solutions to this system by using their method, such as in the Sobolev setting H2(Ω). The present work resolves this problem and the global stability for this system in H2(Ω) is obtained by utilizing the anisotropic interpolation inequalities and the general strong Poincaré type inequalities, which relaxes the result in [1]. In addition, we also show that the oscillation part of the solution converges to zero in H1(Ω) as t → ∞.
A one-dimensional Keller-Segel equation with a piecewise-defined reaction term depending on the density gradient gives rise to a nonsmooth fast-slow system. Using Filippov convexification and geometric singular perturbation theory, we prove the existence of sliding traveling wave solutions for sufficiently small ε > 0 and characterize the corresponding sliding region. In the singular limit ε=0, we construct a singular sliding heteroclinic orbit on the critical manifold and show that it persists for sufficiently small ε > 0 under appropriate conditions on the wave speed and model parameters. The family of perturbed orbits converges to the singular orbit in the Hausdorff distance as ε → 0. These results provide a rigorous construction of sliding traveling waves for a chemotaxis model with gradient-dependent switching.
This paper is devoted to the study of a new partial differential system involving an evolutionary variational inequality, a history-dependent hemivariational inequality and a partial differential equation in Banach spaces. Under mild conditions, by applying Banach’s fixed point theorem, we prove a unique solvability result for the problem. Furthermore, the main results are applied to obtain the unique solvability of a long-memory viscoelastic frictional contact problem that incorporates wear, damage, and a curved contact surface.
Interactions between chemotaxis-driven Hopf bifurcation and other bifurcations can lead to a remarkably rich array of spatiotemporal dynamics. In this paper, the stability of the positive constant equilibrium and the existence of double Hopf bifurcation for a diffusive predator-prey model with chemotactic terms are studied. For the first time, we have formulated computational methods for determining the normal forms associated with double Hopf bifurcation in a reaction-diffusion system that features a chemotactic term and adheres to Neumann boundary conditions. Especially, the developed normal form method is applicable for classifying the dynamics near the double Hopf bifurcation point. Finally, numerical simulations are provided to substantiate the results.
In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter α > 0. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size h tends to zero and the penalization parameter α tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.
