Qualitative analysis on the diffusive Holling-Tanner predator-prey model

We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steadystate. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel’kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1

Global existence of solutions for compressible Navier–Stokes equations with vacuum

AbstractIn this paper, we will investigate the global existence of solutions for the one-dimensional compressible Navier–Stokes equations when the density is in contact with vacuum continuously. More precisely, the viscosity coefficient is assumed to be a power function of density, i.e., μ(ρ)=Aρθ, where A and θ are positive constants. New global existence result is established for 0<θ<1 when the initial density appears vacuum in the interior of the gas, which is the novelty of the presentation

Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion

We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and diffusion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some diffusion coefficient tending to zero is justified. In addition, the $L^2$ convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer

Dirichlet problem for a second order singular differential equation

This article concerns the existence of positive solutions to the Dirichlet problem for a second order singular differential equation. To prove existence, we use the classical method of elliptic regularization

Global stability in a diffusive predator–prey model of Leslie–Gower type

We consider a diffusive predator–prey model of Leslie–Gower type, and obtain a new global stability result by combining the Lyapunov function method and the transformation technique used in Qi and Zhu, (2016). Our result partially answers the question proposed in [Y. H. Du and S. B. Hsu, J. Differential Equations 203(2004) 331–364]. In addition, we extend the result to a class of diffusive systems with a more general type of reaction-terms

Weak solutions for quasilinear degenerate parabolic systems

This paper concerns the initial Dirichlet boundary-value problem for a class of quasilinear degenerate parabolic systems. Due to the degeneracies, the problem does not have classical solutions in general. Combining the special form of the system, a proper concept of a weak solution is presented, then the existence and uniqueness of weak solutions are proved. Moreover, the asymptotic behavior of weak solutions will also be discussed