Yifu Wang

Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source

This paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype (0.1){ut=um(uv)+uur,x,t>0,vt=vv+u,x,t>0,u=v=0,x,t>0,u(x,0)=u0(x),v(x,0)=v0(x)x, in a smooth bounded domain RN(N2) with parameters m,r1 and 0. The PDE system in (0.1) is used in mathematical biology to model the mechanism of chemotaxis, that is, the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space. It is shown that if m{>22Nif11+(N+22r)+N+2ifN+22rN+2N,1ifr>N+22, and the nonnegative initial data (u0,v0)C()W1,()(>0), then (0.1) possesses at least one global bounded weak solution. Apart from this, it is proved that if =0 then both u(,t) and v(,t) decay to zero with respect to the norm in L() as t.

Travelling wave and global attractivity in a competition-diffusion system with nonlocal delays

In this paper, we are concerned with a two-competition model described by a reaction-diffusion system with nonlocal delays which account for the drift of individuals to their present position from their possible positions at previous times. By using the iterative technique recently developed in Wang et al. (2006) [14], the sufficient conditions are established for the existence of travelling wave solutions connecting the semi-trivial steady state to the coexistence steady state of the considered system. When the domain is bounded, we investigate the global attractivity of the coexistence steady state of the system under homogeneous Neumann boundary conditions as well. The approach used is the upper-lower solutions and monotone iteration technique.

Boundedness in a multi-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Abstract We consider the chemotaxis–haptotaxis model { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 in a bounded smooth domain Ω ⊂ R n ( n ≥ 2 ) , where χ , ξ and μ are positive parameters, and the diffusivity D ( u ) is assumed to generalize the prototype D ( u ) = δ ( u + 1 ) − α with α ∈ R . Under zero-flux boundary conditions, it is shown that for sufficiently smooth initial data ( u 0 , v 0 , w 0 ) and α 2 − n n + 2 , the corresponding initial–boundary problem possesses a unique global-in-time classical solution which is uniformly bounded, which improves the previous results.

On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition

We establish the critical Fujita exponents for the solution of the porous medium equation ut = ∆u m , x ∈ R N , t> 0, subject to the nonlinear boundary condition −∂u m /∂x1 = u p , x1 = 0, t> 0,

Boundedness of solutions to a quasilinear chemotaxis-haptotaxis model

We study global solutions of a class of chemotaxis-haptotaxis systems generalizing the prototype { u t = ? ? ( ( u + 1 ) m - 1 ? u ) - ? ? ( u ( u + 1 ) q - 1 ? v ) - ? ? ( u ( u + 1 ) p - 1 ? w ) + H ( u , w ) , 0 = Δ v - v + u , w t = - v w , in a bounded domain ? ? R N ( N ? 1 ) with smooth boundary, H ( u , w ) : = u ( 1 - u r - 1 - w ) , with parameters m ? 1 , r 1 and positive constants p , q . It is shown that either max { q + 1 , p , 2 p - m } < max { m + 2 N , r } or ? max { q + 1 , p , 2 p - m } = r and b 0 is large enough, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results of this paper improve the results of Tao and Winkler (2014)?46,51.

Global dynamics of reaction–diffusion systems with delays☆

Abstract Taking the effect of spatial diffusion into account, we introduce an exponential ordering and give sufficient conditions under which reaction–diffusion systems with delays generate monotone semi-flows on a suitable phase space even if they are not quasi-monotone. The powerful theory of monotone semi-flows is applied to describe the threshold dynamics for a nonlocal delayed reaction–diffusion system modelling the spread of bacterial infections.

Periodic solutions to a heat equation with hysteresis in the source term

This paper is concerned with a parabolic problem with hysteresis effects in the heat source, which models the feedback control. The existence of periodic solutions is proved by the viscosity approach when the heat force changes periodically in time. More precisely, with the help of the subdifferential operator theory and the Poincare map, the existence of solutions to the approximation problem is shown and the solution of the periodic problem is obtained under consideration by using a passage-to-limit procedure.

Minimal wave speed of a nonlocal diffusive epidemic model with temporal delay

This note is concerned with a nonlocal version of the man-environment-man epidemic model in which the dispersion of infectious agents is assumed to follow a nonlocal diffusion law modeled by a convolution operator. The purpose of this note is to show that the minimal wave speeds of properly re-scaled nonlocal diffusion equations can approximate the corresponding one of the classical diffusion equation for this model. As a byproduct, our results indicate that the temporal delay in an epidemic model can reduce the speed of epidemic spread while the nonlocal effect can increase the speed.

Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source , , , , , , and , , which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.