Sheng-Chen Fu

Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions

We study the blow-up behavior for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. The main purpose is to understand how the reaction terms and the absorption terms affect the blow-up properties. We obtain a necessary and sufficient condition for blow-up, derive the upper bound and lower bound for the blow-up rate, and find the blow-up set under certain assumptions.

Oscillation in nonlinear difference equations

Abstract In this paper, we shall discuss oscillatory behavior of the solutions of difference equations, including the self-adjoint second-order linear equation and the discrete version of the nonlinear wave equation. Our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates.

Oscillation and nonoscillation criteria for linear dynamic systems on time scales

In this paper we establish oscillation and nonoscillation criteria for the linear dynamic system u^@D=pv,v^@D=-qu^@s. Here we assume that p and q are nonnegative, rd-continuous functions on a time scale T such that supT=~. Indeed, we extend some known oscillation results for differential systems and difference systems to the so-called dynamic systems.

Wave propagation in predator-prey systems

In this paper, we study a class of predator–prey systems of reaction–diffusion type. Specifically, we are interested in the dynamical behaviour for the solution with the initial distribution where the prey species is at the level of the carrying capacity, and the density of the predator species has compact support, or exponentially small tails near . Numerical evidence suggests that this will lead to the formation of a pair of diverging waves propagating outwards from the initial zone. Motivated by this phenomenon, we establish the existence of a family of travelling waves with the minimum speed. Unlike the previous studies, we do not use the shooting argument to show this. Instead, we apply an iteration process based on Berestycki et al 2005 (Arch. Rational Mech. Anal. 178 57–80) to construct a set of super/sub-solutions. Since the underlying system does not enjoy the comparison principle, such a set of super/sub-solutions is not based on travelling waves, and in fact the super/sub-solutions depend on each other. With the aid of the set of super/sub-solutions, we can construct the solution of the truncated problem on the finite interval, which, via the limiting argument, can in turn generate the wave solution. There are several advantages to this approach. First, it can remove the technical assumptions on the diffusivities of the species in the existing literature. Second, this approach is of PDE type, and hence it can shed some light on the spreading phenomenon indicated by numerical simulation. In fact, we can compute the spreading speed of the predator species for a class of biologically acceptable initial distributions. Third, this approach might be applied to the study of waves in non-cooperative systems (i.e. a system without a comparison principle).

The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction

We investigate the existence of traveling wave solutions to the onedimensional reaction-diffusion system ut = δuxx − 2uv/(β + u), vt = vxx + uv/(β + u), which describes the acidic nitrate-ferroin reaction. Here β is a positive constant, u and v represent the concentrations of the ferroin and acidic nitrate respectively, and δ denotes the ratio of the diffusion rates. We show that this system has a unique, up to translation, traveling wave solution with speed c iff c ≥ 2/ √ β + 1.

Oscillation theorems for fourth-order partial difference equations

The behavior of the solutions of nonlinear partial difference equations of fourth order is discussed. We give some sufficient conditions for the oscillation of nontrivial solutions of the given equation by using the weighted techniques.