Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity

In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wavefronts with the speed c>c_∗, where c_∗>0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x→−∞, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c>c_∗ with sufficiently small delay time r≈0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c>c_∗ and for an arbitrary time-delay r>0.     

Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces

In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the L^p−L^q estimates for in Besov spaces and the perturbation of linearization.     

On camel-like traveling wave solutions in cellular neural networks

This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.     

Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in , while for a general smooth nonlinear flux function, we need to ask for the L²-norm of the initial perturbation to be small but the L²-norm of the first derivative of the initial perturbation can be large and, consequently, the -norm of the initial perturbation can also be large.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.     

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.     

Remark on critical speed of traveling wavefronts for Nicholson's blowflies equation with diffusion

This note is devoted to the study on the traveling wavefronts to the Nicholson's blowflies equation with diffusion, a time-delayed reaction-diffusion equation. For the critical speed of traveling waves, we give a detailed analysis on its location and asymptotic behavior with respect to the mature age.     

REMARK ON CRITICAL SPEED OF TRAVELING WAVEFRONTS FOR NICHOLSON'S BLOWFLIES EQUATION WITH DIFFUSION

This note is devoted to the son＇s blowflies equation with diffusion, a critical speed of traveling waves, we give behavior with respect to the mature age study on the traveling wavefronts to the Nicholtime-delayed reaction-diffusion equation. For the a detailed analysis on its location and asymptotic     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

Global attractor and nonhomogeneous equilibria for a nonlocal evolution equation in an unbounded domain

In this work we consider the nonlocal evolution equation which arises in models of phase separation. We prove the existence of a compact global attractor in some weighted spaces and the existence of a distinguished nonhomogeneous equilibrium: the ‘critical droplet.’     

Nonmonotone travelling waves in a single species reaction-diffusion equation with delay

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction-diffusion equations , when g∈C²(R₊,R₊) has exactly two fixed points: x₁=0 and x₂=K>0. Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass-type equations with diffusion fall within this subclass of (∗). As an example, we consider the diffusive Nicholson's blowflies equation.     

The asymptotic critical wave speed in a family of scalar reaction-diffusion equations

We study traveling wave solutions for the class of scalar reaction-diffusion equations

     

Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces

We shall show existence of global strong solution to the semi-linear Keller-Segel system in Rn, n?3, of parabolic-parabolic type with small initial data and for max{1,n/4}<r<n/2. Our method is based on the perturbation of linealization together with the Lp-Lq estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall prove the decay property of solutions as the time goes to infinity.     

Global well-posedness of the critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t_∂u+ux_∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where .     

Contact discontinuity with general perturbations for gas motions

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier-Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier-Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L^∞ norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.