Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

Entire solutions in nonlocal dispersal equations with bistable nonlinearity

We consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one-dimensional spatial domain. A two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed. Furthermore, we show that such an entire solution is unique up to space-time translations and Liapunov stable. A key idea is to characterize the asymptotic behaviors of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions. We have to emphasize that a lack of regularizing effect occurs.     

Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity

This is the second part of a series of study on the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm our theoretical study.     

Existence and stability of traveling waves for an integro-differential equation for slow erosion

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order nonlinear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at ±∞. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approach the traveling waves asymptotically as t→+∞$t\to +\infty$.     

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.     

Nonuniqueness of the traveling wave speed for harmonic heat flow

Given any wave speed c∈R, we construct a traveling wave solution of ut=Δu+²|∇u|u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x₃=±∞. Here u is a director field with values in S²⊂R³: |u|=1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.     

Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L²-norm. The numerical examples are given to illustrate the theoretical results.     

Large time behavior of disturbed planar fronts in the Allen-Cahn equation

We consider the Allen-Cahn equation in Rn (with n?2) and study how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. We first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. Using this observation, we then show that the planar front is asymptotically stable in L^∞(Rn) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. As a by-product of our analysis, we present a result of a rather general nature, which states that, for a large class of evolution equations, the unique ergodicity of the initial data is inherited by the solution at any later time.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Global stability of traveling wavefronts for nonlocal reaction-diffusion equations with time delay

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts(waves with speeds c c_*, where c = c~* is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x →-∞, but it can be allowed arbitrary large in other locations, which improves the results in [9, 18, 21].     

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.     

Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain

Uniform energy and L² decay of solutions for linear wave equations with localized dissipation will be given. In order to derive the L²-decay property of the solution, a useful device whose idea comes from Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) is used. In fact, we shall show that the L²-norm and the total energy of solutions, respectively, decay like and O(1/t²) as t→+∞ for a kind of the weighted initial data.     

Shift Invariant Subspaces with Arbitrary Indices in ? Spaces

We construct right shift invariant subspaces of index n, 1?n?∞, in ?^p spaces, 2<p<∞, and in weighted ?^p spaces.     

Smooth hypersurfaces with a CR contraction

In this paper we show that a C^∞ real hypersurface in Cn+1 of finite D'Angelo type admitting a weakly contracting local CR automorphism is CR equivalent to a weighted homogeneous hypersurface. As an application, we show that a bounded pseudoconvex domain in Cn+1 with C^∞ boundary of finite D'Angelo type with a hyperbolic orbit accumulation point is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.     

Stability of Traveling Wave Fronts for Nonlocal Diffusive Systems

The paper is concerned with stability of traveling wave fronts for nonlocal diffusive systems. We adopt L1-weighted, L1- and L2-energy estimates for the perturbation systems, and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev space.     

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces H^p(w)$H^p(w)$ with w   belonging to Muckenhoupt class _A∞$A_{\infty }$. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.     

Contractivity of Wasserstein metrics and asymptotic profiles for scalar conservation laws

The aim of this paper is to analyze contractivity properties of Wasserstein-type metrics for one-dimensional scalar conservation laws with nonnegative, L^∞ and compactly supported initial data and its implications on the long time asymptotics. The flux is assumed to be convex and without any growth condition at the zero state. We propose a time-parameterized family of functions as intermediate asymptotics and prove the solutions, after a time-depending scaling, converge toward this family in the d_∞-Wasserstein metric. This asymptotic behavior relies on the aforementioned contraction property for conservation laws in the space of probability densities metrized with the d_∞-Wasserstein distance. Finally, we also give asymptotic profiles for initial data whose distributional derivative is a probability measure.     

Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.