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1.
In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures. 相似文献
2.
Haiyan Wang 《Journal of Differential Equations》2009,247(3):887-905
We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper. 相似文献
3.
This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique. 相似文献
4.
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. 相似文献
5.
We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c 〉 c. is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature. 相似文献
6.
Entire solutions for monostable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results. 相似文献
7.
Traveling Wave Solutions of a Fourth-order Generalized Dispersive and Dissipative Equation
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In this paper, we consider a generalized nonlinear forth-order dispersive-dissipative equation with a nonlocal strong generic delay kernel, which describes wave propagation in generalized nonlinear dispersive, dissipation and quadratic diffusion media. By using geometric singular perturbation theory and Fredholm alternative theory, we get a locally invariant manifold and use fast-slow system to construct the desire heteroclinic orbit. Furthermore we construct a traveling wave solution for the nonlinear equation. Some known results in the literature are generalized. 相似文献
8.
In this paper, a nonlocal reaction–diffusion model with distributed delay is studied. The asymptotic speed of spread is established for this model, and its coincidence with the minimal wave speed for traveling wave fronts is proved. Moreover, the dependence of the asymptotic speed of spread on delay and the nonlocal effect is considered. Our main finding here is that the delay can induce slow asymptotic speed of spread while the nonlocal effect can increase fast asymptotic speed of spread. 相似文献
9.
10.
Wan-Tong Li Zhi-Cheng Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(4):571-591
This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial
effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion
systems with spatio-temporal delays, J. Differential Equations
222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the
zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP
equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra
system with discrete delays.
Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher
Education Institutions of Ministry of Education of China. 相似文献
11.
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. 相似文献
12.
A. Moussaoui 《Applicable analysis》2020,99(13):2307-2321
ABSTRACT A nonlocal reaction–diffusion equation arising in various applications is studied. The speed of traveling waves is determined by means of a minimax representation. It is used to obtain the wave speed estimates and asymptotic values. 相似文献
13.
Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay
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Hai‐Qin Zhao 《Mathematical Methods in the Applied Sciences》2017,40(18):6702-6714
The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c ?. Then we show that the traveling wave fronts with speed c >c ? are exponentially asymptotically stable. 相似文献
14.
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. 相似文献
15.
Kai Zhou 《Journal of Mathematical Analysis and Applications》2010,372(2):598-610
This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems. 相似文献
16.
Shi-Liang Wu Wan-Tong Li San-Yang Liu 《Journal of Mathematical Analysis and Applications》2009,360(2):439-458
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. 相似文献
17.
A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c∗ and no traveling wave with c<c∗. These results are also applied to some nonlocal reaction-diffusion population models. 相似文献
18.
Guo-Bao Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(15):5030-5047
This paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age-structure. If the birth function is monotone, then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness. It is proved that the traveling wavefront is strictly increasing and unique up to a translation. The asymptotic behavior of traveling wavefronts is also obtained. If the birth function is not monotone, the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi-monotonicity. 相似文献
19.
Summary. In this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation
period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal
effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves
for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider
and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally
delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible
occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent
partial differential equations with nonlocal delay. 相似文献
20.
This paper is devoted to the study of bistable traveling waves for a competitive–cooperative reaction and diffusion system with nonlocal time delays. The existence of bistable waves is established by appealing to the theory of monotone semiflows and the finite-delay approximations. Then the global stability of such traveling waves is obtained via a squeezing technique and a dynamical systems approach. 相似文献