非局部扩散方程的单稳行波解
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摘要
近年来,在材料科学、生态学、流行病学、神经网络等学科的研究中导出了许多非局部扩散方程,并已得到了许多学者的关注.我们知道，用积分算子所表示的非局部扩散能够更加准确地描述所考虑的实际问题.然而,由于非局部项的出现导致方程的性质和动力学行为发生了改变，例如，方程的解半流不再是紧的以及解的正则性降低等.这给数学理论的研究带来了新的困难.在非局部扩散方程的研究中,行波解是一个重要分支.行波解可以很好地描述自然界中大量有限速度传播问题及振荡现象.
本文首先研究了退化单稳非局部扩散方程的行波解的单调性、唯一性、稳定性以及初值问题的解的传播速度.通过考虑原方程所对应的线性方程,我们讨论了行波解在实轴两端的指数渐近行为.然后利用滑动平面技术证明了行波解的单调性和唯一性.接着利用基于比较原理的挤压技术，证明了最小波速行波解的渐近稳定性.最后，借助于比较原理和上下解方法,得出紧初值问题解的传播速度与行波解的最小波速相一致的结论.
其次,研究了具有年龄结构的时滞单稳非局部扩散方程的行波解的渐近稳定性.通过加权能量方法结合比较原理的方法，证明了大波速行波解的稳定性.这个稳定性结果表明初值问题的解以时间指数衰减收敛到行波解.特别地,在该类方程中,时滞对行波解的稳定性没有影响.
最后,研究了具有阶段结构的捕食者-食饵系统的行波解的存在性.通过引入部分拟单调条件(PQM)和部分指数拟单调条件(PEQM),利用上下解方法、交错迭代技巧和Schauder不动点定理,对一类抽象的非局部扩散系统建立了行波解的存在性.然后把这些结果运用到所考虑的捕食者-食饵系统，得到了连接平凡平衡态和共存平衡态的行波解的存在性.
In recent years, a lot, of nonlocal dispersal equations have been derived from the research in many disciplines, such as material science, biology, epidemiology and neural network. Although the nonlocal dispersal represented by the integral operator is closer to the reality, it leads to the many new mathematical difficulties and the essential change of dynamics. For example, the solution semi-flows are not usually compact. And the solutions do not have a priori regularity. In the study of nonlocal dispersal equations, one important topic is their traveling wave solutions, which can well model the oscillatory phenomenon and the propagation with finite speed of nature.
Firstly, we study the monotonicity, uniqueness and stability of traveling wave solutions, and spreading speed for a nonlocal dispersal equation with degenerate monostable nonlinearity. By considering the corresponding linear equation, we dis-cuss the exactly exponentially asymptotic behavior of traveling wave solutions at infinity. We then apply the sliding method to obtain the monotonicity and unique-ness of traveling wave solutions. By the squeezing technique, the asymptotic stability of traveling wave solutions with minimal speed is established. Furthermore, we con-sider the spreading speed of the solution of the initial problem with the compact initial value. The result implies that the spreading speed is coincident with the minimal wave speed.
Secondly, we investigate the asymptotic stability of traveling wave solutions for a delayed nonlocal dispersal equation with age structure. By appealing to the weighted energy method together with the comparison principle, we prove the stabil-ity of traveling wave solutions with large speed. The result shows that the solution of the initial problem converges to the corresponding traveling wave solution with an exponential decay in time. In particular, the time delay does not affect the stability of traveling wave solutions.
Finally, we consider the traveling wave solutions for a predator-prey system with nonlocal dispersal and stage structure. By introducing partially quasi-monotone conditions and partially exponentially quasi-monotone conditions for the nonlinear-ity, we establish the existence for a general nonlocal dispersal system. Our methods are to use the cross-iteration scheme together with upper and lower solutions and the Schauder's fixed point theorem. Then we apply the results to the predator-prey system and obtain the existence of traveling wave solutions connecting 0 with coexistence equilibrium.
本文首先研究了退化单稳非局部扩散方程的行波解的单调性、唯一性、稳定性以及初值问题的解的传播速度.通过考虑原方程所对应的线性方程,我们讨论了行波解在实轴两端的指数渐近行为.然后利用滑动平面技术证明了行波解的单调性和唯一性.接着利用基于比较原理的挤压技术，证明了最小波速行波解的渐近稳定性.最后，借助于比较原理和上下解方法,得出紧初值问题解的传播速度与行波解的最小波速相一致的结论.
其次,研究了具有年龄结构的时滞单稳非局部扩散方程的行波解的渐近稳定性.通过加权能量方法结合比较原理的方法，证明了大波速行波解的稳定性.这个稳定性结果表明初值问题的解以时间指数衰减收敛到行波解.特别地,在该类方程中,时滞对行波解的稳定性没有影响.
最后,研究了具有阶段结构的捕食者-食饵系统的行波解的存在性.通过引入部分拟单调条件(PQM)和部分指数拟单调条件(PEQM),利用上下解方法、交错迭代技巧和Schauder不动点定理,对一类抽象的非局部扩散系统建立了行波解的存在性.然后把这些结果运用到所考虑的捕食者-食饵系统，得到了连接平凡平衡态和共存平衡态的行波解的存在性.
In recent years, a lot, of nonlocal dispersal equations have been derived from the research in many disciplines, such as material science, biology, epidemiology and neural network. Although the nonlocal dispersal represented by the integral operator is closer to the reality, it leads to the many new mathematical difficulties and the essential change of dynamics. For example, the solution semi-flows are not usually compact. And the solutions do not have a priori regularity. In the study of nonlocal dispersal equations, one important topic is their traveling wave solutions, which can well model the oscillatory phenomenon and the propagation with finite speed of nature.
Firstly, we study the monotonicity, uniqueness and stability of traveling wave solutions, and spreading speed for a nonlocal dispersal equation with degenerate monostable nonlinearity. By considering the corresponding linear equation, we dis-cuss the exactly exponentially asymptotic behavior of traveling wave solutions at infinity. We then apply the sliding method to obtain the monotonicity and unique-ness of traveling wave solutions. By the squeezing technique, the asymptotic stability of traveling wave solutions with minimal speed is established. Furthermore, we con-sider the spreading speed of the solution of the initial problem with the compact initial value. The result implies that the spreading speed is coincident with the minimal wave speed.
Secondly, we investigate the asymptotic stability of traveling wave solutions for a delayed nonlocal dispersal equation with age structure. By appealing to the weighted energy method together with the comparison principle, we prove the stabil-ity of traveling wave solutions with large speed. The result shows that the solution of the initial problem converges to the corresponding traveling wave solution with an exponential decay in time. In particular, the time delay does not affect the stability of traveling wave solutions.
Finally, we consider the traveling wave solutions for a predator-prey system with nonlocal dispersal and stage structure. By introducing partially quasi-monotone conditions and partially exponentially quasi-monotone conditions for the nonlinear-ity, we establish the existence for a general nonlocal dispersal system. Our methods are to use the cross-iteration scheme together with upper and lower solutions and the Schauder's fixed point theorem. Then we apply the results to the predator-prey system and obtain the existence of traveling wave solutions connecting 0 with coexistence equilibrium.
引文
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