arXiv 1909.124802019-09-27math.AP

具有类前沿初值的时间周期反应扩散方程在$\mathbb{R}$上的动力学

Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$

Weiwei Ding, Hiroshi Matano

摘要：

本文研究了Cauchy问题$$u_t=u_{xx} +f(t,u), \,\,\, x\in\mathbb{R},\,t>0, $$ $$u(0,x)= u_0(x), \,\,\, x\in\mathbb{R},$$，其中$f$是一个相当一般的非线性项，在$t$中是周期的，满足$f(\cdot,0)\equiv 0$，且相应的常微分方程有正周期解$p(t)$ .假设$u_0$是类前沿的，即对于$x\approx -\infty$，$u_0(x)$接近于$p(0)$，对于$x\approx \infty$，接近于$0$，我们的目标是通过使用Ducrot，Giletti和Matano（2014）引入的传播平台的概念来确定解$u(t,x)$的长时间动力学行为。我们建立了一个非常大的类的非线性传播平台的存在性和唯一性，并显示了在$f$或$u_0$的各种条件下，解$u(t,x)$收敛到平台为$t\to\infty$。我们首先考虑$u_0$是Heaviside型函数的特殊情况，并证明了在不要求$f$具有任何非退化性的情况下的收敛结果。此外，如果$u_0$是更一般的，它可以被困在两个Heaviside型函数之间，但不一定单调，我们证明了收敛结果仍然有效，在一个相当温和的非退化假设$f$。最后，在$f$是一个非退化的多稳态非线性的情况下，我们证明了一个更大的类前端初始数据的全局和指数收敛。

英文摘要：

This paper is concerned with the Cauchy problem $$u_t=u_{xx} +f(t,u), \,\,\, x\in\mathbb{R},\,t>0, $$ $$u(0,x)= u_0(x), \,\,\, x\in\mathbb{R},$$ where $f$ is a rather general nonlinearity that is periodic in $t$, and satisfies $f(\cdot,0)\equiv 0$ and that the corresponding ODE has a positive periodic solution $p(t)$. Assuming that $u_0$ is front-like, that is, $u_0(x)$ is close to $p(0)$ for $x\approx -\infty$ and close to $0$ for $x\approx \infty$, we aim to determine the long-time dynamical behavior of the solution $u(t,x)$ by using the notion of propagation terrace introduced by Ducrot, Giletti and Matano (2014). We establish the existence and uniqueness of propagating terrace for a very large class of nonlinearities, and show the convergence of the solution $u(t,x)$ to the terrace as $t\to\infty$ under various conditions on $f$ or $u_0$. We first consider the special case where $u_0$ is a Heaviside type function, and prove the converge result without requiring any non-degeneracy on $f$. Furthermore, if $u_0$ is more general such that it can be trapped between two Heaviside type functions, but not necessarily monotone, we show that the convergence result remains valid under a rather mild non-degeneracy assumption on $f$. Lastly, in the case where $f$ is a non-degenerate multistable nonlinearity, we show the global and exponential convergence for a much larger class of front-like initial data.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

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