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

27 Sep 2019 Ding Weiwei Matano Hiroshi

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)... (read more)

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