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Register a new userLong time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds
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In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(theta_tomega)u(1-u),quad xinmathbb{R} $$ where $omegainOmega$, $(Omega, mathcal{F},mathbb{P})$ is a given probability space, $theta_t$ is an ergodic metric dynamical system on $Omega$, and $a(omega)>0$ for every $omegainOmega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),quad xinmathbb{R}$$ where $a_0(t)$ is a positive locally Holder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $a(omega)$, we show that the constant solution $u=1$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $[2sqrt{underline a}, 2sqrt{bar a}]$, where $underline{a}$ and $bar a$ are the least and the greatest means of $a(cdot)$, respectively, and hence the spreading speed interval is linearly determinant. It is shown that the solution of (1) with the initial function which is bounded away from $0$ for $xll -1$ and is $0$ for $xgg 1$ propagates at the speed $2sqrt {hat a}$, where $hat a$ is the average of $a(cdot)$. Under some assumption on $a_0(cdot)$, we also show that the constant solution $u=1$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $a(omega)$ and $a_0(t)$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher-KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).
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In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(theta_tomega)u(1-u),quad xinR, eqno(1) $$ where $omegainOmega$, $(Omega, mathcal{F},mathbb{P})$ is a given probability space, $theta_t$ is an ergodic metric dynamical system on $Omega$, and $a(omega)>0$ for every $omegainOmega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),quad xinR, eqno(2) $$ where $a_0(t)$ is a positive locally Holder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1) and (2). In this second part of the series, we investigate the existence and stability of transition fronts of (1) and (2). We first study the transition fronts of (1). Under some proper assumption on $a(omega)$, we show the existence of random transition fronts of (1) with least mean speed greater than or equal to some constant $underline{c}^*$ and the nonexistence of ranndom transition fronts of (1) with least mean speed less than $underline{c}^*$. We prove the stability of random transition fronts of (1) with least mean speed greater than $underline{c}^*$. Note that it is proved in the first part that $underline{c}^*$ is the infimum of the spreading speeds of (1). We next study the existence and stability of transition fronts of (2). It is not assumed that $a(omega)$ and $a_0(t)$ are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.
In Cao, Du, Li and Li [8], a nonlocal diffusion model with free boundaries extending the local diffusion model of Du and Lin [12] was introduced and studied. For Fisher-KPP type nonlinearities, its long-time dynamical behaviour is shown to follow a spreading-vanishing dichotomy. However, when spreading happens, the question of spreading speed was left open in [8]. In this paper we obtain a rather complete answer to this question. We find a condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, which is achieved by completely solving the associated semi-wave problem that determines this linear speed; when the kernel function violates this condition, we show that accelerating spreading happens.
This paper is devoted to the study of the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in R N. We first prove a general Freidlin-G{a}rtner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the $Omega$-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmicin-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems.
We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy [ lim_{tto -infty} sup_{|x|le c|t|} u(t,x) = 0 qquad text{for each $c<2sqrt{f(0)},$,} ] and we consider here those satisfying [ lim_{tto -infty} sup_{|x|le c|t|} u(t,x) = 0 qquad text{for some $c>2sqrt{f(0)},$.} ] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.
We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable $x/t$ and in which the time and space derivatives are coupled together. We will first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable $s=x/t$. In terms of the standard Fisher-KPP equation, our results leads to a new class of asymptotically homogeneous environments which share the same spreading speed with the corresponding homogeneous environments.
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