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Register a new userMultidimensional transition fronts for Fisher-KPP reactions
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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.
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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.
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.
We consider the nonlinear Stefan problem $$ left { begin{array} {ll} -d Delta u=a u-b u^2 ;; & mbox{for } x in Omega (t), ; t>0, u=0 mbox{ and } u_t=mu| abla_x u |^2 ;;&mbox{for } x in partialOmega (t), ; t>0, u(0,x)=u_0 (x) ;; & mbox{for } x in Omega_0, end{array}right. $$ where $Omega(0)=Omega_0$ is an unbounded smooth domain in $mathbb R^N$, $u_0>0$ in $Omega_0$ and $u_0$ vanishes on $partialOmega_0$. When $Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $Omega_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Omega_0$.
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.
We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit.
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