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Register a new userSemi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries
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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.
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We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. The problem is monostable in nature, resembling the well known Fisher-KPP equation. Such a system covers various models arising from mathematical biology, with the Fisher-KPP equation as the simplest special case, where a spreading-vanishing dichotomy is known to govern the long time dynamical behaviour. The question of spreading speed is widely open for such systems except for the scalar case. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and traveling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave, and obtain sharp estimates of the semi-wave profile and the spreading speed. For kernel functions that behave like $|x|^{-gamma}$ near infinity, we are able to obtain better estimates of the spreading speed for both the finite speed case, and the infinite speed case, which appear to be the first for this kind of free boundary problems, even for the special Fisher-KPP equation.
We study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and reveal some fundamental differences from its one dimensional version considered in cite{cdjfa} recently. Technically, this high dimensional problem is much more difficult to treat since it involves two kernel functions which arise from the original kernel function $J(|x|)$ in rather implicit ways. By introducing new techniques, we are able to determine the long-time dynamics of the model, including firstly finding the threshold condition on the kernel function that governs the onset of accelerated spreading, and the determination of the spreading speed when it is finite. Moreover, for two important classes of kernel functions, sharp estimates of the spreading profile are obtained. More precisely, for kernel functions with compact support, we show that logarithmic shifting occurs from the finite wave speed propagation, which is strikingly different from the one dimension case; for kernel functions $J(|x|)$ behaving like $|x|^{-beta}$ for $xinR^N$ near infinity, we obtain the rate of accelerated spreading when $betain (N, N+1]$, which is the exact range of $beta$ where accelerated spreading is possible. These sharp estimates are obtained by constructing subtle upper and lower solutions, based on careful analysis of the involved kernel functions.
A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.
In this paper, we study the asymptotic behavior as $varepsilonto0^+$ of solutions $u_varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ frac{1}{varepsilon^m}int_{mathbb{R}^N}J_varepsilon(x-y)(u_varepsilon(y)-u_varepsilon(x))mathrm{d}y+u_varepsilon(x)(a(x)-u_varepsilon(x))=0text{ in }mathbb{R}^N, $$where $varepsilon>0$ and $0leq m<2$. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution $u_varepsilon$ and that $u_varepsilonto a^+$ as $varepsilonto0^+$ where $a^+=max{0,a}$. This generalizes the previously known results and answers an open question raised by Berestycki, Coville and Vo. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.
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 $Omega_0$.
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