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Register a new userEffect of small noise on the speed of reaction-diffusion equations with non-Lipschitz drift
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We consider the $[0,1]$-valued solution $(u_{t,x}:tgeq 0, xin mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise [ partial_t u = partial_x^2 u + f(u) + epsilon sqrt{u(1-u)} dot W. ] Here, $W$ is a space-time white noise, $epsilon > 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $sup_{zin [0,1]}|f(z)|/ sqrt{z(1-z)} < infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. 384 (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,epsilon}$ was derived as the noise strength $epsilon$ approaches $infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,epsilon}$ as the noise strength $epsilon$ approaches $0$: for a given $pin [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,epsilon}$ is comparable to $epsilon^{-2frac{1-p}{1+p}}$ for sufficiently small $epsilon$.
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We study the minimal speed of propagating fronts of convection reaction diffusion equations of the form $u_t + mu phi(u) u_x = u_{xx} +f(u)$ for positive reaction terms with $f(0 >0$. The function $phi(u)$ is continuous and vanishes at $u=0$. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assesment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if $f(u)/sqrt{f(0)} + mu phi(u) < 0$, then the minimal speed is given by the linear value $2 sqrt{f(0)}$, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case $u_t + mu u u_x = u_{xx} + u (1 -u)$. Results are also given for the density dependent diffusion case $u_t + mu phi(u) u_x = (D(u)u_x)_x +f(u)$.
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