Pushed traveling fronts in monostable equations with monotone delayed reaction

Abstract

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_t(t,x)=\Delta u(t,x)-u(t,x)+g(u(t-h,x)),$ with monotone delayed reaction term $g:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and $h>0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x)\le g^{\prime }(0)x,$ $x\ge 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.

17 pages, submitted

References in corpus (1)

• Nonlocal anisotropic dispersal with monostable nonlinearity

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