Existence and Spatial Decay of Forced Waves for the Fisher-KPP Equation with a Degenerate Shifting Environment

Abstract

This paper studies forced waves for the heterogeneous Fisher-KPP equation $u_t=u_{xx}+u(a(x-ct)-u)$, where $c>0$ and $a(z)>0$ satisfies $a(-\infty )=\hat{I}\pm >0=a(+\infty )$, $a^{\prime }(z)\le 0$ ($z\gg 1$). Using ODE asymptotic analysis, we classify all local positive solutions near $z=+\infty$. Exponential decay solutions always exist; non-exponential decay solutions exist if and only if $\exp \left({\int }_{z_0}^z\frac{-c+\sqrt{c^2-4a(s)}}{2}\mathrm{d}s\right)\in L^1([z_0,+\infty ))$. We obtain a complete existence, multiplicity and spatial decay for forced waves. For each $c\in (0,2\sqrt{\hat{I}}\pm )$, there exists a unique exponentially decaying forced wave. This wave is either the unique forced wave or the minimal forced wave, depending on the integrability condition. In the case $\exp \left({\int }_{z_0}^z\frac{-c+\sqrt{c^2-4a(s)}}{2}\mathrm{d}s\right)\in L^1([z_0,+\infty ))$, for any $c>0$ there exist infinitely many non-exponentially decaying forced waves and the maximal wave is not in $L^1([z_0,+\infty ))$. These results provide complete answers to open problems concerning the existence, uniqueness, multiplicity and spatial decay rates of forced waves in Fisher-KPP models with degenerate moving environments.