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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.10922 |
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| _version_ | 1866929272636047360 |
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| author | Needham, D. J. Billingham, J. Ladas, N. M. Meyer, J. C. |
| author_facet | Needham, D. J. Billingham, J. Ladas, N. M. Meyer, J. C. |
| contents | We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-ϕ*u), \] where $ϕ*u$ is a spatial convolution with the top hat kernel, $ϕ(y) \equiv H\left(\frac{1}{4}-y^2\right)$.
After showing that the problem is globally well-posed, we demonstrate that positive, spatially-periodic solutions bifurcate from the spatially-uniform steady state solution $u=1$ as the diffusivity, $D$, decreases through $Δ_1 \approx 0.00297$. We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly-spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at leading order as $D \to 0^+$.
From numerical solutions, we find that for $D \geq Δ_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < Δ_1$, the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created. The structure of these transitional travelling waves is examined in some detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line Needham, D. J. Billingham, J. Ladas, N. M. Meyer, J. C. Analysis of PDEs 35B36 We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-ϕ*u), \] where $ϕ*u$ is a spatial convolution with the top hat kernel, $ϕ(y) \equiv H\left(\frac{1}{4}-y^2\right)$. After showing that the problem is globally well-posed, we demonstrate that positive, spatially-periodic solutions bifurcate from the spatially-uniform steady state solution $u=1$ as the diffusivity, $D$, decreases through $Δ_1 \approx 0.00297$. We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly-spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at leading order as $D \to 0^+$. From numerical solutions, we find that for $D \geq Δ_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < Δ_1$, the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created. The structure of these transitional travelling waves is examined in some detail. |
| title | The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line |
| topic | Analysis of PDEs 35B36 |
| url | https://arxiv.org/abs/2304.10922 |