Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918460110405632 |
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| author | Barker, William Van Minh, Nguyen |
| author_facet | Barker, William Van Minh, Nguyen |
| contents | We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form $\partial u(x,t)/\partial t = Δu(x,t-τ_1)+f(u(x,t),u(x,t-τ_2))$, where $τ_1,τ_2$ are positive constants. We extend the monotone iteration method to systems that satisfy typical monotone conditions by thoroughly studying the sign of the Green function associated with a linear functional differential equation. Namely, we show that for small positive $r$ the functional equation $x''(t)-ax'(t+r)-bx(t+r)=f(t)$, where $a\not=0, b>0$ has a unique bounded solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is sufficiently small, $f(t)\ge 0$ for $t\in {\mathbb R}$, then the unique bounded solution $x_f(t)\le 0$ for all $t\in {\mathbb R}$. In the framework of the monotone iteration method that is developed based on this result, upper and lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to show that traveling waves exist for these equations when delays are small in both diffusion and reaction terms. The obtained results appear to be new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11504 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms Barker, William Van Minh, Nguyen Dynamical Systems 35C07, 35K57 We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form $\partial u(x,t)/\partial t = Δu(x,t-τ_1)+f(u(x,t),u(x,t-τ_2))$, where $τ_1,τ_2$ are positive constants. We extend the monotone iteration method to systems that satisfy typical monotone conditions by thoroughly studying the sign of the Green function associated with a linear functional differential equation. Namely, we show that for small positive $r$ the functional equation $x''(t)-ax'(t+r)-bx(t+r)=f(t)$, where $a\not=0, b>0$ has a unique bounded solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is sufficiently small, $f(t)\ge 0$ for $t\in {\mathbb R}$, then the unique bounded solution $x_f(t)\le 0$ for all $t\in {\mathbb R}$. In the framework of the monotone iteration method that is developed based on this result, upper and lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to show that traveling waves exist for these equations when delays are small in both diffusion and reaction terms. The obtained results appear to be new. |
| title | Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms |
| topic | Dynamical Systems 35C07, 35K57 |
| url | https://arxiv.org/abs/2301.11504 |