Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations
In this paper we use a dynamical systems approach to prove the existence of a unique critical value c * of the speed c for which the degenerate density-dependent diffusion equation $u_t = [D(u)u_x]_x + g(u)$ has: 1. no travelling wave solutions for 0 < c < c *, 2. a travelling wave sol...
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| Format: | Journal article |
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1994
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| _version_ | 1797061764946329600 |
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| author | Sánchez-Garduño, F Maini, P |
| author_facet | Sánchez-Garduño, F Maini, P |
| author_sort | Sánchez-Garduño, F |
| collection | OXFORD |
| description | In this paper we use a dynamical systems approach to prove the existence of a unique critical value c * of the speed c for which the degenerate density-dependent diffusion equation $u_t = [D(u)u_x]_x + g(u)$ has: 1. no travelling wave solutions for 0 < c < c *, 2. a travelling wave solution $u(x, t) = \phi(x - c^{*} t)$ of sharp type satisfying $\phi(-\infty) = 1$, $\phi(\tau) = 0$ $\forall \tau\geq\tau^{*}$; $\phi'(\tau^{*-}) = -c^{*}/D'(0),\phi'(\tau^{*+}) = 0$ and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c *. These fronts satisfy the boundary conditions $\phi(–\infty) = 1$, $\phi'(–\infty)=\phi(+\infty) = \phi'(+\infty) = 0$. We illustrate our analytical results with some numerical solutions. |
| first_indexed | 2024-03-06T20:35:57Z |
| format | Journal article |
| id | oxford-uuid:32a1c4a4-f60c-41ef-95d7-1ae8e7edfa50 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:35:57Z |
| publishDate | 1994 |
| record_format | dspace |
| spelling | oxford-uuid:32a1c4a4-f60c-41ef-95d7-1ae8e7edfa502022-03-26T13:15:16ZExistence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:32a1c4a4-f60c-41ef-95d7-1ae8e7edfa50Mathematical Institute - ePrints1994Sánchez-Garduño, FMaini, PIn this paper we use a dynamical systems approach to prove the existence of a unique critical value c * of the speed c for which the degenerate density-dependent diffusion equation $u_t = [D(u)u_x]_x + g(u)$ has: 1. no travelling wave solutions for 0 < c < c *, 2. a travelling wave solution $u(x, t) = \phi(x - c^{*} t)$ of sharp type satisfying $\phi(-\infty) = 1$, $\phi(\tau) = 0$ $\forall \tau\geq\tau^{*}$; $\phi'(\tau^{*-}) = -c^{*}/D'(0),\phi'(\tau^{*+}) = 0$ and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c *. These fronts satisfy the boundary conditions $\phi(–\infty) = 1$, $\phi'(–\infty)=\phi(+\infty) = \phi'(+\infty) = 0$. We illustrate our analytical results with some numerical solutions. |
| spellingShingle | Sánchez-Garduño, F Maini, P Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title | Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title_full | Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title_fullStr | Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title_full_unstemmed | Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title_short | Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
| title_sort | existence and uniqueness of a sharp travelling wave in degenerate non linear diffusion fisher kpp equations |
| work_keys_str_mv | AT sanchezgardunof existenceanduniquenessofasharptravellingwaveindegeneratenonlineardiffusionfisherkppequations AT mainip existenceanduniquenessofasharptravellingwaveindegeneratenonlineardiffusionfisherkppequations |