Travelling wave phenomena in some degenerate reaction-diffusion equations
In this paper we study the existence of travelling wave solutions (t.w.s.), $u(x, t)=\phi(x−ct)$ for the equation $u_t=[D(u)u_x]_x+g(u) (*)$ where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion term D(u). Both functions D a...
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| Format: | Journal article |
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1994
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| _version_ | 1826307179860197376 |
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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 study the existence of travelling wave solutions (t.w.s.), $u(x, t)=\phi(x−ct)$ for the equation $u_t=[D(u)u_x]_x+g(u) (*)$ where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion term D(u). Both functions D and g are defined on the interval [0, 1]. The existence problem is analysed in the following two cases. Case 1. D(0)=0, D(u)>0 $\forall u \in (0, 1]$, D and $g\in C^{2}_{[0,1]}$, $D'(0)\neq0$ and $D''(0)\neq 0$. We prove that if there exists a value of c, c*, for which the equation (*) possesses a travelling wave solution of sharp type, it must be unique. By using some continuity arguments we show that: for 0<c<c*, are="" c="" for="" no="" t.w.s.,="" there="" while="">c*, the equation (*) has a continuum of t.w.s. of front type. The proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations, which is also proved. Case 2. $D(0)=D'(0)=0$, D and $g \in C^{2}_{[0,1]}$, $D''(0)\neq 0$. If, in addition, we impose $D''(0)>0$ with $D(u)>0$ $\forall u(0, 1]$, We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if $D''(0)<0$ with $D(u)<0$ $\forall u\in (0, 1]$ we analyse just one example ($D(u)=-u^2$, and $g(u)=u(1-u)$) which has oscillatory t.w.s. for $0<c\leq2$ and="" c="" for="" front="" of="" t.w.s.="" type="">2. In both the above cases we use higher order terms in the Taylor series and the Centre Manifold Theorem in order to get the local behaviour around a non-hyperbolic point of codimension one in the phase plane.</c\leq2$></c<c*,> |
| first_indexed | 2024-03-07T06:59:03Z |
| format | Journal article |
| id | oxford-uuid:ff1c4d2e-8e66-4a42-94df-d79a8ce5c2c5 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:59:03Z |
| publishDate | 1994 |
| record_format | dspace |
| spelling | oxford-uuid:ff1c4d2e-8e66-4a42-94df-d79a8ce5c2c52022-03-27T13:42:04ZTravelling wave phenomena in some degenerate reaction-diffusion equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ff1c4d2e-8e66-4a42-94df-d79a8ce5c2c5Mathematical Institute - ePrints1994Sánchez-Garduño, FMaini, PIn this paper we study the existence of travelling wave solutions (t.w.s.), $u(x, t)=\phi(x−ct)$ for the equation $u_t=[D(u)u_x]_x+g(u) (*)$ where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion term D(u). Both functions D and g are defined on the interval [0, 1]. The existence problem is analysed in the following two cases. Case 1. D(0)=0, D(u)>0 $\forall u \in (0, 1]$, D and $g\in C^{2}_{[0,1]}$, $D'(0)\neq0$ and $D''(0)\neq 0$. We prove that if there exists a value of c, c*, for which the equation (*) possesses a travelling wave solution of sharp type, it must be unique. By using some continuity arguments we show that: for 0<c<c*, are="" c="" for="" no="" t.w.s.,="" there="" while="">c*, the equation (*) has a continuum of t.w.s. of front type. The proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations, which is also proved. Case 2. $D(0)=D'(0)=0$, D and $g \in C^{2}_{[0,1]}$, $D''(0)\neq 0$. If, in addition, we impose $D''(0)>0$ with $D(u)>0$ $\forall u(0, 1]$, We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if $D''(0)<0$ with $D(u)<0$ $\forall u\in (0, 1]$ we analyse just one example ($D(u)=-u^2$, and $g(u)=u(1-u)$) which has oscillatory t.w.s. for $0<c\leq2$ and="" c="" for="" front="" of="" t.w.s.="" type="">2. In both the above cases we use higher order terms in the Taylor series and the Centre Manifold Theorem in order to get the local behaviour around a non-hyperbolic point of codimension one in the phase plane.</c\leq2$></c<c*,> |
| spellingShingle | Sánchez-Garduño, F Maini, P Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title | Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title_full | Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title_fullStr | Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title_full_unstemmed | Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title_short | Travelling wave phenomena in some degenerate reaction-diffusion equations |
| title_sort | travelling wave phenomena in some degenerate reaction diffusion equations |
| work_keys_str_mv | AT sanchezgardunof travellingwavephenomenainsomedegeneratereactiondiffusionequations AT mainip travellingwavephenomenainsomedegeneratereactiondiffusionequations |