Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion
We consider the nonlinear eigenvalue problem Duu′′+λfu=0, u(t)>0, t∈I≔(0,1), u(0)=u(1)=0, where D(u)=uk, f(u)=u2n-k-1+sinu, and λ>0 is a bifurcation parameter. Here, n∈N and k (0≤k<2n-1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and...
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| Format: | Article |
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Wiley
2018-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2018/5053415 |
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| author | Tetsutaro Shibata |
| author_facet | Tetsutaro Shibata |
| author_sort | Tetsutaro Shibata |
| collection | DOAJ |
| description | We consider the nonlinear eigenvalue problem Duu′′+λfu=0, u(t)>0, t∈I≔(0,1), u(0)=u(1)=0, where D(u)=uk, f(u)=u2n-k-1+sinu, and λ>0 is a bifurcation parameter. Here, n∈N and k (0≤k<2n-1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and λ is parameterized by the maximum norm α=uλ∞ of the solution uλ associated with λ and is written as λ=λ(α). Since f(u) contains both power nonlinear term u2n-k-1 and oscillatory term sinu, it seems interesting to investigate how the shape of λ(α) is affected by f(u). The purpose of this paper is to characterize the total shape of λ(α) by n and k. Precisely, we establish three types of shape of λ(α), which seem to be new. |
| format | Article |
| id | doaj-art-334abb5ce1da45d9b346c21ef0a688b8 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-334abb5ce1da45d9b346c21ef0a688b82025-02-03T06:44:32ZengWileyInternational Journal of Differential Equations1687-96431687-96512018-01-01201810.1155/2018/50534155053415Global and Local Structures of Bifurcation Curves of ODE with Nonlinear DiffusionTetsutaro Shibata0Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, JapanWe consider the nonlinear eigenvalue problem Duu′′+λfu=0, u(t)>0, t∈I≔(0,1), u(0)=u(1)=0, where D(u)=uk, f(u)=u2n-k-1+sinu, and λ>0 is a bifurcation parameter. Here, n∈N and k (0≤k<2n-1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and λ is parameterized by the maximum norm α=uλ∞ of the solution uλ associated with λ and is written as λ=λ(α). Since f(u) contains both power nonlinear term u2n-k-1 and oscillatory term sinu, it seems interesting to investigate how the shape of λ(α) is affected by f(u). The purpose of this paper is to characterize the total shape of λ(α) by n and k. Precisely, we establish three types of shape of λ(α), which seem to be new.http://dx.doi.org/10.1155/2018/5053415 |
| spellingShingle | Tetsutaro Shibata Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion International Journal of Differential Equations |
| title | Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion |
| title_full | Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion |
| title_fullStr | Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion |
| title_full_unstemmed | Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion |
| title_short | Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion |
| title_sort | global and local structures of bifurcation curves of ode with nonlinear diffusion |
| url | http://dx.doi.org/10.1155/2018/5053415 |
| work_keys_str_mv | AT tetsutaroshibata globalandlocalstructuresofbifurcationcurvesofodewithnonlineardiffusion |