A Positive Solution on RNto a System of Elliptic Equations of FitzHugh–Nagumo Type
A Positive Solution on RNto a System of Elliptic Equations of FitzHugh–Nagumo Type
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FitzHugh-Nagumo型椭圆方程组RN的正解
DOI：
10.1006/jdeq.1998.3560
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发表时间：
1999
影响因子：
2.4
通讯作者：
G. Sweers
中科院分区：
文献类型：
--
作者：
C. Reinecke;G. Sweers
&2v=$(u&#v)在RN中，(1)u(X)=0，如果|x，v(X)；0，其中$，#>；0和f(U)=&u(u&1)(u&a)，其中0<；a<；1和2。Klaasen和Troy[12]研究了当N=1时的这个系统。在某些条件下，他们证明了关于参数的非平凡解的存在性。他们还证明了(1)中的方程存在无穷多个周期解。在光滑有界域上，Klaasen和Mitidieri[11]在齐次Dirichlet边界条件下研究了相应的系统。他们的结果表明，区域和参数都对解的存在和不存在起作用。De Figueiredo和Mitidieri[6]得到了关于这些解的正性的结果。在适当的尺度下，有界域上的解被看作是系统ut=d1 2u+f(U)&v in(0，)_0，(2)vt=d2 2v+=(u&#v)in(0，)_0，其中d1，d2，=>；0的稳态解。当d2=0时，(2)称为Fitzhugh Nagumo方程。这些方程被用作神经传导和其他化学和生物系统的模型[7，10]。
&2v= $(u&# v) in RN,(1) u (x) Ä 0, if| x| Ä, v (x) Ä 0, if| x| Ä, where $,#> 0 and f (u)= &u (u&1)(u&a) with 0< a< 1Ā2. Klaasen and Troy [12] investigated this system when N= 1. Using a shooting argument they proved under certain conditions on the parameters the existence of a nontrivial solution. They also proved the existence of an infinite number of periodic solutions to the equations in (1). On smooth bounded domains, Klaasen and Mitidieri [11] studied the corresponding system, subjected to homogeneous Dirichlet boundary conditions. Their results show that both the domain and the parameters play a role in the existence and nonexistence of solutions. Results about the positivity of these solutions were obtained by De Figueiredo and Mitidieri [6]. With a suitable rescaling, solutions on a bounded domain, say 0, are seen to be steady state solutions of the system ut= D1 2u+ f (u) &v in (0,) _0,(2) vt= D2 2v+=(u&# v) in (0,) _0, with D1, D2,=> 0. With D2= 0,(2) is known as the FitzHugh Nagumo equations. These equations are used as a model for nerve conduction and other chemical and biological systems [7, 10].