APPLICATION OF THE INVARIANCE PRINCIPLE TO REACTION-DIFFUSION EQUATIONS
APPLICATION OF THE INVARIANCE PRINCIPLE TO REACTION-DIFFUSION EQUATIONS
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DOI：
10.1016/0022-0396(79)90088-3
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发表时间：
1979-01-01
影响因子：
2.4
通讯作者：
ALIKAKOS, ND
中科院分区：
文献类型：
--
作者：
ALIKAKOS, ND
要研究的问题是系统-= v，dc，+ c，B，（c，）.，c '。g）、。f？acz at 'I an aR= 0，i= 1，.，开的，光滑的，有界的。在E&k上的食物金字塔条件的假设下，证明了常微分方程的凸Liapunov函数的存在性，并在yi与Liapunov函数相联系的额外假设下，证明了（*）的解逼近于（* a）的最大不变集的一个子集--紧连通集.将上述结果应用于具有扩散的Volterra-Lo&a方程组，得到了对于任意扩散系数Ye，Y* 和任意维数，（*）的任何解都指数地逼近经典Volterra-Lo&a方程的周期解.研究了经典Volterra-Lo&a方程的周期解在（*）解类中的流形的稳定性.
The problem to be studied is the system-= v, dc,+ c, B,(c,)..., c ‘. g),. f?. acz at’I an aR= 0, i= l,..., N;(* I vp z 0, n open, smooth, bounded in (w”. Under rhe assumption of a food pyramid condition on E&k, the existence of a convex Liapunov function for the ordinary differential equations, and under an extra hypothesis relating the yi’s with the Liapunov function it is shown that the solutions of (*) approach a compact connected set, a subset of the maximal invariant set of (* a). Applying the above to the system of the Volterra-Lo&a equations with diffusion, me obtain that any solution of (***) approaches exponentially a periodic solution of the classical Volterra-Lo&a equations for arbitrary diffusion coefficients Ye, Y* and for any dimension. The stability of the manifold of the periodic solutions of the classical Volterra-Lo&a equations in the class of solutions of (***) is investigated.