Extremal equilibria for dissipative parabolic equations in locally uniform spaces
Extremal equilibria for dissipative parabolic equations in locally uniform spaces
复制标题
局部均匀空间中耗散抛物线方程的极值平衡
DOI：
10.1142/s0218202509004029
复制
发表时间：
2009
影响因子：
3.5
通讯作者：
A. Rodríguez
中科院分区：
文献类型：
--
作者：
J. Cholewa;A. Rodríguez
考虑一类反应扩散方程ut = Δu + f（x，u），初值在局部一致空间q ∈ [1，∞）中，耗散非线性项满足sf（x，s）≤ C（x）s2 + D（x）.| S|，在哪里，而且肯定。本文构造了一个全局吸引子，并证明了它实际上包含在有序区间[φm，φM]中，其中是一对平稳解，分别为极小解和极大解，它们一致满足φm ≤ lim inft→∞ u（t; u 0）≤ lim supt→∞ u（t; u 0）≤ φM，其中u 0在[φ m，φM]的有界子集中.给出了系统存在最小正平衡态且自下渐近稳定的一个充分条件。某些充分条件也讨论了确保解决方案是渐近小，|X| → ∞。在这种情况下，解渐近地进入可积函数的Lebesgue空间，吸引子在一致收敛拓扑中吸引，并且对于某些r > N/2，它是W2，r（<$N）的有界子集。讨论了正解的唯一性和渐近稳定性。应用到一些模型问题，包括一些数学生物学。
We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space , q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where and for certain . We construct a global attractor and show that is actually contained in an ordered interval [φm, φM], where is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of . A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.