Asymptotic stability for some systems of semilinear Volterra diffusion equations
Asymptotic stability for some systems of semilinear Volterra diffusion equations
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DOI：
10.1016/0022-0396(84)90165-7
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发表时间：
1984-03
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影响因子：
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通讯作者：
山田 義雄
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文献类型：
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作者：
山田 義雄
设 0 为 R” 中的有界域，边界光滑 X！。我们考虑以下问题，对于 U&X, t), i= 1, 2,..., N，其中 x=(x*, x*,..., x,) Ea 和 tER*： au.--=/@ ui++ at] yl pi.fuj+ il, Cij (t-S> uj (s) ds)+ fi (ul, l>~ 2, t,*** r uN, t),(i= 1, 2,..., N) in 0 X (0,+ co),(1.1) aui av-0(i= 1, 2,..., N) on aB x (0,+ co),(1.2) ui= fji(i= 1, 2,..., N) in BX (-co, 01,(1.3) 其中每个 pi 是正常数，A 是拉普拉斯算子 (CJ= i a'/ax,')，对于任何 t> 0，每个 u~,~ 表示由 ui, l (x, 0)= ui (x, t+ S) 定义的 R x (-co, 0] 上的函数，其中 x ER 和 0 E (-co, 01, a/& 表示 aR 的外向正规导数，每个# i 是 0 X (-co, 01. 在 (ll) 中，B 和 C,(t) 是某些函数空间上的有界线性算子，C,(t) 在算子范数上可在 [0,+ co) 上可积，fi (v,, v2,..., vN) 在某种意义上是关于 (w,> V2Y’, w,,,) 的高阶项，满足 fi (O, 0,..., 0)= 0 和某种要精确的小条件许多作者研究了类似 Volterra 积分微分方程解的存在性、唯一性和稳定性（例如，参见有限维情况下的 [1, 2, 5, 13, 14, 27, 291 和无限维情况下的 [15, 17, 18, 20-25, 28, 301]，其中包括 Schiafftno 和 Tesei [22, 23, 251 用（1.2）（或零狄利克雷边界条件）处理形式为（1.1）的方程，以获得局部295的一些充分条件
Let 0 be a bounded domain in R” with smooth boundary X!. We consider the following problem for U&X, t), i= 1, 2,..., N, with x=(x*, x*,..., x,) Ea and tER*: au.--=/@ ui++ at] yl pi. fuj+ il, Cij (t-S> uj (s) ds)+ fi (ul, l>~ 2, t,*** r uN, t),(i= 1, 2,..., N) in 0 X (0,+ co),(1.1) aui av-0(i= 1, 2,..., N) on aB x (0,+ co),(1.2) ui= fji(i= 1, 2,..., N) in BX (-co, 01,(1.3) where each pi is a positive constant, A is the Laplace operator (CJ= i a’/ax,‘), for any t> 0 each u~,~ represents a function on R x (-co, 0] defined by ui, l (x, 0)= ui (x, t+ S) with x ER and 0 E (-co, 01, a/& denotes the outward normal derivative to aR and each# i is a given function on 0 X (-co, 01. In (ll), B, and C,(t) are bounded linear operators on some function spaces, C,(t) are integrable over [0,+ co) in the operator norm and fi (v,, v2,..., vN) are, in a sense, higher-order terms with respect to (w,> V2Y’, w,,,), which satisfy fi (O, 0,..., 0)= 0 and a certain type of smallness condition to be made precise later.Existence, uniqueness, and stability of solutions for similar Volterra integrodifferential equations have been studied by many authors (see, eg,[1, 2, 5, 13, 14, 27, 291 in the finite dimensional case and [15, 17, 18, 20-25, 28, 301 in the infinite dimensional case). Among others, Schiafftno and Tesei [22, 23, 251 have treated equations of the form (1.1) with (1.2)(or zero Dirichlet boundary condition) to get some sufficient conditions for the local 295