奇异半线性发展方程的局部Cauchy问题  

The Local Cauchy Problem of Singular Semilinear Evalution Equations in Banach Spaces

作　　者：蹇素雯[1] 

机构地区：[1]云南师范大学数学系

出　　处：《数学学报（中文版）》1997年第5期793-800,共8页Acta Mathematica Sinica：Chinese Series

摘　　要：本文在Ｂａｎａｃｈ空间Ｅ中讨论如下问题ｄｕｄｔ＋１ｔσＡｕ＝Ｊ（ｕ），０＜ｔＴ，ｌｉｍｔ→０＋ｕ（ｔ）＝０，其中ｕ：（０，Ｔ］Ｅ，Ａ是与ｔ无关的线性算子．（－Ａ）是Ｅ上Ｃ０半群｛Ｔ（ｔ）｝ｔ０的无穷小生成元，常数σ１，Ｊ是一个非线性映射ＥＪ→Ｅ．它满足局部Ｌｉｐｓｃｈｉｔｚ条件．我们证明了当其Ｌｉｐｓｃｈｉｔｚ常数ｌ（ｒ）满足一定条件时．问题（Ｓ）有局部解，且在某函类中解唯一．设Ｊ（ｕ）＝｜ｕ｜γ－１ｕ＋ｆ（ｘ）（γ＞１），Ｅ＝Ｌｐ，ＥＪ＝Ｌｐγ时得到了与Ｗｅｉｓｌｅｒ［２］在非奇异情形类似的结果．In this paper, we study the following cauchy problem  dudt + 1t σ Au =J(u), 0<tT, lim t→0 +u(t)=0,in the Banach space E, where u(0,T] E, A is a linear operator with independent of t. σ1, (-A) is infinitesimal generator of continuous semi group {T(t)} t0 on E, and J is a nonlinear function from a subset E J of E into E. We assume that J:E J→E is locally Lipschitz. As the Lipscitz constant l(r) of J satisfying some conditions. We prove the locally existence and uniqueness of the problem (S). In addition, if A=-n i=1  2x 2 i, J(u) =|u| γ-1 u+ f(x)(γ>1), E=L p, E J=L pγ , we have the similar results with the results of Fed B. Weissler in .

关 键 词：半线性 发展方程 C0半群 初值问题 巴拿赫空间 

分 类 号：O175.26[理学—数学]