一维热方程奇异初边值问题

A Singular initial-Bondary- Value Problem for the One-Dimensional Heat Equation

本文证明了形如ｕ＿（ｘｘ）＝ｕ＿ｔ，ｕ（ｘ，０）＝０，ｕ（０，ｔ）＝Ｕｔ￣（－（ｋ＋１）／２）），ｕ（∞，ｔ）＝０，的奇异初边值问题，当ｋ＝１，３，５，…时没有相似解；而当ｋ＞一１且ｋ≠１，３，５，…时相似解一定存在。第一个断言推翻了Ｐｈａｎ－Ｔｈｉｅｎ于文［１］中提出的一个重要结论。

The present paper proves the singular initial-boundary-value problem u_(xx)=u_t,u(x,0)=0,u_(0,t)=Ut￣(-(k+1)/2),u(+∞,t)=0,does not possess any similarity solution when k=1,3,5,…;when k>一1and k≠1,3,5,…,similarity solutions exist.The first assertion shows that the important conclusion presentedby Phan-Thien in[1] is not true。