一类电报方程解的Sobolev指数

THE SOBOLEV EXPONENTS OF A TELEGRAM EQUATION

对一般的偏微分方程，其Ｓｏｂｏｌｅｖ指数为（ｎ２＋１），Ｐｏｎｃｅ和Ｓｉｄｅｒｉｓ（Ｃｏｍｍｕｎ．ｉｎＰＤＥ，１９９３，１８（２～４）：１６９）证明了对一些具特殊非线性项的半线性方程，如ｕｔ－Δｕ＝ｕｋ（Ｄｕ）α（ｘ∈Ｒｎ，ｋ∈Ｚ＋，ρ＝｜α｜≥２），其Ｓｏｂｏｌｅｖ指数会在ｎ２与（ｎ２＋１）之间，本文研究半线性电报方程ｕｔ－Δｕ＋ｐｕ＝ｕｋ（Ｄｕ）α（ｘ∈Ｒｎ，ｋ∈Ｚ＋，ρ＝｜α｜≥２，ｐ≠０），得到了其Ｓｏｂｏｌｅｖ指数仍然在区间（ｎ２，ｎ２＋１）之内。

It is well known that the Sobolev exponent for general partial differential equations is n2+1. Ponce and Sideris (Commun. in PDE,1993,18(2～4):169) obtained that the sobolev exponent belongs to the interal (n2,n2+1) for some special semilinear equations, such as u tt - Δ u=u k(Du) α(x∈R n,k∈Z +,ρ=|α|≥2). This paper studies the semilinear telegram equation u tt - Δ u+pu=u k(Du) α(x∈R n,k∈Z +,ρ=|α|≥2,p≠0), and proves that the Sobolev exponent is still in the interal (n2,n2+1) . Therefore, the result can be regarded as a continuation of Ponce and Sideris work.