一类半线性波动方程的Sobolev指数

The Sobolev Exponents for a Semilinear Wave Equation

Ｇａｕｓｔａｖｏ Ｐｏｎｃｅ与 Ｔｈｏｍａｓ Ｃ． Ｓｉｄｅｒｉｓ猜测：对一些具有特殊非线性项的半线性波动方程，如 其 Ｓｏｂｏｌｅｖ指数会在 中．他们在 ｘ ∈ Ｒ３时回答了这一问题．本文在 Ｒｎ（ｎ≥ ４）中得到了半线性波动方程的Ｓｏｂｏｌｅｖ指数为 ｍａｘ，此数确实在区间 中．特别当 ｐ ≤ ｎ－ １时，我们得到了此半线性波动方程的Ｓｏｂｏｌｅｖ指数为．

Gustavo Ponce and Thomas C. Sideris conjectured that the Sobolev exponents for semilinear wave equations with special nonlinear terms, such as nit - △u = uk(△u)a(x ∈ R',k ∈ Z+, p = |a| ≥ 2) belong to the interval . If x ∈ R3, the answer can be found in [l]. In this paper, we obtain that the Sobolev exponent for the equation nit - △u = the This result is in some sense better than that of [3].