任意维数半线性拟抛物方程的整体W2,p(2＜p＜∞)解

摘要

研究有界域上的任意维数的半线性拟抛物方程的初边值问题ut-△ut=f(u) x∈Ω, t＞0 (1.1)u(x, 0)= u0(x) x∈Ω (1.2)u| Ω=0 t≥0 (1.3)利用逐次磨光法,证明了,若f∈C1,f(u)上方有界,且满足(H): |f′u)|≤A1|u|γ1+B1, 0≤γ1＜∞ ifn=4; 0≤γ1＜4-4 if n＞4u0(x)∈W2,p(Ω)∩W1,p 0(Ω)(2＜p＜∞),则对任一T(x),问题(1.1)-(1.3)存在唯一整体解u(x,t)∈W2,∞(0,T;W2,p(Ω)∩W1,p 0(Ω)).从实质上改进和推广了文献[1-3]的结果.%The initial boundary value problem on bounded domain for semilinear pseudoparabolic equations in arbitrary dimensions is studied By using successive smoothing method, it is proved that if f∈C′, f′(u) is bounded above and satisfies (H): |f′(u)|≤A1|u|γ1+B1, 0≤γ1＜∞ if n=4; 0≤γ1 ＜ 4-4 if n＞4and u0(x)∈W2,p(Ω)∩ W1,p 0(Ω)(2＜p ＜∞), then for any T(x) problem (1.1)-(1.3) admits a unique global solution u(x, t)∈W2,∞ (0, T; W2,p(Ω)∩W1,p 0(Ω)). The results of [1-3] are generalized and improved essentially.