广义Kawahara方程的Cauchy问题

On the Cauchy Problem of Generalized Kawahara Equations

对初值在Besov空间中的广义Kawahara方程(?)_tu+αu^k(?)_xu+β(?)_x^3u+γ(?)_x^5u=0进行了研究,其中k是大于4的正整数,证明了对任意的1≤q≤∞,其Cauchy问题在Besov空间B_(2,q)^(sk)(R)和B_(2,q)~s(R)中局部适定,这里s_k=(k-8)/2k,s＞max(0,s_k)；对小初值问题几乎整体适定．并证明了如果β=0或βγ＜0,对小初值问题整体适定．

This paper studies the Cauchy problem of the generalized Kawahara equations 偏倒dtu＋αu^k偏倒dxu＋β偏倒d^3xu＋γ偏倒du^5xu=0 where k is an integer greater than 4, with initial data in Besov spaces. It is proved that for any 1 ≤ 〈 q≤∞ the Cauchy problem of this equation is locally well-posed in the Besov spaces B2,q（R） and B^8k 2,q（R） where sk = k-8/2k and s 〉 Sk, and almost globally well-posed in these spaces if initial data are small, and also proves that if either β = 0 or βγ〈 0, then global well-posedness holds in these spaces for small initial data problem.