一类Boussinesq方程的Sobolev指数

Sobolev exponent of the damped Boussinesq equation

研究了如下Boussinesq方程Cauchy问题的整体解:utt-aΔutt-2bΔut=-cΔ2u+Δu-αu+βΔ(up),u(x,0)=ε2(x),ut(x,0)=ε2ψ(x).其中x∈Rn,n≥2,t>0,a,b,c,α是正常数,β∈R,ε>0是小参数,p≥2是正整数.当a+c-b2>0时,得到了上面问题整体解的存在性,而且得到方程的Sobolev指数是n/2-1/(p-1).

The authors studied the global solution of the Cauchy problem for the following Boussinesq equation： utt-αΔutt-2bΔut=-cΔ^2u＋Δu-αu＋βΔ（u^p）,u（x,0）=ε^2φ（x）,ut（x,0）=ε^2ψ（x）,where x∈R^n,n≥2,t〉0,α,b,c and α are positive constants, β ∈ R,eis a small positive parameter,p≥2 is positive integer. For the case α ＋ c - b^2 〉 0, the well-posedness of the global solution for the equation is studied. It is proved that the Sobolev exponent of the equation is n/2-1/p-1.