一类非线性薛定谔方程解的爆破

Blowup of Solutions to a Class of Nonlinear Schrödinger Equations

考虑非线性薛定谔方程i∂_(t)u=-Δu+i(-t)^(a(p-1))|u|^(p-1)u,这里p>1,满足(n-2)(p-1)≤4,a≥0是已知实数,(t,x)∈(-∞,0)×R^(n),u=u(t,x)是未知的复值函数.第一,证明了反向方程解的整体适定性;第二,构造了所研究方程的一个近似解,主要想法是构造一个显函数Ф(t,x)=(C(-t)^(a(p-1)+1)+φ(x))^(1/(p-1)),其中C=(p-1)/[a(p-1)+1],(t,x)∈(-∞,0)×R^(n),且函数Φ满足常微分方程Φ_(t)=(-t)^(a(p-1)|Φ|p-1)Φ,对φ加以一系列假设,使得当t→0^(-)时,‖Φ‖L^(2)(R)^(n)→∞;第三,利用能量方法及已知不等式对误差项进行估计;第四,利用紧致性理论找到了一个逼近近似解Φ的解析解,利用对近似解的估计证明最终的爆破结果.

Study the following nonlinear Schrödinger equation i∂_(t)u=-Δu+i(-t)^(a(p-1))|u|^(p-1)u,where p>1,(n-2)(p-1)≤4,a≥0 is a real number,∈(-∞,0)×R^(n),u=u(t,x)is an unknown complex value function.Firstly,the global well-posedness of the solution of the inverse equation is proved.Secondly,an approximate solution of the equation studied in this paper is constructed.The idea is to construct a explicit func-tionФ(t,x)=(C(-t)^(a(p-1)+1)+φ(x))^(1/(p-1)),where C=(p-1)/[a(p-1)+1],(t,x)∈(-∞,0)×R^(n).And the functionΦsatisfies the ordinary differential equation ofΦ_(t)=(-t)^(a(p-1)|Φ|p-1)Φ,with a series of assumptions aboutφ,such that‖Φ‖L^(2)(R)^(n)→∞.Thirdly,the energy method and some important inequalities are used to estimate the error term.Finally,we find an analytic solution close toΦby using the compactness theorem,and prove the final blow-up result by using the previous estimate.