Blowup of Solutions to a Class of Nonlinear Schr?dinger Equations
Study the following nonlinear Schrödinger equationi∂tu=-Δu+i(-t)a(p-1)|u|p-1u,wherep>1,(n-2)(p-1)≤4,a≥0 is a real number,(t,x)∈(-∞,0)×Rn,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-tion Φ(t,x)=(C(-t)a(p-1)+1+φ(x))-1/p-1,where C=(p-1)/[a(p-1)+1],(t,x)∈(-∞,0)×Rn.And the function Φ satisfies the ordinary differential equation of Φt=(-t)a(p-1)|Φ|p-1Φ,with a series of assumptions about φ,such that‖Φ‖L2(Rn)→∞ when t→0-.Thirdly,the energy method and some important inequalities are used to estimate the error term.Finally,we find an analytic solution close to Φ by using the compactness theorem,and prove the final blow-up result by using the previous estimate.
Nonlinear schrödinger equationGlobal well-posedness of the solution of the inverse equationApproximate solutionFinite time blow-up
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