Studies On The Properties Of The Solutions Of Inhomogeneous Schr?dinger Equations

In this doctoral dissertation,we study the following inhomogeneous nonlinear Schrodinger equations i(?)tu-(-?)su+|x|k|u|2?u=0,(t,x)?R×RN,where 1/2<s?1,k=±b(b>0).We mainly study the sufficient conditions for global existence and blow up in finite time,dynamical behavior,limit behavior for blow-up solutions,the stability and instability of standing waves and so on.We deal with the following four cases:1)s=1,k=±b,(b>0),2+k/N??<min{2,2+k/N-2,2(1+k)+N/2N}(N?3,k>-2);2)s=1,k=-b(0<b<1),?=2+k/N,(N?2,K>-2);3)1/2<s<1,k=-b<0,2s-b/N??<2s-b/N-2s(N?2,b<2s);4)1/2<s<1,k=-b<0,0<??2s-b/N(N?2,b<2s).For Case 1),we investigate the limit behavior of the H1 blow-up solution at blow up time.If ||u(t)||H1 blows up at T*,we show that when ?= 2+k/N,u(t)has no L2-limit as t?T*.In particular,for a radially symmetric blow-up solution,it has the L2 con-centration at the origin.Furthermore,if 2+k/N<?<min{2,2+k/N-2,2(1+k)+N/2N},we show that there exists a unique u*?L2(RN)such that ?(-t)u(t)??(-T*)u*in Lr(RN)(r?[2,2*))as ?T*.Our results extend those for k=0 given by F.Merle around 1990.For Case 2),we study the dynamical behavior for blow-up solutions when initial date ??H1(RN)and satisfies ||?||L2?||Q||L2,where Q is the ground state solution of our problem.Furthermore,we obtain the determination of the blow-up solutions when ||?||L2= ||Q||L2.For Case 3),we prove a Gagliardo-Nirenberg-type estimate and use it to estab-lish sufficient conditions for global existence in Hs(RN).In addition,we derive a localized Virial estimate for inhomogeneous fractional nonlinear Schrodinger equa-tion in RN.which uses Balakrishnanns formula for the fractional Laplacian(-?)s from semigroup theory.By these estimates,we give the blowup criterion of radial solutions in RN for L2-critical,L2-supercritical and Hs-subcritical power.For Case 4),we consider the stability and instability of standing waves.Using a Gagliardo-Nirenberg type inequality and the profile decomposition,we obtain that in the L2-subcritical case,i.e.0<?<2s-b/N,the solution exist globally and the standing waves are orbitally stable.In the L2-critical case,i.e.?=2s-b/N,we obtai the ground state solitary waves are strongly unstable.