共查询到20条相似文献,搜索用时 15 毫秒
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Chunyan Huang 《Journal of Functional Analysis》2008,255(3):681-725
In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg-Landau equation in Rn, n?3. In lower dimension case (3?n?6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in , T>0, s=0,1, as a by-product, we get the regularity of solutions in H3 for the nonlinear Schrödinger equation. In higher dimension case (n>6), we get the similar convergent behavior in C(0,T,L2(Rn)). In both cases we obtain the optimal convergent rate. 相似文献
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We study the long-time behavior of solutions of semilinear parabolic equation of the following type t∂u−Δu+a0(x)uq=0 where , d0>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators. 相似文献
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U.-W. Schmincke 《Journal of Mathematical Analysis and Applications》2003,277(1):51-78
Starting from a selfadjoint Schrödinger operator A=−d2/dx2+q with a gap G in its spectrum F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267-324] succeed in constructing another Schrödinger operator that is unitarily equivalent (and thus isospectral) to A. As the means they apply come from the Weyl-Titchmarsh theory the connections prove to be intricate, in particular the relation between A and . We show that a central assertion in GST's paper rests substantially on factorizations of the form
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We consider the nonlinear nonelliptic Schrödinger equation defined by with initial datum in L2(R2). We show that if the solution blows up in finite time, then there is a mass concentration phenomenon near the blow-up time. The key ingredient is a refinement of the Strichartz inequality on the saddle. 相似文献
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Benjamin Dodson 《Journal of Functional Analysis》2010,258(7):2373-2421
We prove the nonlinear Schrödinger equation has a local solution for any energy - subcritical nonlinearity when u0 is the characteristic function of a ball in Rn. Additionally, we establish the existence of a global solution for n?3 when and α?2. Finally, we establish the existence of a global solution when the initial function is radial, the nonlinear Schrödinger equation has an energy subcritical nonlinearity, and the initial function lies in Hρ+?(Rn)∩H1/2+?(Rn)∩H1/2+?,1(Rn). 相似文献
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Consider the family of Schrödinger operators (and also its Dirac version) on ?2(Z) or ?2(N)
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We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations. 相似文献
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In this article we prove the property of unique continuation (also known for C∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds? 相似文献
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Xiangqing Zhao Cuihua Guo Wancheng Sheng 《Journal of Mathematical Analysis and Applications》2011,382(1):97-109
In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x1,x2,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s1,s2 satisfy the conditions: s1?0, s2?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces . 相似文献
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Xiaoyi Zhang 《Journal of Differential Equations》2006,230(2):422-445
We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ1p|u|u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ1 is. Scattering will occur either when the perturbation is defocussing and or when the mass of the solution is small enough and . When the quintic term is focusing, we show the blow up for certain solutions. 相似文献
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Lionel Rosier 《Journal of Differential Equations》2009,246(10):4129-97
This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if
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Tai-Peng Tsai 《Journal of Differential Equations》2003,192(1):225-282
We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H1, and that its ground state component is larger than n3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity. 相似文献
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Changxing Miao 《Journal of Differential Equations》2009,246(9):3715-3402
We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d?5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)‖2<‖ΔW‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters. 相似文献