ar X iv : 0 90 4 . 13 17 v 1 [ m at h - ph ] 8 A pr 2 00 9 MINIMAL BLOW - UP SOLUTIONS TO THE MASS - CRITICAL INHOMOGENEOUS NLS EQUATION
نویسنده
چکیده
We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrödinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.
منابع مشابه
ar X iv : 0 90 4 . 13 17 v 2 [ m at h - ph ] 2 0 Ju l 2 01 0 MINIMAL BLOW - UP SOLUTIONS TO THE MASS - CRITICAL INHOMOGENEOUS NLS EQUATION
We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the crit...
متن کاملar X iv : 0 80 4 . 34 08 v 1 [ m at h . A P ] 2 1 A pr 2 00 8 SMOOTHNESS OF LIPSCHITZ MINIMAL INTRINSIC GRAPHS IN HEISENBERG GROUPS
We prove that Lipschitz intrinsic graphs in the Heisenberg groups H , with n > 1, which are vanishing viscosity solutions of the minimal surface equation are smooth.
متن کاملm at h . A P ] 1 6 O ct 2 00 6 MINIMAL - MASS BLOWUP SOLUTIONS OF THE MASS - CRITICAL NLS
We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = μ|u|4/du to blow up. If m0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in Lx(R d) is compact after quotienting out by the symmetries of the equation. A similar res...
متن کامل. A P ] 2 9 A ug 2 00 5 NON - GENERIC BLOW - UP SOLUTIONS FOR THE CRITICAL FOCUSING NLS IN 1 -
We consider the critical focusing NLS in 1-d of the form (1.1) i∂ t ψ + ∂ 2 x ψ = −|ψ| 4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is well-known that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ 0 (x, α), α > 0 Indeed, requiring positivity and evenness in x for φ 0 (x, α) implies for example φ 0 (x, α) = α 1 2 (3 2) 1 4 cosh 1 2 (α 2 x) Another remarkable...
متن کاملExistence and Uniqueness of Minimal Blow-up Solutions to an Inhomogeneous Mass Critical Nls
(1.1) (NLS) { i∂tu = −Δu− k(x)|u|u, (t, x) ∈ R× R, u(t0, x) = u0(x), u0 : R 2 → C, for some smooth bounded inhomogeneity k : R → R+ and some real number t0 < 0. This kind of problem arises naturally in nonlinear optics for the propagation of laser beams. From the mathematical point of view, it is a canonical model to break the large group of symmetries of the k ≡ 1 homogeneous case. From standa...
متن کامل