1,720,980 research outputs found
On the supercritical Schrödinger equation on the exterior of a ball
No full textWe consider the mixed problem on the exterior of the unit ball in (Formula presented.) for a defocusing Schrödinger equation with a power nonlinearity (Formula presented.) with zero boundary data. Assuming that the initial data are non-radial, sufficiently small perturbations of large radial initial data, we prove that for all powers (Formula presented.) the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class
On the supercritical defocusing NLW outside a ball
Full text linkWe study a defocusing semilinear wave equation, with a power nonlinearity | u| p-1u, defined outside the unit ball of Rn, n≥ 3 , with Dirichlet boundary conditions. We prove that if p> n+ 3 and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a Strichartz estimate for the exterior linear wave equation perturbed with a large, time dependent potential
On large potential perturbations of the Schrödinger, Wave and Klein–Gordon equations
Full text linkWe prove a sharp resolvent estimate in scale invariant norms of Amgon–Hörmander type for a magnetic Schrödinger operator on Rn, n ≥ 3 L = −(∂ + iA)2 + V with large potentials A, V of almost critical decay and regularity. The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein–Gordon flows associated to L
A Short Proof of Commutator Estimates
Full text linkThe goal of this note is to give, at least for a restricted range of indices, a short proof of homogeneous commutator estimates for fractional derivatives of a product, using classical tools. Both Lp and weighted Lp estimates can be proved by the same argument. When the space dimension is 1, we obtain some new estimates in the unexplored range 1 / 3 < r≤ 1 / 2
Global solvability for the degenerate Kirchhoff equation with real analytic data
No full textWe prove global solvability for real analytic data of a Kirchhoff equation with degenerate hyperbolicity
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