On the supercritical defocusing NLW outside a ball

We study a defocusing semilinear wave equation, with a power nonlinearity $$|u|^{p-1}u$$ | u | p - 1 u , defined outside the unit ball of $$\mathbb {R}^{n}$$ R n , $$n\ge 3$$ n ≥ 3 , with Dirichlet boundary conditions. We prove that if $$p>n+3$$ 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.

Strong Exponential Attractors for Weakly Damped Semilinear Wave Equations

In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain of ℝ3. The exponential attractor is investigated in a strong energy space for the case of subquintic nonlinearity, which is based on the recent extension of the Strichartz estimate for the case of a bounded domain. The results obtained complete some previous works.

A Sharp Lorentz-Invariant Strichartz Norm Expansion for the Cubic Wave Equation in ℝ1+3

We provide an asymptotic formula for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant for the linear Strichartz estimate. We calculate the constant in the second term, which differs depending on whether the equation is focussing or defocussing. The sign of this coefficient also changes accordingly.

Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

By [Formula: see text]-method, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation [Formula: see text] where [Formula: see text], and [Formula: see text], [Formula: see text], with radial data in [Formula: see text] for [Formula: see text]. It is a sharp global result except the critical case [Formula: see text], which is a very difficult open problem.

Causal geometry of rough Einstein CMCSH spacetime

This is the second (and last) part of a series in which we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in constant mean curvature and spatial harmonic (CMCSH) gauge, and we obtain a local well-posedness result in Hs with s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough Einstein metric g, we manage to implement the commuting vector field approach and prove a Strichartz estimate for the geometric wave equation □g ϕ = 0 in a direct manner. This direct treatment would not work without gaining sufficient regularity on the background geometry. In this paper, we analyze the geometry of null hypersurfaces in rough Einstein spacetimes in terms of Hs data. We provide an integral control on the spatial supremum of the connection coefficients [Formula: see text], ζ, which is crucially tied to the Strichartz estimates established in the first part.