Article

# The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

02/2010; DOI:abs/1002.1756
Source: arXiv

ABSTRACT We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

0 0
·
0 Bookmarks
·
24 Views
• ##### Article:Regularity for the wave equation with a critical nonlinearity
Communications on Pure and Applied Mathematics 10/2006; 45(6):749 - 774. · 2.58 Impact Factor
• Source
##### Article:Endpoint Strichartz Estimates
[show abstract] [hide abstract]
ABSTRACT: . We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrodinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation. 1. Introduction In this paper we shall prove a Strichartz estimate in the following abstract setting 1 . Let (X; dx) be a measure space and H a Hilbert space. We'll write the Lebesgue norm of a function f : X ! C by kfk p j kfk L p (X) j Gamma Z X jf(x)j p dx Delta 1 p : Suppose that for each time t 2 R we have an operator U (t) : H ! L 2 (X) which obeys the energy estimate: ffl For all t and all f 2 H we have kU (t)fk L 2 x . kfkH (1) and that for some oe ? 0, one of the following decay estimates: 1991 Mathematics Subject Classification. 35L05, 35J10, 42B15, 46B70. 1 In the ...
08/1997;
• Source
##### Article:Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation
[show abstract] [hide abstract]
ABSTRACT: We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy holds: a) if the homogeneous Sobolev norm H^1 of u_0 is smaller than that of W, we have global well-posedness and scattering, b) if the homogeneous Sobolev norm H^1 of u_0 is larger than that of W, there is blow-up in finite time. Our general approach is the one we introduced in our previous work on the corresponding problem for NLS (math.AP/0610266, Inventiones Math 2006, Online First), where we proved the corresponding result for NLS in the radial case. In the case of the wave equation we are able to treat general data by using a further conservation law in the energy space, the finite speed of propagation and Lorentz transformations to establish a crucial orthogonality property for " energy critical " elements. To prove the required rigidity theorem in the case of blow-up in finite time, we cannot use the invariance of the L^2 norm as in the case of NLS. Instead, (following earlier work of Merle-Zaag and of Giga-Kohn in the parabolic case) we introduce self-similar variables. We thus find a further Liapunov function which allows us to reduce matters to a degenerate elliptic problem with critical non-linearity. We use unique continuation to rule out the existence of non-zero solutions for the degenerate elliptic problem, thus completing the proof.
11/2006;

Available from

### Keywords

bounded critical Sobolev norm

equivalent formulation

principal result

spherically-symmetric initial data