Critical Nonlinear Wave Equation Research Articles

Strichartz estimates and blowup stability for energy critical nonlinear wave equations

We prove Strichartz estimates in similarity coordinates for the radial wave equation with a self-similar potential in dimensions d ≥ 4 d\geq 4 . As an application of these, we establish the asymptotic stability of the ODE blowup profile of the energy critical radial nonlinear wave equation for 4 ≤ d ≤ 6 4\leq d\leq 6 .

Almost sure scattering for the energy critical nonlinear wave equation

We study the defocusing energy critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random perturbations of the initial data. The random perturbation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the perturbation.

Asymptotic simplification for solutions of the energy critical nonlinear wave equation

In this article, we survey recent results (with Duyckaerts and Merle) on the long-time behavior of radial solutions of the energy critical nonlinear wave equation in odd dimensions.

On stability of type II blow up for the critical nonlinear wave equation on ℝ³⁺¹

On stability of type II blow up for the critical nonlinear wave equation on ℝ³⁺¹

Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations

We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.

Almost-sure scattering for the radial energy-critical nonlinear wave equation in three dimensions

We study the Cauchy problem for the radial energy critical nonlinear wave equation in three dimensions. Our main result proves almost sure scattering for radial initial data below the energy space. In order to preserve the spherical symmetry of the initial data, we construct a radial randomization that is based on annular Fourier multipliers. We then use a refined radial Strichartz estimate to prove probabilistic Strichartz estimates for the random linear evolution. The main new ingredient in the analysis of the nonlinear evolution is an interaction flux estimate between the linear and nonlinear components of the solution. We then control the energy of the nonlinear component by a triple bootstrap argument involving the energy, the Morawetz term, and the interaction flux estimate.

The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) $\partial_t^2 u - \Delta u +\partial_t u = |u|^{\frac{4}{d-2}}u$, $(t,x) \in [0,T) \times \mathbb{R}^d$, where $3 \leq d \leq 5$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.

The method of energy channels for nonlinear wave equations

This is a survey of some recent results on the asymptotic behavior of solutions to critical nonlinear wave equations.

The threshold theorem for geometric nonlinear wave equations

The aim of these notes is to provide a brief overview of a large body recent work whose aim was to prove the Threshold Theorem for energy critical geometric nonlinear wave equations. Within the class of geometric wave equations we include nonlinear wave evolutions which have a geometric structure and origin, including a nontrivial gauge group. The problems discussed here include Wave Maps, Maxwell–Klein–Gordon, as well as the hyperbolic Yang–Mills flow. In a nutshell, the Threshold theorem asserts that these problems are globally well-posed for initial data below the ground state energy.

Scattering for a critical nonlinear wave equation in two space dimensions

We show that the solutions to the Cauchy problem for a wave equation with critical exponential nonlinearity in 2 space dimensions scatter for arbitrary smooth, compactly supported initial data.

Scattering for a massless critical nonlinear wave equation in two space dimensions

We prove scattering for a massless wave equation which is critical in two space dimensions. Our method combines conformal inversion with decay estimates from Struwe's previous work on global existence of a similar equation.

ASYMPTOTICS OF THE CRITICAL NONLINEAR WAVE EQUATION FOR A CLASS OF NON-STAR-SHAPED OBSTACLES

Scattering for the energy critical nonlinear wave equation for domains exterior to non-trapping obstacles in 3+1 dimension is known for the star-shaped case. In this paper, we extend the scattering for a class of non-star-shaped obstacles called illuminated from exterior. The main tool we use is the method of multipliers with weights that generalize the Morawetz multiplier to suit the geometry of the obstacle.

The critical nonlinear wave equation in two space dimensions

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

Global dynamics of the nonradial energy-critical wave equation above the ground state energy

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in H(over dot) x L-2 with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as t -> +/-infinity.

Norm Concentration for the -Critical Nonlinear Wave Equation

Norm Concentration for the -Critical Nonlinear Wave Equation

Global Existence of Critical Nonlinear Wave Equation with Time Dependent Variable Coefficients

In this paper, we establish global existence of smooth solutions for Cauchy problem of the critical nonlinear wave equation with time dependent variable coefficients in three space dimensions where (g ij (t, x)) is a regular function valued in the spacetime of 3 × 3 positive definite matrix and (g ij (t, x)) its inverse matrix. Here and in the sequence, a repeated sum on an index in lower and upper position is never indicated. In the constant coefficients case, the result of global existence is due to Grillakis [5]; and in the time-independent variable coefficients case, the result of global existence and regularity is due to Ibrahim and Majdoub [7]. The key point of our proofs is to show that the energy cannot concentrate at any point. For that purpose, following Christodoulou and Klainerman [3], we use a null frame associated to an optical function to construct a geometric multiplier similar to the well-known Morawetz multiplier. Then we use comparison theorem originated from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality due to [15] as Ibrahim and Majdoub [7], we obtain global existence.

Dynamics for the energy critical nonlinear wave equation in high dimensions

T. Duyckaerts and F. Merle (2008) studied the variational structure near the ground state solution W W of the energy critical wave equation and classified the solutions with the threshold energy E ( W , 0 ) E(W,0) in dimensions d = 3 , 4 , 5 d=3,4,5 . In this paper, we extend the results to all dimensions d ≥ 6 d\ge 6 . The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W W .

On the equipartition of energy for the critical NLW

We study some qualitative properties of global solutions to the following focusing and defocusing critical NLW: □ u + λ u | u | 2 ∗ − 2 = 0 , λ ∈ R , u ( 0 ) = f ∈ H ˙ 1 ( R n ) , ∂ t u ( 0 ) = g ∈ L 2 ( R n ) on R × R n for n ⩾ 3 , where 2 ∗ ≡ 2 n n − 2 . We will consider the global solutions of the defocusing NLW whose existence and scattering property is shown in [J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. (7) (1994) 303–309 (electronic); H. Bahouri, J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (6) (1998) 783–789] and [H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175], without any restriction on the initial data ( f , g ) ∈ H ˙ 1 ( R n ) × L 2 ( R n ) . As well as the solutions constructed in [H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (2) (1984) 261–270] to the focusing NLW for small initial data and to the ones obtained in [C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, preprint], where a sharp condition on the smallness of the initial data is given. We prove that the solution u ( t , x ) satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see [C. Morawetz, Time decay for the nonlinear Klein–Gordon equation, Proc. Roy. Soc. London Ser. A 306 (1968) 291–296]). As a by-product we deduce that any global solution to critical NLW belonging to a natural functional space satisfies: lim R → ∞ 1 R ∫ R ∫ | x | < R | ∇ x u ( t , x ) | 2 d x d t = lim R → ∞ 1 2 R ∫ R ∫ | x | < R ( | ∇ t , x u ( t , x ) | 2 + 2 λ 2 ∗ | u ( t , x ) | 2 ∗ ) d x d t = ∫ R n ( | ∇ t , x u ( 0 , x ) | 2 + 2 λ 2 ∗ | u ( 0 , x ) | 2 ∗ ) d x .

On the critical nonlinear damped wave equation with large initial data

We study the nonlinear damped wave equation (0.1) { u t t + u t − Δ u = − | u | σ u , x ∈ R n , t > 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , in the critical case of σ = 2 n . Our aim is to prove the large time asymptotic formulas for solutions of the Cauchy problem (0.1) without any restriction on the size of the initial data and on the spatial dimension.

Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality

We show uniqueness of sufficiently regular solutions to critical semilinear wave equations and wave maps in the (a priori) much larger class of distribution solutions with finite energy, assuming only that the energy is nonincreasing in time. © 1999 John Wiley & Sons, Inc.