Wave Equations In Higher Dimensions Research Articles

Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation (1.1) with dimension n ⩾ 4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form Δ u - ∂ t 2 u + | u | p = 0 . The rest of the cases, the lower-dimensional case n ⩽ 3 , and the sub or super critical cases were settled many years earlier by the work of several authors.

Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions

In this article we consider an initial--boundary value problem for a wave equation in high dimensions with a nonlinear damping term that is not Lipschitz in $u_t$. We establish the existence and uniqueness of a global solution by using a compactness method and by exploiting the monotonicity property of the nonlinearity.

Exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in higher dimensions

Exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in higher dimensions

A self-adjoint problem for the wave equation in higher dimensions

A self-adjoint problem for the wave equation in higher dimensions

Factorization of the wave equation in higher dimensions

The factorization of the wave equation into a coupled system involving up-and down-going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables, but the velocity c is a function of the z variable only. The form of the reflection operator is derived and the quadratic differential-integral equation satisfied by its kernel is obtained.

Finite-time blow-up for nonlinear wave equations in high dimensions

On considere le probleme de Cauchy pour l'equation d'onde non lineaire u tt -Δu=G(u r ,u t ), (XeR n , t>0) avec des dimensions d'espace superieures a un et avec des donnees initiales de support compact

On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions

On etudie la propriete de decroissance des solutions des equations d'onde dissipatives non lineaires de la forme: ∂ 2 u/∂t 2 -Δu+g(∂u/∂t)=f(x,t), (x,t)∈Ω×R + avec les conditions u(x,o)=u 0 (x), u t (x,o)=u 1 (x), x∈Ω et u(x,t)=0 sur ∂Ω×R + , ou Ω est un domaine borne de R n avec une frontiere lisse ∂Ω et g(v) est une fonction du genre k/v/ α v,k>o,α≥o

Non-resonance for a strongly dissipative wave equation in higher dimensions

Under some regularity conditions, a non-resonance property is established for a semi-linear forced wave equation with a strong local damping term and Dirichlet boundary conditions in a bounded open domain. In dimension less than or equal to six, the damping term can grow at infinity like an arbitrarily large power of the velocity. If a viscosity term is added, in dimension less or equal to four a stronger result is obtained, and this property allows to construct almost periodic solutions for an arbitrary forcing term in a suitable regularity class.

Lower bounds for the life span of solutions of nonlinear wave equations in higher dimensions

when m > 0. Writing (1) as a quasilinear symmetric hyperbolic system, it is known that a classical solution exists locally in time. provided that f, g are “small” and sufficiently smooth. The time interval [O. T] for which this classical solution exists depends on the size of the initial data. More precisely, if f = .@,, and g = &go where fO, go are fixed, the largest time T = T(E) for which the classical solution of (1) is assured by the Local Existence Theorem is of order l/~. For the one dimensional genuine nonliner wave equation. i.e. m = 0, n = 1 in (1) and cy= 1 in (2). Lax [l] has shown that this estimate is sharp. In this case when the genuine nonlinearity condition is violated. i.e. LY> 1, Klainerman and Majda [2] have shown that T(E) = 0( l/.?) and that this estimate is also sharp for E sufficiently small. In contrast to the one dimensional case, John has shown that in higher dimensions the existence time of classical solutions can be significantly improved. In fact, he proved for the case m = 0, (Y = 1, that T(E) = 0( l/s) for n = 4,5 [3] and recently T(E) = U( l/8) for

Nonexistence of global solutions to semilinear wave equations in high dimensions

On considere le probleme de Cauchy pour l'equation d'onde semi lineaire (ο 2 /∂t 2 -Δ)u(x,t)=|u(x,t)| p (p>1), u(x,o)=f(x), u t (x,o)=g(x) avec f,g∈C o ∞ (R n )

Lower bounds for the life span of solutions of nonlinear wave equations in three dimensions

Communications on Pure and Applied MathematicsVolume 36, Issue 1 p. 1-35 Article Lower bounds for the life span of solutions of nonlinear wave equations in three dimensions† Fritz John, Fritz JohnSearch for more papers by this author Fritz John, Fritz JohnSearch for more papers by this author First published: January 1983 https://doi.org/10.1002/cpa.3160360102Citations: 30 † This is a continuation of the author's paper Delayed Singularity Formation of Solutions of Nonlinear Wave Equations in Higher Dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682, referred to as (*) in the sequel. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume36, Issue1January 1983Pages 1-35 RelatedInformation

Delayed singularity formation in solution of nonlinear wave equations in higher dimensions

It is well known that solutions u(t, x) of quasi-linear homogeneous hyperbolic equations in one space dimension tend to develop singularities after a finite time (see [1], [2]).