Nontrivial Global Solutions to Some Quasilinear Wave Equations in Three Space Dimensions

In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu's seminal work [G. Luli, S. Yang, P. Yu, On one-dimension semi-linear wave equations with null conditions, Adv. Math.329 (2018) 174-188] from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu's weighted energy estimates with positive weights, introduce some space-time weighted energy estimates and pay some special attentions to the highest order energies, then use some suitable bootstrap process to close the argument.

We discuss the existence of a global small solution to the Cauchy problem for a system of quasilinear wave equations in three space dimensions, when its nonlinear term have a critical exponent. Global existence is established on the null condition which is extended to the condition for systems of wave equations with different propagation speeds.

We give a global existence theorem to systems of quasilinear wave equations in three space dimensions, especially for the multiple-speed cases. It covers a wide class of quadratic nonlinearities which may depend on unknowns as well as their first and second derivatives. Our proof is achieved through total use of pointwise and L2-estimates concerning unknowns and their first and second derivatives.

We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

We deal with systems of quasilinear wave equations which contain quadratic nonlinearities in 2-dimensional space. We have already known that such the system has a smooth solution till the time t0 = Ce-2 for sufficiently small e > 0, where e is the size of initial data. In this paper, we shall show that if quadratic and cubic nonlinearities satisfy so-called Null-condition, then the smooth solution exists globally in time. In the proof of the theorem, we use the Alinhac ghost weight energy.

We consider the Cauchy problem for systems of quasi-linear wave equations in two and three space dimensions. We investigate the small data global existence and the asymptotic behavior of solutions under a condition in between the standard and weak null conditions.

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman-Sideris type weighted estimates and space-time L2 estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n ≥ 5 of small amplitude solutions are presented.

Global solutions to systems of quasilinear wave equations with low regularity data and applications

Exact boundary controllability and its applications for a coupled system of quasilinear wave equations with dynamical boundary conditions

A transmission problem for quasi-linear wave equations

The Cauchy problem is studied for systems of quasi-linear wave equations with multiple speeds. We pursue the extension of the excellent method of Klainerman and Sideris to its limit, and a unified proof is given to previous results of Agemi-Yokoyama, Hoshiga-Kubo, Kovalyov, and Yokoyama.

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary (null) controllability and the local boundary (weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.

Small Data Blowup for Systems of Semilinear Wave Equations with Different Propagation Speeds in Three Space Dimensions

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4D anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang–Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.