The Existence of Global Solutions to Systems of Quasilinear Wave Equations with Quadratic Nonlinearities in 2-Dimensional Space

Abstract

Abstract. 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 t 0 ¼ Ce À2 for su‰ciently 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.

Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

“…Lemma can be proved by propositions 4.1 and 4.2 in Hoshiga and Kubo (see also Hoshiga or Kong et al).…”

“…Proof. The local existence of the solution to the Cauchy problem (19) can be derived by the standard Picard iteration method; we omit the details here. For the global existence of f(t), we use the continuity method.…”

“…Thus, we can see that f(t) exists on [0, + ∞). For the second inequality of (20), by applying m t to the left hand side of (19) and by Leibniz role, we have…”

Global existence of smooth solutions to 2D Chaplygin gases on curved space

“…where C is a constant. Lemmas 3.1-3.5 can be found in [6,23,24] and references therein; we omit the proof here.…”

Stability in exponential time of Minkowski space-time with a space-like translation symmetry

“…Global existence for a semi-linear wave equation with the quadratic and cubic terms satisfying the null condition has been shown by Godin in [10] using an algebraic trick to remove the quadratic terms, which does however not extend to systems. The global existence in the case of systems of semi-linear wave equations with the null structure has been shown by Hoshiga in [11]. It requires the use of L ∞ −L ∞ estimates for the inhomogeneous wave equations, introduced in [17] (see also Proposition 4.3).…”