Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions

Abstract

We discuss the global existence of small solutions to the Cauchy problem for systems of quasilinear wave equations in three space dimensions, when their nonlinear terms have quadratic nonlinearity. A global existence theorem is established on the null condition which is extended to the condition for systems of wave equations with di¨erent propagation speeds.

Asymptotic behavior of solutions to semilinear systems of wave equations

“…Yokoyama [21] proved that when c 1 = c 2 , the problem admits a unique global smooth solution when p = q = 2 and the initial data are in C ∞ 0 (R 3 ) and sufficiently small. On the other hand, Deng showed in Theorem 3.3 of [6] that if c 1 = c 2 and q(p−1) ≤ 2, then, in general, a classical solution to the problem blows up in finite time however small the initial data are.…”

“…In order to extend the existence result due to [21] for general p, q > 1, we consider only radially symmetric solution to the Cauchy problem for (2.1). To be more specific, we seek solutions to the problem in X × X , where X is defined by…”

“…. , c N do not necessarily coincide with each other) so that the small data global existence for (1.1) holds (see Kovalyov [12], Agemi-Yokoyama [1], Yokoyama [21], Sideris -Tu [18], Kubota -Yokoyama [15], Katayama [7], [8], [9], KatayamaYokoyama [10] and so on). For example, in addition to null forms, terms like (∂ a u j )(∂ b u k ) with c j = c k are allowed to be included for the multiple speeds case.…”

Global Existence for Systems of Nonlinear Wave Equations in 3D with Multiple Speeds

“…The case of nonrelativistic systems has been considered in 3D [22] and in 2D [5], [4]. We mention also the early work [16], [15], [17] which deals with nonresonant interactions.…”

On Systems of Semilinear Wave Equations with Unequal Propagation Speeds in Three Space Dimensions

“…[2,11,13,14,15,21,26,28]. In [28] the case where all F j does not depend explicitly on u itself was studied and the result in it yields, in particular, the small data global existence when F j ðqu; q 2 uÞ ¼ P N k; l¼1 A j; k; l ðq t u k Þðq t u l Þ with constants A j; k; l satisfying A j; j; j ¼ 0, which corresponds to all cases (a), (b) and (c). Therefore the discrepancy of the propagation speeds is more e¤ective in this case.…”