Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions (Q2911985)

The paper deals with the Cauchy problem for the system of semilinear wave equations NEWLINE\[NEWLINE \frac{\partial ^2u_j}{\partial t^2}-\Delta u_j=F_j(u,\partial u),\; j=1,\dots ,N \; \text{ in } (0,+\infty )\times \mathbb{R}^3 NEWLINE\]NEWLINE with small initial data, where \(F_1,\dots ,F_N\) are homogeneous polynomials of the degree 2. \textit{S. Alinhac} [Indiana Univ. Math. J. 55, No. 3, 1209--1232 (2006; Zbl 1122.35068)] considered the case of \(F_j=F_j(\partial u)\) and introduced a sufficient condition for global existence of the solution and its asymptotic behavior. The author investigates the asymptotic pointwise behavior of global solutions to the above problem under a certain condition, which is slightly weaker than Alinhac's condition. Several examples are given.