Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions*

Abstract

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t → −∞ in the energy norm, and to show it has a free profile as t → +∞. Our approach is based on the work of [11]. Namely we use a weighted L ∞ norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.

Large time behavior of solutions to semilinear systems of wave equations

“…We also remark that the estimates (4.15) and (4.18) are extensions of the part B) of Theorem 2.3 in [8] to the three space dimensional case.…”

“…We remark that the part (i) of Theorem 7 is an extension of the part A) of Theorem 2.3 in [8] to the three space dimensional case. On the other hand, the part (ii) and (iii) of Theorem 7 contain essentially new conclusions, because they concern with the case γ < 0.…”

“…Let c 1 = c 2 , 1 < p ≤ 2 and (1. 8) hold. Suppose that (f j , g j ) are as in the above and that there exists a positive number r 0 such that…”

“…However, such an improvement is not useful in the application to the system (1.1), hence we omit the detail and only refer to Theorem 2.3 and its proof in [8]. We also remark that the estimates (4.15) and (4.18) are extensions of the part B) of Theorem 2.3 in [8] to the three space dimensional case.…”