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Existence and asymptotic behavior of radially symmetric solutions to a semilinear hyperbolic system in odd space dimensions. (English) Zbl 1129.35045
The authors study the following semilinear hyperbolic systems in odd space dimensions: \[ \partial_t^2 u_1-c^2_1\Delta u_1=F(u_2),\qquad \partial_t^2 u_2-c^2_2\Delta u_2=G(u_1) \] for \((x,t)\in {\mathbb R}^n\times {\mathbb R}\) and for nonlinearities \(F\) and \(G\) of a polynomial growth. The main goal of the paper is to prove the existence of a small amplitude solution which is asymptotic to the free solution as \(t\to-\infty\) in the energy norm, and to show it has a free profile as \(t\to +\infty\). Here, the approach is based on the previous work of the authors [Hokkaido Math. J. 24, No. 2, 287–336 (1995; Zbl 0840.35063)], which consists in using a weighted \(L^2\)-norm to get suitable a priori estimates as well as in the restriction of the attention to radially symmetric solutions. The obtained results extend the previous three-dimensional theory published in [H. Kubo, K. Kubota, Adv. Differ. Equ. 7, No. 4, 441–468 (2002; Zbl 1223.35232)].

MSC:
35L55 Higher-order hyperbolic systems
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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