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Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

Authors:

Institution:

(1) Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China;(2) Department of Mathematics, Pohang University of Science and Technology, Pohang, 790–784, Republic of Korea

Abstract:

This paper is concerned with a nonlocal hyperbolic system as follows:

$\begin{array}{*{20}l} {{\left| {u_{{tt}} = \Delta u + {\left( {{\int_\Omega {vdx} }} \right)}^{p} {\text{for}}x \in \mathbb{R}^{N} ,t > 0,} \right.} \hfill} \\ {{v_{{tt}} = \Delta v + {\left( {{\int_\Omega {udx} }} \right)}^{q} {\text{for}}x \in \mathbb{R}^{N} ,t > 0,} \hfill} \\ {{u{\left( {x,0} \right)} = u_{0} {\left( x \right)},u_{t} {\left( {x,0} \right)} = u_{{01}} {\left( x \right)}{\text{for}}x \in \mathbb{R}^{N} ,} \hfill} \\ {{v{\left( {x,0} \right)} = v_{0} {\left( x \right)},v_{t} {\left( {x,0} \right)} = v_{{01}} {\left( x \right)}{\text{for}}x \in \mathbb{R}^{N} ,} \hfill} \\ \end{array}$

where 1 ≤ N ≤ 3, p ≥ 1, q ≥ 1 and pq > 1. Here the initial values are compactly supported and Ω ⊂ ℝ^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed. This work is supportd by the Com² MaC-SRC/ERC program of MOST/KOSEF (grant R11-1999-054)

Keywords:

hyperbolic system nonlocal term blow-up profile

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