Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping

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Abstract

This article concerns the Cauchy problem for the damped nonlinear hyperbolic system Ïμutt+(-Δ)αu+ut=vp,t>0,x∈RN, u>0,v>0,Ïμvtt+(-Δ)αv+vt=uq,t>0,x∈RN,u>0, v>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈RN,v(x,0)=v0(x),vt(x,0)=v1(x), x∈RN, where Ïμ > 0 is a small parameter, 0 <α ≤ 1,0 <β ≤ 1,p,q ≥ 1 satisfying pq > 1, and N ≥ 1 is an integer.It is proved that if N/2α <max((p + 1)/(pq - 1),(q + 1)/(pq - 1)), then every weak solution does not exist globally whenever the initial data satisfy ∫RN{u0(x)+u1(x)}dx>0 or ∫RN(v0(x)+v1(x))dx>0.

Original languageEnglish
Pages (from-to)621-626
Number of pages6
JournalMathematical Methods in the Applied Sciences
Volume36
Issue number6
DOIs
Publication statusPublished - Apr 2013

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Nonlinear Hyperbolic Systems
Global Solution
Small Parameter
Damped
Nonexistence
Cauchy Problem
Damping
Integer

Keywords

  • hyperbolic systems
  • linear damping
  • nonexistence
  • nonlocal spatial operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Cite this

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title = "Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping",
abstract = "This article concerns the Cauchy problem for the damped nonlinear hyperbolic system {\"I}μutt+(-Δ)αu+ut=vp,t>0,x∈RN, u>0,v>0,{\"I}μvtt+(-Δ)αv+vt=uq,t>0,x∈RN,u>0, v>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈RN,v(x,0)=v0(x),vt(x,0)=v1(x), x∈RN, where {\"I}μ > 0 is a small parameter, 0 <α ≤ 1,0 <β ≤ 1,p,q ≥ 1 satisfying pq > 1, and N ≥ 1 is an integer.It is proved that if N/2α 0 or ∫RN(v0(x)+v1(x))dx>0.",
keywords = "hyperbolic systems, linear damping, nonexistence, nonlocal spatial operator",
author = "S. Kerbal",
year = "2013",
month = "4",
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volume = "36",
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journal = "Mathematical Methods in the Applied Sciences",
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TY - JOUR

T1 - Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping

AU - Kerbal, S.

PY - 2013/4

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N2 - This article concerns the Cauchy problem for the damped nonlinear hyperbolic system Ïμutt+(-Δ)αu+ut=vp,t>0,x∈RN, u>0,v>0,Ïμvtt+(-Δ)αv+vt=uq,t>0,x∈RN,u>0, v>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈RN,v(x,0)=v0(x),vt(x,0)=v1(x), x∈RN, where Ïμ > 0 is a small parameter, 0 <α ≤ 1,0 <β ≤ 1,p,q ≥ 1 satisfying pq > 1, and N ≥ 1 is an integer.It is proved that if N/2α 0 or ∫RN(v0(x)+v1(x))dx>0.

AB - This article concerns the Cauchy problem for the damped nonlinear hyperbolic system Ïμutt+(-Δ)αu+ut=vp,t>0,x∈RN, u>0,v>0,Ïμvtt+(-Δ)αv+vt=uq,t>0,x∈RN,u>0, v>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈RN,v(x,0)=v0(x),vt(x,0)=v1(x), x∈RN, where Ïμ > 0 is a small parameter, 0 <α ≤ 1,0 <β ≤ 1,p,q ≥ 1 satisfying pq > 1, and N ≥ 1 is an integer.It is proved that if N/2α 0 or ∫RN(v0(x)+v1(x))dx>0.

KW - hyperbolic systems

KW - linear damping

KW - nonexistence

KW - nonlocal spatial operator

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