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Solution blow‐up for a fractional in time acoustic wave equation
Author(s) -
Jleli Mohamed,
Kirane Mokhtar,
Samet Bessem
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6400
Subject(s) - mathematics , bounded function , mathematical analysis , nonlinear system , domain (mathematical analysis) , initial value problem , wave equation , acoustic wave equation , cauchy problem , boundary value problem , fractional calculus , time derivative , acoustic wave , physics , acoustics , quantum mechanics
We consider the fractional in time acoustic wave equation1c 0 2∂ 0 | t α u − Δ u = εc 0 4ρ 0∂ 0 | t αu 2 , where 1< α <2,∂ 0 | t αis the Caputo fractional derivative of order α , u = u ( t , x ), t >0, x ∈ R 3 , is the pressure in the medium, ε is the nonlinear acoustic parameter, ρ 0 is the equilibrium density in the medium, and c 0 is the equilibrium sound velocity. We study a Cauchy problem for this equation and a mixed boundary value problem in a bounded domain. For each problem, sufficient conditions for the blow‐up of solutions are derived. Moreover, we provide a class of initial data for which there are no classical solutions even locally in time. Our approach is based on the nonlinear capacity method.