Well-posedness, asymptotic stability and blow-up results for a nonlocal singular viscoelastic problem with logarithmic nonlinearity

Considered herein is the well-posedness, asymptotic stability and blow-up of the initial-boundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity u tt - 1 x ( x u x ) x - 1 x ( x u xt ) x + ∫ 0 t m ( t - λ ) 1 x ( x u x ( x , λ ) ) x d λ = | u | r - 2 u l...

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Veröffentlicht in:Zeitschrift für angewandte Mathematik und Physik 2024-04, Vol.75 (2), Article 34
Hauptverfasser: Di, Huafei, Qiu, Yi
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Sprache:eng
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Zusammenfassung:Considered herein is the well-posedness, asymptotic stability and blow-up of the initial-boundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity u tt - 1 x ( x u x ) x - 1 x ( x u xt ) x + ∫ 0 t m ( t - λ ) 1 x ( x u x ( x , λ ) ) x d λ = | u | r - 2 u ln | u | subject to a nonlocal boundary condition. Through the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differential-integral inequality techniques, we firstly demonstrate the global existence and uniqueness of weak solutions in certain weighted Sobolev spaces; Secondly, we establish the explicit polynomial and exponential energy decay estimates under some suitable conditions; Finally, we investigate the finite time blow-up criterion and then derive its upper and lower bounds of blow-up time. The above conclusions extend and improve some results in the literatures.
ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-023-02177-5