WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with Variable- Exponent Nonlinearities
Authors: ,
Abstract: This paper aims to study the model of a nonlinear viscoelastic wave equation with damping and
source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially,
and thus in $$L^{p_{2}} $$ and $$L^{p_{1}} $$ norms. For the case 2 ≤ k(. ) < p(. ), we reach the exponential growth result of a blowup
in finite time with positive initial energy and get the upper bound for the blow-up time. For the case k(. ) =
2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up
time. Furthermore, for the case k(. ) ≥ 2, under some conditions on the data, we give a lower bound for the
blow-up time when the blow-up occurs.
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Keywords: Viscoelastic wave equations, Exponential Growth, Blow-up, Lower and upper bound, Sobolev
spaces with variable exponents, variable nonlinearity
Pages: 451-465
DOI: 10.37394/23206.2023.22.51