Abstract
In this work, we investigate blowup phenomena for nonlinearly damped viscoelastic equations with logarithmic source effect and time delay in the velocity. Owing to the nonlinear damping term instead of strong or linear dissipation, we cannot apply the concavity method introduced by Levine. Thus, utilizing the energy method, we show that the solutions with not only non-positive initial energy but also some positive initial energy blow up at a finite point in time.
Highlights
1 Introduction We discuss the viscoelastic wave equation with nonlinear damping, logarithmic source, and delay terms utt – u + k ∗ u + c1 ut(t) q–2ut(t) + c2 ut(t – τ ) q–2ut(t – τ ) = |u|p–2u ln |u| in × (0, T), (1.1)
We discuss the viscoelastic wave equation with nonlinear damping, logarithmic source, and delay terms utt – u + k ∗ u + c1 ut(t) q–2ut(t) + c2 ut(t – τ ) q–2ut(t – τ ) = |u|p–2u ln |u| in × (0, T), (1.1)u = 0 on ∂ × (0, T), (1.2)u(0) = u0, ut(0) = u1 in, (1.3)ut(t – τ ) = j0(t – τ ) for t ∈ (0, τ ), (1.4) here⊂ Rn is a bounded domain with smooth boundary ∂,k∗
Most work dealing with wave equations with logarithmic nonlinearity is associated with a strongly or linearly damped mechanism, and blowup results are investigated by virtue of the potential well method and Levine’s concavity technique [12]
Summary
1 Introduction We discuss the viscoelastic wave equation with nonlinear damping, logarithmic source, and delay terms utt – u + k ∗ u + c1 ut(t) q–2ut(t) + c2 ut(t – τ ) q–2ut(t – τ ) = |u|p–2u ln |u| in × (0, T), (1.1) Most work dealing with wave equations with logarithmic nonlinearity is associated with a strongly or linearly damped mechanism, and blowup results are investigated by virtue of the potential well method and Levine’s concavity technique [12]. Kafini and Messaoudi [8] considered the wave equation with linear damping and delay terms utt – u + c1ut(t) + c2ut(t – τ ) = k|u|p–2u ln |u|
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