Upper and Lower Bounds of Blowup Time to a Strongly Damped Wave Equation of Kirchhoff Type with Memory Term and Nonlinear Dissipations
Upper and Lower Bounds of Blowup Time to a Strongly Damped Wave Equation of Kirchhoff Type with Memory Term and Nonlinear Dissipations
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DOI：
10.1007/s00009-018-1161-5
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发表时间：
2018-05
影响因子：
1.1
通讯作者：
A. Peyravi;F. Tahamtani
中科院分区：
文献类型：
--
作者：
A. Peyravi;F. Tahamtani
In this paper, a wave equation of Kirchhoff type of the form $$\begin{aligned}&u_{tt}-M(\Vert \nabla u\Vert _{2}^{2})\Delta u + \int _{0}^{t}g(t-s)\Delta u(s)\mathrm{d}s+h(u_{t})-\mu \Delta u_{t}\\&\quad =|u|^{\alpha }u,\quad \mathrm {in}\quad \Omega \times (0,+\infty ) \end{aligned}$$is considered. Under suitable assumptions on the viscoelastic term and initial data, we prove that the solutions blow up at a finite time. Our result improves the previous work by Chen and Liu (IMA J Appl Math 1–29, 2015) in which the authors obtained an exponential growth of solutions where the memory kernel depends on initial energy and we show that it is not necessary to have such a restriction. This also extends the work by Hu et al. (Bound Value Probl 2017:112, 2017) where the relaxation function has not been considered. Some estimates for lower bounds of blowup time are also given.
In this paper, a wave equation of Kirchhoff type of the form $$\begin{aligned}&u_{tt}-M(\Vert \nabla u\Vert _{2}^{2})\Delta u + \int _{0}^{t}g(t-s)\Delta u(s)\mathrm{d}s+h(u_{t})-\mu \Delta u_{t}\\&\quad =|u|^{\alpha }u,\quad \mathrm {in}\quad \Omega \times (0,+\infty ) \end{aligned}$$is considered. Under suitable assumptions on the viscoelastic term and initial data, we prove that the solutions blow up at a finite time. Our result improves the previous work by Chen and Liu (IMA J Appl Math 1–29, 2015) in which the authors obtained an exponential growth of solutions where the memory kernel depends on initial energy and we show that it is not necessary to have such a restriction. This also extends the work by Hu et al. (Bound Value Probl 2017:112, 2017) where the relaxation function has not been considered. Some estimates for lower bounds of blowup time are also given.