摘要：
The blow-up of solutions of a class of nonlinear viscoelastic Kirchhoff equation with suitable initial data and Dirichlet boundary conditions is discussed. By constructing a suitable auxiliary function to overcome the difficulty of gradient estimation and making use of differential inequality technique, we establish a finite time blow-up result when the initial data is at arbitrary energy level. Moreover, a lower bound of the lifespan is also derived by constructing a control function with both nonlocal term and memory kernel. Compared with the previous literature, our approach to estimate the lifespan does not require the initial energy to control some norms of the solution.
期刊：
ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS,2016年2016 ISSN：1072-6691
通讯作者：
Gong, Zhaogang
作者机构：
[Yang, Zhifeng; Gong, Zhaogang] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.
通讯机构：
[Gong, Zhaogang] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.
关键词：
Viscoelastic equation;blow-up;arbitrary positive initial energy
摘要：
We consider the viscoelastic equation u(tt)(x, t) - M(vertical bar vertical bar del u vertical bar vertical bar(2)(2))Delta u(x,t) +integral(t)(0)g(t-s)Delta u(x,s)ds + u(t) = vertical bar u vertical bar(p-1)u with suitable initial data and boundary conditions. Under certain assumptions on the kernel g and the initial data, we establish a new blow-up result for arbitrary positive initial energy, by using simple analysis techniques.
摘要：
In this article, we consider the Euler-Bernoulli viscoelastic equation u(tt)(x,t) + Delta(2)u(x,t) - integral(t)(0) g(t-s)Delta(2)u(x,s)ds = vertical bar u vertical bar(p-1)u together with some suitable initial data and boundary conditions in Omega x (0, +infinity). Some sufficient conditions on blow-up of solutions are obtained under different initial energy states. And from these results we can clearly understand the competitive relationship between the viscoelastic damping and source.
摘要：
In the article, the variational iteration algorithm LFVIA-II is implemented to solve the diffusion equation occurring in non-differentiable heat transfer. The operators take in sense of the local fractional operators. The obtained results show the fractal behaviors of heat transfer with non-differentiability.
摘要：
In this paper, we consider initial-boundary value problem of Euler–Bernoulli viscoelastic equation with a delay term in the internal feedbacks. Namely, we study the following equation
$$u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s){\rm d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0 $$
together with some suitable initial data and boundary conditions in
$${\Omega\times (0,+\infty)}$$
. For arbitrary real numbers μ
1 and μ
2, we prove that the above-mentioned model has a unique global solution under suitable assumptions on the relaxation function g. Moreover, under some restrictions on μ
1 and μ
2, exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.
摘要：
In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation
$$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$
together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers μ
1 and μ
2, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ
1 and μ
2. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ
1 = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).
