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Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy

  • * Corresponding author: Runzhang Xu

    * Corresponding author: Runzhang Xu 

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday

R. Xu is partially supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation(2013M540270) and the Fundamental Research Funds for the Central Universities. M. Zhang is partially supported by the National Natural Science Foundation of China (11871199)

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  • In this paper, we study blow up and blow up time of solutions for initial boundary value problem of Kirchhoff-type wave equations involving the fractional Laplacian

    $\left\{ \begin{align} & {{u}_{tt}}+[u]_{s}^{2(\theta -1)}{{(-\Delta )}^{s}}u=f(u),\ \ \ \ \text{in}\ \Omega \times {{\mathbb{R}}^{+}}, \\ & u(x,0)={{u}_{0}},\ \ {{u}_{t}}(x,0)={{u}_{1}},\ \ \ \ \ \ \text{in}\ \Omega , \\ & u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ ({{\mathbb{R}}^{N}}\backslash \Omega )\times \mathbb{R}_{0}^{+}, \\ \end{align} \right.$

    where $ [u]_s $ is the Gagliardo seminorm of $ u $, $ s\in(0, 1) $, $ \theta\in[1, 2_s^*/2) $ with $ 2_s^* = \frac{2N}{N-2s} $, $ (-\Delta)^s $ is the fractional Laplacian operator, $ f(u) $ is a differential function satisfying certain assumptions, $ \Omega\subset\mathbb{R}^N $ is a bounded domain with Lipschitz boundary $ \partial \Omega $. By introducing a new auxiliary function and an adapted concavity method, we establish some sufficient conditions on initial data such that the solutions blow up in finite time for the arbitrary positive initial energy. Moreover, as $ f(u) = |u|^{p-1}u $, we estimate the upper and lower bounds for blow up time with arbitrary positive energy.

    Mathematics Subject Classification: Primary: 35R11, 35L05; Secondary: 47G20.


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