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Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms

  • Ning Pan
  • Patrizia Pucci
  • Runzhang Xu
  • Binlin ZhangEmail author
Article

Abstract

In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian,
$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{tt} +[u]^{2\gamma -2}_{s}(-\Delta )^su+|u_t|^{a-2}u_t+u=|u|^{b-2}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\ \ \ \ u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N{\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array} \right. \end{aligned}$$
where \((-\Delta )^s\) is the fractional Laplacian, \([u]_{s}\) is the Gagliardo semi-norm of u, \(s\in (0,1)\), \(2<a<2\gamma<b<2_s^*=2N/(N-2s)\), \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \Omega \). Under some natural assumptions, we obtain the global existence, vacuum isolating, asymptotic behavior and blowup of solutions for the problem above by combining the Galerkin method with potential wells theory. The significant feature and difficulty of the problem are that the coefficient of \((-\Delta )^s\) can vanish at zero.

Keywords

Degenerate Kirchhoff type Wave equations Fractional Laplacian Potential wells 

Mathematics Subject Classification

35R11 35L20 35L70 47G20 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNortheast Forestry UniversityHarbinChina
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  3. 3.College of Mathematical SciencesHarbin Engineering UniversityHarbinChina
  4. 4.School of Mathematics and StatisticsNortheast Petroleum UniversityDaqingChina

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