Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Abstract

<p style='text-indent:20px;'>In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document}</tex-math></inline-formula> is a nondecreasing and continuous function, <inline-formula><tex-math id="M2">\begin{document}$ [u]_{\alpha, 2} $\end{document}</tex-math></inline-formula> is the Gagliardo-seminorm of <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> are the fractional Laplace operators, <inline-formula><tex-math id="M6">\begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document}</tex-math></inline-formula> is a positive nonincreasing function and <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.</p>