Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity

Abstract

<p style='text-indent:20px;'>In this paper, we study the fractional pseudo-parabolic equations <inline-formula><tex-math id="M1">\begin{document}$ u_{t} + \left(-\Delta\right)^{s} u + \left(-\Delta\right)^{s} u_{t} = u\log \left| u \right| $\end{document}</tex-math></inline-formula>. Firstly, we recall the relationship between the fractional Laplace operator <inline-formula><tex-math id="M2">\begin{document}$ \left(-\Delta\right)^{s} $\end{document}</tex-math></inline-formula> and the fractional Sobolev space <inline-formula><tex-math id="M3">\begin{document}$ H^{s} $\end{document}</tex-math></inline-formula> and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of weak solution: for the low initial energy case (i.e., <inline-formula><tex-math id="M4">\begin{document}$ J(u_{0}) < d $\end{document}</tex-math></inline-formula>), the solution is global in time with <inline-formula><tex-math id="M5">\begin{document}$ I(u_{0}) >0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M6">\begin{document}$ \Vert u_{0}\Vert_{{X_{0}(\Omega)}} = 0 $\end{document}</tex-math></inline-formula> and blows up at <inline-formula><tex-math id="M7">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ I(u_{0}) < 0 $\end{document}</tex-math></inline-formula>; for the critical initial energy case (i.e., <inline-formula><tex-math id="M9">\begin{document}$ J(u_{0}) = d $\end{document}</tex-math></inline-formula>), the solution is global in time with <inline-formula><tex-math id="M10">\begin{document}$ I(u_{0}) \geq0 $\end{document}</tex-math></inline-formula> and blows up at <inline-formula><tex-math id="M11">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ I(u_{0}) < 0 $\end{document}</tex-math></inline-formula>. The decay estimate of the energy functional for the global solution is also given.</p>