Cauchy problem for a damped generalized IMBq equation

In this paper, we prove that the Cauchy problem for the following damped generalized IMBq equation, \documentclass[12pt]{minimal}\begin{document}$u_{tt}-u_\textit{\scriptsize xx}-u_\textit{\scriptsize xxtt}+\nu _2 u_\textit{\scriptsize xxt}=f(u)_\textit{\scriptsize xx}, x\in \mathbb {R}, t>0,$\end{document}utt−uxx−uxxtt+ν2uxxt=f(u)xx,x∈R,t>0, admits a unique global generalized solution in \documentclass[12pt]{minimal}\begin{document}$C^3([0,\infty );W^{m,p}(\mathbb {R})\cap L^\infty (\mathbb {R})\cap L^2 (\mathbb {R})) (1\break \leq p \leq \infty,m\ge 0)$\end{document}C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(1≤p≤∞,m≥0) and a unique global classical solution in \documentclass[12pt]{minimal}\begin{document}$C^3([0,\infty );W^{m,p}(\mathbb {R})\cap L^\infty (\mathbb {R}) \cap L^2(\mathbb {R}) ) (m>2+\frac{1}{p})$\end{document}C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(m>2+1p). Moreover, the blow up of the solution for the Cauchy problem of damped generalized IMBq equation is studied. We also prove that the Cauchy problem of the above-mentioned equation has a unique global generalized solution in \documentclass[12pt]{minimal}\begin{document}$C^2([0,\infty );H^s(\mathbb {R}) )(s > \frac12)$\end{document}C2([0,∞);Hs(R))(s>12) and a unique global classical solution in \documentclass[12pt]{minimal}\begin{document}$C^2([0,\infty );H^s(\mathbb {R}))(s>\frac{5}{2})$\end{document}C2([0,∞);Hs(R))(s>52), and discuss the blow up of the solution.