A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

Abstract

<p style='text-indent:20px;'>In this paper, we study the blow – up of solutions to the semilinear Moore – Gibson – Thompson (MGT) equation with nonlinearity of derivative type <inline-formula><tex-math id="M1">\begin{document}\(|u_t|^p \)\end{document}</tex-math></inline-formula> in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow – up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent <inline-formula><tex-math id="M2">\begin{document}\(p \)\end{document}</tex-math></inline-formula> for the nonlinear term satisfies <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p\leqslant (n+1)/(n-1) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}\(n\geqslant2 \)\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}\(p&gt;1 \)\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}\(n = 1 \)\end{document}</tex-math></inline-formula>. In particular, we find the same blow – up range for <inline-formula><tex-math id="M7">\begin{document}$ p $\end{document}</tex-math></inline-formula> as in the corresponding semilinear wave equation with nonlinearity of derivative type.</p>