A better bound on blow-up rate for the superconformal semilinear wave equation

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $\mathrm{\mid }\log (T-t){\mathrm{\mid }}^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.