We consider blow-up solutions of a semilinear wave equation with a \(\log \log \) perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. As for the upper bound, the main difficulty comes from the fact that the PDE is not scaling invariant. Our argument is a delicate adaptation of the original approach introduced by Hamza and Zaag for the \(\log \) perturbation of the pure power nonlinearity (see [9]). It mainly relies on the design of a Lyapunov functional and the use of a blow-up criterion in the similarity variables’ setting, together with some energy, nonlinear, and interpolation estimates. As for the proof of the lower bound, we consider our argument as a major contribution, since the argument for the pure power case cannot be adapted when the scale invariance breaks down.
