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We consider wave equations with a special type of log-fractional damping. Westudy the Cauchy problem for this model in the whole space, and we obtain anasymptotic profile and optimal estimates of solutions as time goes to infinityin L^2-sense. A maximal discovery of this note is that under the effectivedamping, in case of n = 1 L^2-norm of the solution blows up in infinite time,and in case of n = 2 L^2-norm of the solution never decays and never blows upin infinite time. The latter phenomenon seems to be a rare case.

Asymptotic profile of L^2-norm of solutions for wave equations with critical log-damping