We consider the Cauchy problem for the damped wave equation with absorptionwith . The behavior of as is expected to be the Gauss kernel in the supercritical case . In fact, this has been shown by Karch  (Studia Math., 143 (2000), 175--197) for , Hayashi, Kaikina and Naumkin  (preprint (2004)) for and by Ikehata, Nishihara and Zhao  (J. Math. Anal. Appl., 313 (2006), 598--610) for and . Developing their result, we will show the behavior of solutions for , . For the proof, both the weighted -energy method with an improved weight developed in Todorova and Yordanov  (J. Differential Equations, 174 (2001), 464--489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for , because the semilinear term is not in and the second derivatives are necessary when the explicit formula of solutions is estimated.
"Global asymptotics for the damped wave equation with absorption in higher dimensional space." J. Math. Soc. Japan 58 (3) 805 - 836, July, 2006. https://doi.org/10.2969/jmsj/1156342039