  We consider a singular limit problem from the damped wave equation with a
power type nonlinearity to the corresponding heat equation. We call our
singular limit problem non-delay limit. Our proofs are based on the argument
for non-relativistic limit from the nonlinear Klein-Gordon equation to the
nonlinear Schr\"{o}dinger equation by the second author, Nakanishi, and Ozawa
(2002), Nakanishi (2002), and Masmoudi and Nakanishi (2002). We can obtain
better results for the non-delay limit problem than that for the
non-relativistic limit problem due to the dissipation property. More precisely,
we get the better convergence rate of the $L^2$-norm and we also obtain the
global-in-time uniform convergence of the non-delay limit in the
$L^2$-supercritical case.
