For the strongly damped wave equation, D(t) is bounded by C t^{1/4} ||g||_L1 in 1D (free wave is effective), but has a logarithmic lower bound in 2D when initial velocity mass is nonzero (wave approximation fails).
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On the $L^{2}$ estimates of the diffusion waves
For the strongly damped wave equation, D(t) is bounded by C t^{1/4} ||g||_L1 in 1D (free wave is effective), but has a logarithmic lower bound in 2D when initial velocity mass is nonzero (wave approximation fails).