In this paper we study the Cauchy problem to the linear damped wave equation in . It has been asserted that the above equation has the diffusive structure as . We give the precise interpolation of the diffusive structure, which is shown by estimates. We apply the above estimates to the Cauchy problem for the semilinear damped wave equation in . If the power is larger than the critical exponent (Fujita critical exponent) and it satisfies when , then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.
" estimates for damped wave equations and their applications to semi-linear problem." J. Math. Soc. Japan 56 (2) 585 - 626, April, 2004. https://doi.org/10.2969/jmsj/1191418647