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Consider the following Cauchy problem: $$u_{tt} + u_t - \Delta u = 0,$$ $$u(0,x)=u_0(x) \in L^1 \cap L^2,$$ $$u_t(0,x) = u_1(x) \in L^1 \cap L^2.$$

Question: By means of Fourier transform and Plancherel theorem, what decay estimates can we obtain for $\Vert u(t,\cdot)\Vert_{L^2}$?

With some calculations I can obtain energy estimates (that is, estimates for $\Vert \nabla_x u\Vert_{L^2}$ and $\Vert \partial_t u \Vert_{L^2}$), but I can't really figure out the right approach to obtain the decay estimate for the solution itself. My guess is it should decay as $t^{-\frac{N}{4}}$, where $N$ is the space dimension.

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Since you tagged reference request:

An early paper is Matsumura, Akitaka, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, 169-189 (1976). ZBL0356.35008. The first part of the paper is not interesting to you, since it deals with wave equations with no damping, but the second part deals with the damped counterpart and proves linear decay among other things.

You can also look at

Ikehata, Ryo, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Methods Appl. Sci. 27, No. 8, 865-889 (2004). ZBL1049.35135. Ikehata has done a lot of work on the damped wave equation, going through his list of published papers would be a good start.

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  • $\begingroup$ Thank you. Lemma 1 of the first paper (page 10) is what I'm looking for. However, I don't understand the last few steps of the proof: specifically, the last three inequalities of the argument and the final sentence. Could you have a look at it and let me know? $\endgroup$
    – Jun
    Commented Jun 23, 2017 at 20:01

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