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.