# Strichartz estimates of damped wave equation

If $w(t,x)$ is a solution of wave equation $$w_{tt}-\triangle w = 0, w(0)=w_0, w_t(0)=w_1,$$ then $w$ satisfies the following Strichartz esitmates $$\|w\|_{L^q_tL^r_x} \lesssim \|w_0\|_{H^1} + \|w_1\|_{L^2}$$ where $L^q_tL^r_x = L^q_t(0,\infty;L^r_x)$, $(q,r)$ is a wave admissable pair, and satifies $$\frac{1}{q} + \frac{n}{r} = \frac{n}{2} - 1.$$

Now, let us consider the damped wave equation $$w_{tt} + w_t -\triangle w = 0, w(0)=w_0, w_t(0)=w_1.$$ It's easy to check that the above Strichartz estimates hold. My question is that, whether these estimates hold for some other pair $(q,r)$ not concluded above. In other words, whether the strichartz estimates of solution for damped wave equation hold for larger range of $(q,r)$?

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Yes, the range of $q$ will be enlargeed. –  Wang Ming May 7 '12 at 4:24
The Strichartz estimate you have written is wrong, both the admissibility condition on the indices and the loss of derivatives w.r.to the initial data. Please check the relevant literature. Moreover, I do not think you can get a scale invariant estimate for a non-scale-invariant equation –  Piero D'Ancona May 7 '12 at 15:01