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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
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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

1 Answer 1

Not sure, but it seems to me that one hardly can expect new Strichartz exponents for the weakly damped wave equation ( the equation you wrote). The reason is that such damped wave equation can be transformed to pure wave equation just by change of variables $v(t)=e^\frac{t}{2}w(t)$. Thus, solutions of the weakly damped wave equation and pure wave equation differ just by factor $e^\frac{t}{2}$. Nevertheless this dissipation term ( $\partial_t w$) plays important role in the long time dynamics of semi-linear damped wave equations ( see http://arxiv.org/abs/1309.6272 )

Also new Strichartz exponents can be obtained by strengthening the dissipation term. For example, one can change term $\partial_t w$ to $(-\Delta)^\frac{1}{2}\partial_t w$ Then the corresponding solution possesses also $L^2([0,T];H^\frac{3}{2}(\Omega))$ regularity. Furthermore, this extra space-time norm can be controlled by energy norm of initial data even in semi-linear case ( see http://arxiv.org/abs/1306.2294 ).

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