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Strichartz estimates show that the solution $u$ of the Cauchy problem for several equations of physical interest do belong to an $L^p_t(L^q_x)$ if the initial data is appropriate. When $p=q$, this just means that $u\in L^p$.

For instance, consider the wave equation $$\partial_t^2u-\Delta_xu=0,\qquad t\in\mathbb R,x\in\mathbb R^d,$$ in which $n=d+1$. Say that $d\ge3$. Let the initial data be $$u(0,x)=a(x),\qquad \partial_tu(0,x)=b(0,x),$$ where $a\in H^1(\mathbb R^d)$ and $b\in L^2(\mathbb R^d)$. Then $u\in L^p(\mathbb R^{1+d})$ with $$p=2\frac{d+1}{d-2}.$$ There are variants of this result, but this is too a rich topic to be developped here.

Edit. This phenomenon is called a dispersion effect. It is related to the fact that the curvature of the characteristic cone of $L$ (here $\xi_0^2=\xi_1^2+\cdots+\xi_d^2$) is non-zero.

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Strichartz estimates show that the solution $u$ of the Cauchy problem for several equations of physical interest do belong to an $L^p_t(L^q_x)$ if the initial data is appropriate. When $p=q$, this just means that $u\in L^p$.

For instance, consider the wave equation $$\partial_t^2u-\Delta_xu=0,\qquad t\in\mathbb R,x\in\mathbb R^d,$$ in which $n=d+1$. Say that $d\ge3$. Let the initial data be $$u(0,x)=a(x),\qquad \partial_tu(0,x)=b(0,x),$$ where $a\in H^1(\mathbb R^d)$ and $b\in L^2(\mathbb R^d)$. Then $u\in L^p(\mathbb R^{1+d})$ with $$p=2\frac{d+1}{d-2}.$$ There are variants of this result, but this is too a rich topic to be developped here.