The solution of many initial value problems for linear partial differential equations is of the form $K*u_0$, where $u_0$ is the initial value and $K$ is a kernel associated with the equation. Young's inequality gives estimates of the of the solution in $L^p$ spaces in terms of the size of the initial value.
Consider for instance the heat equation in $\mathbb{R}^n\times[0,\infty)$:
$$
u_t-\Delta u=0,\quad u(x,0)=u_0(x).
$$
The solution is $u(x,t)=K_t*u_0(x)$, where
$$
K_t(x)=(4\,\pi\,t)^{-\tfrac{n}{2}}\,e^{-\tfrac{|x|^2}{4t}}
$$
is the heat or Gauss kernel. It is in $L^P(\mathbb{R}^n)$ for all $p\ge1$, and
$$
\|K_t\|_p\le C_p\,t^{-\tfrac{n}{2}\bigl(1-\tfrac{1}{p}\bigr)}.
$$
Then, if $u_0\in L^q(\mathbb{R}^n)$ and $p^{-1}+q^{-1}=1+r^{-1}$, the decay in time of the solution in the $L^r$ norm is
$$
\|u(\,\cdot\,,t)\|_r\le C_pt^{-\tfrac{n}{2}\bigl(1-\tfrac{1}{p}\bigr)}\|u_0\|_q.
$$