Consider $p(u,x)=(4\pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x\in \mathbb{R}.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[u,v],\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|dx \leq C\frac{v-u}{u-r}$
Any ideas how to prove this inequality?
I tried computing the integral without sucess.