I see a Harnack inequation in a Chinese book. But there is not proof in it. I try to find the proof. But in other book, only proof of different form Harnack inequality can be found. So I ask here for the proof of following Harnack inequality.
Harnack inequality : For any $\lambda >1, R>0, x_0\in \mathbb R^n,t_0\ge 0$, $u$ is the non-negative weak solution of $$ u_t=\Delta u ~~~~~~(x,t)\in B_R(x_0)\times(t_0, t_0+\lambda R^2) $$ then, there is constant $C(n,\lambda )$ such that $$ \inf\limits_{B_{_{R/2}}~~(x_0)} u(x,t_0+\lambda R^2)\ge Cu(x_0, t_0 +R^2) $$ where $B_{_{R/2}}~~(x_0)=\{x\in \mathbb R^n : |x-x_0|<\frac{R}{2}\}$.
I know the proof of Harnack inequality is a little fussy. So just tell me which book has the proof of above Harnack inequality is enough. Thanks.