Maximum principle for heat equation with smooth initial datum

Question

Let $u_0 \in C_c^\infty(B(0,2))$ with $supp \ u_0 \in B(0,2)\setminus B(0,1)$ such that $u_0 \ge 0$ and $u_0$ is not the identic $0$ function. Let $u$ be the solution of the heat equation $u_t -\Delta u=0$ in $(0,\infty)\times B(0,2)$ with $u=0$ on $[0,\infty)\times \partial B(0,2)$ and $u(0,\cdot)=u_0$. Show that there exists $M>0$ and $c>0$ so that $$\inf\limits_{B(0,1)} u(t,x) \le Me^{ -\frac{c}{t} } $$