  We study the macroscopic fractal properties of the deep valleys of the
solution of the $(1+1)$-dimensional parabolic Anderson equation $${\partial
\over \partial t}u(t,x) =\frac{1}{2} {\partial^2 \over \partial x^2} u(t,x) +
u(t,x)\dot{W}(t,x),t>0, x\in {\bf R},\quad
  u(0,x) \equiv u_0(x),x\in {\bf R}, $$ where $\dot{W}$ is the time-space white
noise and $0<\inf_{x\in {\bf R}} u_0(x)\leq \sup_{x\in {\bf R}} u_0(x)<\infty.$
Unlike the macroscopic multifractality of the tall peaks, we show that valleys
of the parabolic Anderson equation are macroscopically monofractal. In fact,
the macroscopic Hausdorff dimension (introduced by Barlow and Taylor [J. Phys.
A 22 (1989) 2621--2628; Proc. Lond. Math. Soc. (3) 64 (1992) 125--152]) of the
valleys undergoes a phase transition at a point which does not depend on the
initial data. The key tool of our proof is a lower bound to the lower tail
probability of the parabolic Anderson equation. Such lower bound is obtained
for the first time in this paper and will be derived by utilizing the
connection between the parabolic Anderson equation and the Kardar-Parisi-Zhang
equation. Our techniques of proving this lower bound can be extended to other
models in the KPZ universality class including the KPZ fixed point.
