Let $u\_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u\_0(x)$. How one can apply the maximum principle to prove that the number of critical points of $u(t,x)$ at any given time is not bigger than the number of critical points of $u\_0$? 



Heat equation and evolution of number of critical points
Since $u$ is nothing but a $2\pi$-periodic solution of $u\_t=u\_{xx}$, looking at critical points amounts to looking at the zeroes of $v:=u\_x$, which is another solution of the same equation. Then your question is solved by H. Matano : Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. *J. Fac. Sci. Univ. Tokyo Sect. IA Math.* **29** (1982), no. 2, 401–441. 


