Let $u(t, x)$ be the unique solution of the heat equation on the unit interval with Dirichlet boundary conditions and initial data $u_0$:$$\left\{ \begin{array}{l}\partial_t u(t, x) = \partial_x^2 u(t, x), \quad t > 0,\ x \in [0, 1] \\u(t, 0) = u(t, 1) = 0, \quad t > 0 \\u(0, x) = u_0(x), \quad x \in [0, 1]\end{array}\right. $$The steady state solution of this problem is $u_s(x) = 0$.
Question:Assume that $|u_0^\prime(x)| \leq \delta$ on $[0, 1]$. Does it remain true for all later times, that is $|\partial_x u(t, x)| \leq \epsilon$ on $[0, 1]$?
I would guess yes. Intuitively, this means that the solution to the heat equation, started close to the steady state solution, remains close. The maximum principle seems to be useful, but I can not control the derivative on the boundary. An other relevant result is that both $\| u(t, \cdot) \|_{L^2}$ and $\| \partial_x u(t, \cdot) \|_{L^2}$ vanish as $t \to \infty$.
I would also like to know if this is true for
- the heat equation on the $d$-dimensional unit box $[0, 1]^d$
- the quasilinear heat equation on the unit interval:$$\left\{ \begin{array}{l}\partial_t u(t, x) = \frac{d}{dx} g(\partial_x u(t, x)), \quad t > 0,\ x \in [0, 1] \\u(t, 0) = u(t, 1) = 0, \quad t > 0 \\u(0, x) = u_0(x), \quad x \in [0, 1]\end{array}\right. $$
I asked this question also here: https://math.stackexchange.com/q/2164593/420667