The following is my question.
Let $F$ be smooth, and $u$ a bounded smooth solution in $\mathbb R^+ \times \mathbb R^n$ to the parabolic equation $$u_t+\partial_xF(u)=\Delta u,\ \ \ \ \ u(0)=u_0$$ Show that if $a\le u_0\le b$, then $a\le u\le b$.
I wanted to derive the dependence pattern between $u$ and $u_0$, but I failed. Can someone help me find the relationship between $u$ and $u_0$ so that we can attain the result that $a\le u\le b$ ?
Thanks for help :)