I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem: $$ \begin{cases} u_t = F(u_{xx}),\\ u(0,x) = u_0(x). \end{cases} $$ Let suppose that $F$ is uniformly elliptic and assume that $u_0$ is not linear on all non-empty intervals of $\mathbb{R}$ (i). Is it true that for all $t\in[0,T]$, the function $x\mapsto u(t,x)$ is not linear on all non-empty intervals of $\mathbb{R}$ ?
Any hint or references will be highly appreciated !
(i) : What I mean is that, you cannot find $a>b$ such that $u_0$ is linear on $[a,b]$.
Thank you very much.
