in 1955 Ennio De Giorgi constructed an example of parabolic-type linear equation, whose Cauchy problem has non-unique solution. An English translation of this paper appears in De Giorgi's collected works.
To be more specific, he constructs 4 smooth functions $a(x,t)$, $b(x,t)$, $c(x,t)$, and $u(x,t)$ defined on the strip $\mathbb{R}\times[0,1]$, such that
$$ \frac{\partial^8u}{\partial t^8} = a\frac{\partial^4u}{\partial x^4} + b\frac{\partial^2u}{\partial x^2}+cu, $$
on the whole strip,
$$ \frac{\partial^nu}{\partial t^n} = 0, $$
identically on the line $t=0$ for $n=0,\ldots,7$, and $u$ not identically zero.
