It seems that non-uniqueness is the main source of counter-example, at least in the above answers. So, one more:

Consider the heat equation for harmonic *maps*:
$$
u_t-\Delta u+|\nabla u|^2u=0,\qquad |u(x,t)|\equiv1,\qquad(1)
$$
with prescribed boundary data $u=g$. A steady solution is a $\phi$ (a *harmonic map*) such that
$$-\Delta \phi+|\nabla u|^2\phi=0,\qquad |\phi(x,t)|\equiv1$$
and $\phi=g$ on the boundary. It is a critical point of the functional
$$I[z]:=\int_\Omega|\nabla z|^2dx$$
under the constraints that $|z|\equiv1$ in $\Omega$ and $z=g$ on the boundary. 

One may choose $g$ such that there exists a harmonic map $\phi$ that does not minimize locally $I[z]$. In this case, the Cauchy problem for (1), with initial data $\phi$, has **two solutions**. One is $\phi$, and the other one is time-dependent, with $I[u(t)]$ non-constant (it decays).

This result was due to [Bethuel, Coron, Ghidaglia, and Soyeur][1]. See also the work of [Coron][2] and later [Bertsch, Dal Passo, and van der Hout][3].

 [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1167832
 [2]: http://www.numdam.org/item/?id=AIHPC_1990__7_4_335_0
 [3]: https://doi.org/10.1007/s002050100171