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. See also the work of Coron and later Bertsch, Dal Passo, and van der Hout.
