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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.

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.

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 follow-up work of Coron and later Bertsch, Dal Passo, and van der Hout.

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.

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 must be due to Bethuel and coll.

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 follow-up work of Coron and later Bertsch, Dal Passo, and van der Hout.