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Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

Here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-F(x))\partial_{x}-x \partial_{y}$$ where $F$ is an even polynomial. One can replace this initial system with any other polynomial vector field with a band of closed orbits. For example a polynomial hamiltonian vector field $$H_{y}\partial_{x}- H_{x}\partial_{y}$$

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? What periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number of limit cycles of $U(x,y,t)$, for sufficiently small $t$?

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

Here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-F(x))\partial_{x}-x \partial_{y}$$ where $F$ is an even polynomial. One can replace this initial system with any other polynomial vector field with a band of closed orbits. For example a polynomial hamiltonian vector field $$H_{y}\partial_{x}- H_{x}\partial_{y}$$

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? What periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number of limit cycles of $U(x,y,t)$, for sufficiently small $t$?

Edit: According to the comment of Willie Wong I realize that the previos version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-F(x))\partial_{x}-x \partial_{y}$$. Where $F$ is an even polynomial. One can replace this initial system with any other polynomial vector field with a band of closed orbits. For example a polynomial hamiltonian vector field $$H_{y}\partial_{x}- H_{x}\partial_{y}$$

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? what periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number f limit cycles of $U(x,y,t)$, for sufficiently small $t$?

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

Here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-F(x))\partial_{x}-x \partial_{y}$$ where $F$ is an even polynomial. One can replace this initial system with any other polynomial vector field with a band of closed orbits. For example a polynomial hamiltonian vector field $$H_{y}\partial_{x}- H_{x}\partial_{y}$$

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? What periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number of limit cycles of $U(x,y,t)$, for sufficiently small $t$?

Edit: According to the comment of Willie Wong I realize that the previos version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-x^{2})\partial_{x}-x \partial_{y}$$. One can replace this initial system with any other polynomial vector field with a band of closed orbits.

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? what periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number f limit cycles of $U(x,y,t)$, for sufficiently small $t$?

Edit: According to the comment of Willie Wong I realize that the previos version was trivial. I thank him for his comment. Now I revise it.

We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=F(x,y)$$

here $U$ is not a real valued function in $x,y$ but for every $t$, $U$ is a vector field on the $x-y$ plane. Actually the unknown $U$ is a map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$.(Not to $\mathbb{R}$) In this question we are interested in the following particular initial condition $$F(x,y)=(y-F(x))\partial_{x}-x \partial_{y}$$. Where $F$ is an even polynomial. One can replace this initial system with any other polynomial vector field with a band of closed orbits. For example a polynomial hamiltonian vector field $$H_{y}\partial_{x}- H_{x}\partial_{y}$$

What can be said about this PDE? Is it true to say that, for sufficiently small $t$, there is a solution $U$ which is defined for all $(x,y)$? if the answer is yes, how is the dynamic of the vector field $U(x,y,t)$, for each fixed and sufficiently small $t$? Are there some limit cycles? How many limit cycles does $U$ have? what periodic orbits of the initial vector fields can generate "limit cycles' for sufficiently small $t$? The later question can be considered as a heat analogy of abelian integrals described here:

The perturbation of non-Hamiltonian algebraic vector fields

What type of criterion can be introduced for counting the number f limit cycles of $U(x,y,t)$, for sufficiently small $t$?