Working in optimal control of PDEs I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state Here is a simplified version of the problem:

$$ \begin{cases}\dfrac{\partial y}{\partial t}(t,x)=d\Delta y(t,x)+ f(y(t,x)),\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial\nu}=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y(T,x)+g(x),\ x\in\Omega \end{cases} $$

We have that $\Omega\subset\mathbb{R}^N$ is a bounded and smooth domain.

Is there some theory regarding the existence, uniqueness and regularity of the solution? I will be thankful if you can indicate me some references (article or books) from where I can learn something. It's the first time I see this type of problem and I did not succeed in finding something about it.

At a first glance there are many cases in which there is no solution.

Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state. Here is a simplified version of the problem:

$$ \begin{cases}\dfrac{\partial y}{\partial t}(t,x)=d\Delta y(t,x)+ f(y(t,x)),\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial\nu}=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y(T,x)+g(x),\ x\in\Omega \end{cases} $$

Hereabove $\Omega\subset\mathbb{R}^N$ is a bounded and smooth domain.

Is there some theory regarding the existence, uniqueness and regularity of the solution? I will be thankful if you can indicate me some references (article or books) from where I can learn something. It's the first time I see this type of problem and I did not succeed in finding something about it.

At a first glance there are many cases in which there is no solution.

Working in optimal control of PDE's I came across a type of evolution problem that has instead of a initial condition a link between the initial state and the final state Here is a simplified version of the problem:

$$ \begin{cases}\dfrac{\partial y}{\partial t}(t,x)=d\Delta y(t,x)+ f(y(t,x)),\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial\nu}=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y(T,x)+g(x),\ x\in\Omega \end{cases} $$

We have that $\Omega\subset\mathbb{R}^N$ is a bounded and smooth domain.

Is there some theory regarding the existence, uniqueness and regularity of the solution? I will be thankful if you can indicate me some references (article or books) from where I can learn something. It's the first time I see this type of problem and I did not succed of finding something about it.

At a first glance there are many cases in which there is no solution.

Working in optimal control of PDEs I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state Here is a simplified version of the problem:

$$ \begin{cases}\dfrac{\partial y}{\partial t}(t,x)=d\Delta y(t,x)+ f(y(t,x)),\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial\nu}=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y(T,x)+g(x),\ x\in\Omega \end{cases} $$

We have that $\Omega\subset\mathbb{R}^N$ is a bounded and smooth domain.

Is there some theory regarding the existence, uniqueness and regularity of the solution? I will be thankful if you can indicate me some references (article or books) from where I can learn something. It's the first time I see this type of problem and I did not succeed in finding something about it.

At a first glance there are many cases in which there is no solution.