If I have the following PDE system:
$\frac{\delta}{\delta t}x(t,r)=-\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)$
$\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)-y(t,r)$
$x(0,r)=a(r), y(0,r)=1-a(r)$
where $G:\mathbb{R}^+\to\mathbb{R}$ is $C^\infty$ with compact support.
There exist a teorem for the existence and the uniqueness of the solution $(x(t,r),y(t,r))$ with $(t,r)\in[0,T]\times[0,1]$?
Thank you