I am wondering whether it is possible to convert the following integral equation to a partial differential equation.
where $J_0(t,x)$ is some given nonnegative function and $\nu>0$ is a constant. It is clear $t\ge 0$.
The aim is to solve this equation. To convert it to PDE is just one possible way to solve it, since latter we can use the hopefully the fundamental solutions.
My current solution is
But I am not sure whether it is right or not.
Thanks for any comments or hints!