Solving a PDE with the following boundary problem with arbitrary constant $b$:
$$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$
Hint given in the question is as follows:
Split this integral in two as it is a non-local boundary condition:
$$F(t)=b\int_0^tu(a,t)\mathrm{d}a+b\int_t^\infty u(a,t)\mathrm{d}a$$
As it turns out, the solution for $u(a,t)$ is as follows;
$$u(a,t)=u_0(a-t)e^{-\frac{1}{2}\mu t^2}$$
Where $\mu $ is an arbitrary constant.
So, the integral is now of the form:
$$u_0(-t)=b\int_0^t u_0(a-t)\mathrm{d}a+b\int_t^\infty u_0(a-t)\mathrm{d}a$$
Should I use Leibniz's integral rule, or is there another integral rule to use here? I'm not sure.
I need to show it is a well-posed problem, so I need to do something with this integral.