Need help solving a convolution problem involving Bessel function

Question

so I've been asked to solve $$\int_{0}^\infty J_0(u)J_0(t-u)du$$ I know that $$L[J_0(u)] = \frac{1}{\sqrt{s^2+a^2}}$$ and from that I can guess it has to do something with substituting $u=at$ and seems to me that the answer would be along the lines of $$f * g = L^{-1}[F(s)G(s)]$$but I'm just not quite sure how to land the idea, would really appreciate the help.