How to solve the following partial integro-differential equation
\begin{eqnarray}u_t(x,t)=\int\limits_{\mathbb{R}}u(y,t)dy\\ u(x,0)=u_0(x) \in L^1(\mathbb{R}) \end{eqnarray} Is there any well-posedness theory for such equations? What are the conditions on the initial data $u_0$ for the existence of such a solution. Please suggest me the references.
It is clear that if $\int u_0=0$ then $u(x,t)=u_0(x)$ is the solution.