First order PDE with two conditions?

Hello,

I have a problem in the form

$$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+e^{x}u=0$$

with conditions

$$u(x,0)=u_0(x)$$
$$u(0,t)=\int_{0}^{\infty}f(x)u(x,t)dx$$

Im confused, because in first order PDE i require only 1 condition. How to solve this for two conditions?