First order PDE with two conditions?

[stanley.st]

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?

 

[HallsofIvy]

You have only one condition- in each variable. Since this equation have the first derivative with respect to both x and t, you need one condition for each variable. If this were a "diffusion" (heat) equation, which involves the second derivative with respect to x and first derivative with respect to t, you need two conditions on x and on on t. If it were a "wave" equation, which involves the second order in both x and t, you would need two conditions on each variable.