Partial differential equations — a transport problem on a half-line

Consider the initial-boundary value problem (IBVP)

$$u_t+u_x=0, 0<x<\infty, t>0,\qquad u(x,0)=e^{x}, 0<x<\infty,\qquad u(0,t)=1-\sin t, t>0 .$$

Then which of the following statements is true?