Krdytkyu

I stumbled upon a transport equation of the form
$$u_t(x,t)=u_x(x,t) + u_x(1,t).$$
Since I can write it in the form $u_t(x,t) = Lu(x,t)$ where L is some linear operator I thought that there must be some theory behind this type of equations. Unfortunately, so far my research was unsuccessful.
Is there a way to solve the problem explicitly, e.g., by using method of characteristics in some way?

Thank you in advance!

$$u_t(x,t)-u_x(x,t) = u_x(1,t).$$
$u_x(1+t)$ is a function of $t$ only. Let $u_x(1+t)=f(t)$. Of course, this is an unknown function.
$$u_t(x,t)-u_x(x,t) = f(t).$$
Solving for the general solution. Charpit-Lagrange equations :
$$frac{dt}{1}=frac{dx}{-1}=frac{du}{f(t)}$$
A first family of characteristic curves comes from $frac{dt}{1}=frac{dx}{-1}$
$$x+t=c_1$$
A second family of characteristic curves comes from $frac{dt}{1}=frac{du}{f(t)}$
$$u-int f(t)dt=c_2$$
General solution of the PDE : $u-int f(t)dt=Phi(x+t)$

$Phi(X)$ is an arbitrary function of one variable only. In the above equation $X=(x+t)$.
$$u(x,t)= Phi(x+t)+int f(t)dt$$
$u_x(x,t)=left(frac{dPhi(X)}{dX}right)_{(X=x+t)}=Phi'(x+t)$

$u_x(1,t)=left(frac{dPhi(X)}{dX}right)_{(X=1+t)}=Phi'(1+t)$.
$$f(t)=u_x(1,t)=Phi'(1+t).$$
$int f(t)dt=int_{t_0}^t f(zeta)dzeta+C= int_{t_0}^t Phi'(1+zeta)dzeta +C =Phi(1+t)+C$.

Finally the solution is :
$$u(x,t)= Phi(x+t)+Phi(1+t)+C$$
$Phi$ is an arbitrary function. $C$ is an arbitrary constant.
They have to be determined to fit some boundary and initial conditions. One cannot go further since those conditions are not specified in the wording of the question.

CHECKING :
$$u_x(x,t)=Phi'(x+t)$$
$$u_t(x,t)=Phi'(x,t)+Phi'(1+t)$$
$$u_t(x,t)-u_x(x,t)=Phi'(1+t)=u_x(1,t)$$
The PDE is satisfied. The above result is correct.

IN ADDITION after the comments :

Case of condition $u(x,0)=g(x)=Phi(x)+Phi(1)+C=Phi(x)+C_2$

$Phi(x)=g(x)-C_2$

$u(x,t)= (g(x+t)-C_2)+(g(1+t)-C_2)+C$

Thus $quad -2C_2+C+g(1)=0quad;quad C=2C_2-g(1)$

$u(x,t)=(g(x+t)-C_2)+(g(1+t)-C_2)+2C_2-g(1)$

$$u(x,t)=g(x+t)+g(1+t)-g(1)$$

Example :

$$g(x)=x^mqquad:qquad u(x,t)=(x+t)^m+(1+t)^m-1$$