Does $\partial_t u(x,t) - \partial_x u(x,t) = 0$ have a non-trivial solution? Consider the linear PDE $$\partial_t u(x,t) - \partial_x u(x,t) = 0$$ on the domain $$\Omega=\{(x,t)\;\;|\;\;0\leq x \leq 1,\;0\leq t \leq T\}$$
with boundary conditions $$u(0,t)=u(1,t)=c$$ and the initial condition $$u(x,0)=u_0(x).$$
This is a special case of the advection-diffusion equation or the heat equation with additional terms. In trying to solve this PDE using separation of variables and specifying $u(0,t)=u(1,t)=0$, I find only the trivial solution. In trying to use Fourier methods to solve the more general problem \begin{align}\partial_{t} u(x,t)&=k\,\partial_{xx}u(x,t)+A\,\partial_x u(x,t)+B\,u(x,t)\\u(0,t)&=u(L,t)=0\\u(x,0)&=u_0(x)\end{align}
on the domain $$\Omega=\{(x,t)\;\;|\;\;0\leq x \leq L,\;0\leq t \leq T\},$$ I find the solution to be $$u(x,t)=e^{-\frac{A}{2k}x+(-\frac{A^{2}}{4k}+B)t}\sum_{n=1}^{\infty}b_{n}sin\left(\frac{n\pi x}{L}\right)e^{-n^{2}\pi^{2}kt/L^{2}}$$where$$
b_{n}=\int_{0}^{L}e^{\frac{A}{2k}\xi}f(\xi)\,sin\left(\frac{n\pi\xi}{L}\right)\,d\xi$$
and $f(x)=u(x,0)$ (see P. 52 of O'Neil, Beginning Partial Differential Equations, 3rd Edition, Wiley (2014)). However, for the case of $k=B=0$, the Fourier coefficients blow up.
Does this equation have a non-trivial solution?

For your first equation, the characteristics are given by $dx/dt = -1$. So on $t$-vs-$x$ plane, they "travel" backwards with slope 1.
Thus, if there exists $x \in (0,1)$ such that $u_0(x) \neq c$, then there will be a discontinuity at some $t$ and at $x = 0$. As I understand it, these are sometimes called weak solutions (my terminology might be wrong here).
If you strictly want continuous solutions, then there is only the trivial solution.
