In Chi-wang Shu's article Discontinuous Galerkin Methods: General Approach and Stability, "compactly supported boundary conditions" is mentioned in Proposition 3.2 and Proposition 3.3. I don't know its exact definition and after searching for it through Bing, I fail to get any useful information from the Internet.
I guess that "compact supported boundary conditions" may mean that for $t\in [0,T]$, the solution of the PDE $$u_t+f(u)_x=0 \ (x,t)\in[0,1]\times [0,T]$$ equals to 0 on the boundary, namely $$u(0,t)=0$$and $$u(1,t)=0.$$ I am not sure of my guess, as I have not find any information from published books or papers supporting it.