What is the definition of a global weak solution to a parabolic PDE?
Is it a solution $u \in L^2_{loc}(0,\infty;V)$ with $u' \in L^2_{loc}(0,T;V')$ or is it a solution $u \in L^2(0,\infty;V)$ with $u' \in L^2(0,T;V')$ ?
If the latter, how is the weak derivative defined on the interval $(0,\infty)?$