# Don't understand proof of global solution to parabolic PDE

Let $u \in L^p(0,T;V)$ denote the solution to the parabolic PDE $$u_t + \Delta u = f\qquad\text{a.e t \in [0,T]}$$ where $u_t \in L^q(0,T;V^*).$ We have the usual assumptions on $V \subset H \subset V^*$ and $H$.

Now this PDE has a unique solution, i.e. $u$ is unique.

$T$ is arbitrary, so we have a set of solutions $u_n \in L^p(0,n;V)$ that solve the PDE a.e in $[0,n]$.

Define $$v(t) = u_n(t) \qquad \text{if t \leq n}$$ Then $v \in L^p(0,\infty;V)$ solves the PDE on $[0,\infty).$
Is this obvious that it solves the PDE on the real line? furthermore, to show that $v$ has finite norm in $L^p(0,\infty;V)$ am I right that we need a uniform bound independent of $n$ on $u_n$. These a priori estimates come from considering Galerkin method for example, but I thought that the constant depends on the end time, which in this cases happens to be $n$. So I am not sure how to show it has finite norm.