[proofplan]
Use the continuation property built into local well-posedness. The local theory gives a lifespan depending only on an upper bound for the energy norm of the Cauchy data. If the forward maximal time were finite while the energy norm stayed bounded, data taken close enough to the maximal time would generate a new local solution beyond it. Uniqueness would glue this new solution to the old one, contradicting maximality.
[/proofplan]
[step:Record the local continuation property]
Let the energy space be
\begin{align*}
\mathcal H:=H^1(\mathbb R^n)\times L^2(\mathbb R^n).
\end{align*}
The phrase maximal solution in the local well-posedness class means the following standard consequence of local well-posedness. For every $R>0$ there exists a time $\tau_R>0$ such that whenever Cauchy data $(v_0,v_1)\in\mathcal H$ satisfy
\begin{align*}
\|v_0\|_{H^1(\mathbb R^n)}+\|v_1\|_{L^2(\mathbb R^n)}\le R,
\end{align*}
there is a solution $v$ of the same equation on an interval of length at least $\tau_R$ forward from the initial time, with initial data $(v_0,v_1)$. The same local theory also gives uniqueness in the local well-posedness class, so two solutions with the same Cauchy data agree on the common part of their intervals of existence.
For the exponent range
\begin{align*}
1<p\le \frac{n}{n-2},
\end{align*}
the nonlinearity is locally Lipschitz on bounded subsets of the energy space in the sense required by this local theory. We use only the resulting local lifespan and uniqueness properties.
[/step]
[step:Contradict finite forward maximal time]
Assume, for contradiction, that $T_+<\infty$ and that
\begin{align*}
M:=\sup_{t\in[0,T_+)}\left(\|u(t)\|_{H^1(\mathbb R^n)}+\|\partial_tu(t)\|_{L^2(\mathbb R^n)}\right)<\infty.
\end{align*}
Choose $R>M+1$. By the local continuation property there exists $\tau_R>0$ such that every datum with $\mathcal H$-norm at most $R$ generates a forward solution for at least time $\tau_R$.
Pick $t_0\in[0,T_+)$ with
\begin{align*}
T_+-t_0<\frac{\tau_R}{2}.
\end{align*}
The Cauchy data at time $t_0$ satisfy
\begin{align*}
\|u(t_0)\|_{H^1(\mathbb R^n)}+\|\partial_tu(t_0)\|_{L^2(\mathbb R^n)}\le M<R.
\end{align*}
Therefore local well-posedness gives a solution $v$ on $[t_0,t_0+\tau_R]$ satisfying
\begin{align*}
v(t_0)=u(t_0),\qquad \partial_tv(t_0)=\partial_tu(t_0).
\end{align*}
By uniqueness, $v$ agrees with $u$ on the overlap $[t_0,T_+)$. Define the extended map
\begin{align*}
\widetilde u:(T_-,t_0+\tau_R)\times\mathbb R^n\to\mathbb R
\end{align*}
by setting $\widetilde u(t,x):=u(t,x)$ for $t<T_+$ and $\widetilde u(t,x):=v(t,x)$ for $t\ge t_0$. The agreement on the overlap makes this a well-defined solution in the same local well-posedness class on
\begin{align*}
(T_-,t_0+\tau_R).
\end{align*}
Since $t_0+\tau_R>T_+$, this strictly extends the original maximal forward interval, a contradiction. Hence $T_+=\infty$.
[guided]
The contradiction uses only a uniform local lifespan. If the energy norm is bounded by $M$ on $[0,T_+)$, then all Cauchy data
\begin{align*}
(u(t),\partial_tu(t))
\end{align*}
with $t<T_+$ lie in the radius-$R$ ball of $\mathcal H$ for any $R>M$. Local well-posedness supplies one positive time $\tau_R$ that works for every datum in that ball. Choosing $t_0$ so close to $T_+$ that $T_+-t_0<\tau_R/2$ produces a new solution from the data at $t_0$ all the way to $t_0+\tau_R$, and
\begin{align*}
t_0+\tau_R>T_+.
\end{align*}
Uniqueness identifies this new solution with the old one before $T_+$, so it is a genuine continuation past $T_+$. That contradicts the definition of $T_+$ as the maximal forward existence time.
[/guided]
[/step]
[step:Obtain the backward statement]
The equation is invariant under time reversal. If a solution $u$ is defined on $(T_-,T_+)$, then for a fixed time $a\in(T_-,T_+)$ define the function
\begin{align*}
w:(a-T_+,a-T_-)\times\mathbb R^n\to\mathbb R
\end{align*}
by
\begin{align*}
w(\theta,x):=u(a-\theta,x)
\end{align*}
for $(\theta,x)\in(a-T_+,a-T_-)\times\mathbb R^n$. Then $w$ solves the same equation on the reversed interval, with velocity $\partial_\theta w(\theta,x)=-\partial_tu(a-\theta,x)$. The energy norm of $(w,\partial_\theta w)$ is the same as the energy norm of $(u,\partial_tu)$ at the corresponding time. Applying the forward argument to $w$ gives the analogous criterion at $T_-$. Thus bounded energy norm prevents finite-time breakdown in either time direction.
[/step]
