I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear Wave Equations), but I got stuck in the last part of the proof regarding the blow-up... Allow me to first state the theorem and then my question.

Consider the equation $$ \left\{\begin{array}{ll}\square\, u(t,x) = F(u(t,x)),\; t>0\\ u(0,x) = f(x),\; \partial_tu(0,x)=g(x) \end{array}\right.\qquad\qquad(a)$$

Theorem 5.1. Assume that $F\in C^k$, $F(0)=0$, and that $f\in C_0^{k+1}(\mathbb{R}^3)$, $g\in C_0^k(\mathbb{R}^3)$, with $k = 1,2,\ldots$. then there is a $T > 0$ so that $(\ast)$ has a unique solution $u\in C^k([0,T]\mathbb{R}^3)$. If the supremum, $T_\ast$, of such times $T$ is finite then $\sup_x \lvert u(t,x)\rvert\to \infty$ as $t \to T_\infty$.

Question. I am having difficulties understanding the blow up part of the theorem. What is the best way of proving this blow-up phenomenon (the second half of Theorem 5.1)?

To this end, let us make the following assumption (this is proved in the book):

Assumption. Given a $C^k$ solution $u$ of ($a$) in $[0,T)\times\mathbb{R}^3$. If $\sup_{\{(t,x)\colon 0\leq t<T\}}\lvert u(t,x)\rvert <\infty$ then $u$ extends to a $C^k$ function in the closed strip $[0,T]\times\mathbb{R}^3$.

Attempt. Under the assumption of local existence, according to the definition of $T_\ast$, there is a $u$ being a $C^k$ solution of ($a$) in $[0,T_\ast)\times\mathbb{R}^3$. To prove the second half of the theorem, let us show that if $$\lim_{t\to T_\ast}\sup_x\lvert u(t,x)\rvert \text{ exists in } \mathbb{R}$$ then $T_\ast = \infty$, that is, u in $C^k$ can be extended indefinitely. Now, it can be shown, according to our assumption above, that $u$ can be extended to a $C^k$ solution in the closed strip $[0,T_\ast]$. To prove the result, it remains to show that there exists $T > T_\ast$ such that $u$ extends to $[0,T_\ast]\cup [T_\ast,T)$.

My idea was then to use the first half of the theorem (local existence) on the following problem:

$$ \left\{\begin{array}{ll}\square\, \tilde{u}(t,x) = F(\tilde{u}(t,x)),\; t>T_\ast\\ \tilde{u}(T_\ast,x) = u(T_\ast,x),\; \partial_t\tilde{u}(T_\ast,x)=u(T_\ast,x) \end{array}\right.\qquad\qquad(b)$$ This is not a viable strategy, as can be seen easily: The local existence of a solution $u$ in $C^k$ solving $(a)$ requires that $f$ be $C^{k+1}$-smooth, but the local existence theorem guarentees no more than $u(T_\ast,x)$ being of class $C^k$, that is, the initial data in $(b)$ does not have the sufficient regularity needed.

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http://math.stackexchange.com/questions/1198583/blow-up-for-semi-linear-wave-equation