Blow-Up for Semi-Linear Wave Equations

Question

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 $(a)$ has a unique solution $u\in C^k([0,T]\times\mathbb{R}^3)$. If the supremum, $T_\ast$, of such times $T$ is finite then $\sup_x \lvert u(t,x)\rvert\to \infty$ as $t \to T_\ast$.

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.

This is migrated from:

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

Answer 1

The regularity mentionned in the theorem is not accurate, for two reasons. The first is that the spaces ${\cal C}^k$ don't behave well with PDEs. Often, it is better to work with ${\cal C}^{k,\alpha}$ with $\alpha\in(0,1)$.

The second and deeper reason is that the solutions have what is called a co-normal regularity. Physicists say that the solution is polarized. Let me explain this with the one-dimensional case ($x\in\mathbb R$). Then the equation reads $$(\partial_t+\partial_x)(\partial_t-\partial_x)u=F(u).$$ from this, you deduce that the quantity $w:=(\partial_t-\partial_x)u$ has a better regularity in the direction of $\partial_t+\partial_x$ than in other directions. In other words, $(\partial_t-\partial_x)w$ is more regular than $\partial_tw$ and $\partial_x w$ separately.

In three space dimensions, this can be quantified using pseudo-differential operators. But again, appropriate regularity results are directional, or polarized. The important theorems for propagation of regularity are due to Egorov and Taylor.