Consider the semilinear wave equation in $[0,t\_0] \times \mathbb R^d$ :


$$\square u = \pm |u|^{p-1}u$$


With subcritical/critical power $1 < p \leq \frac{d+2}{d-2}$.


It is easy to show by energy method that any two $C^2$ classical solutions $u(t,x)$ & $v(t,x)$ with same initial conditions on $\{0\} \times B(x\_0, t\_0)$ will coincide on the backwards light cone $K(x\_0,t\_0) = \{(t,x) : 0 \leq t \leq t\_0, 0 \leq |x-x\_0| \leq t\_0-t\}$ (c.f. T. Tao, Nonlinear dispersive equations, Proposition 3.3).


I've seen this property being used for solutions in energy space $C^0([0,T], \dot{H}^1 \times L^2)$. I guess one should be able to generalize the above proposition using some approximation argument but I could not find a reference with a proof.


My main problem is that the proof using energy method uses the fact that a $C^2$ classical solution is locally bounded. This does not hold for energy space solutions.


My second problem is that if $u(t,x)$ is an energy space solution, then we can approximate $u(0,x)$, $\partial\_t u(0,x)$ and $\square u(t,x)$ by smooth functions $u\_{0,n}(x)$, $u\_{1,n}(x)$, $g\_n(t,x)$, and get an approximate classical solution $u\_n(t,x)$ to the inhomogeneous equation
\begin{align\*}
\square u\_n(t,x) &= g\_n(t,x), \quad (t,x) \in [0,t\_0] \times \mathbb R^d \\
u\_n(0,x) &= u\_{0,n}(x), \quad x \in \mathbb R^d \\
\partial\_t u\_n(0,x) &= u\_{1,n}(x), \quad x \in \mathbb R^d
\end{align\*}


but I don't know whether we can approximate $u(t,x)$ by classical solutions $u\_n(t,x)$ solving
\begin{align\*}
\square u\_n(t,x) &= \pm |u\_n|^{p-1}u\_n, \quad (t,x) \in [0,t\_0] \times \mathbb R^d \\
u\_n(0,x) &= u\_{0,n}(x), \quad x \in \mathbb R^d \\
\partial\_t u\_n(0,x) &= u\_{1,n}(x), \quad x \in \mathbb R^d
\end{align\*}


Any help or reference is welcomed.



Finite propagation speed for non-smooth solutions to nonlinear wave equation
You can approach this dually using that for classical solutions the finite propagation speed holds. This argument is similar in spirit to [this answer of mine for low regularity uniqueness for the linear wave equation](https://mathoverflow.net/a/361009/3948).


1
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First write $w = u-v$. Then you see that $w$ solves a linear wave equation of the form


$$ \Box w = G(u,v) w $$


where the potential $G(u,v) \lesssim |u|^{p-1} + |v|^{p-1}$. **Pretend now that $G$ is a smooth function of $(t,x)$.**


2
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Suppose for convenience that $x\_0 = 0$. You want to prove that $w(t,\cdot) |\_{B(0,t\_0 - t)} = 0$ for $t\in (0,t\_0)$. So it is enough to show that $\int w(t,x) f(x) ~dx = 0$ for every $f\in C^\infty\_c(B(0,t\_0-t))$. We will try to do so using a duality argument.


Fix $T\in (0,t\_0)$, and take $f\in C^\infty\_c(B(0,t\_0-T))$. Solve the wave equation
$$ \Box \varpi = G \varpi $$
with initial data $\varpi(0,T) = 0$ and $\partial\_t\varpi(0,T) = f$. **Under the assumption that $G$ is smooth**, we have that $\varpi$ is a smooth function with compact support for all time. In fact, for $t\in (0,T)$ we have that $\mathrm{supp}~ \varpi \subset B(0,t\_0 - t)$ using the finite speed of propagation for classical solutions.


3
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That $w$ is an energy solution implies that the following identity holds for any test function $\varpi$:


$$ \int\_0^T \int\_{\mathbb{R}^d} \Box w \varpi - w \Box \varpi = \int\_{\mathbb{R}^d} w \partial\_t\varpi - \partial\_t w \varpi \Big|\_{t = 0}^T $$


Using the support property for $\varpi$ as derived above, you have that the right hand integral vanishes at $t = 0$ since $w,\partial\_t w$ vanishes on $B(0,t\_0)$. The integral at $t = T$ has only one term, and that is $\int w f$.


The left hand side however vanishes: since the two functions solve the same (linear) equation you have $LHS = \int Gw~v - w ~Gv = 0$.


4
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The above works assuming that $G$ is smooth. In general, $G$ is not. Replace $G$ by $G\_\epsilon$ through mollification. And replace $\varpi$ correspondingly by $\varpi\_\epsilon$. Then it suffices to show that


$$ \int\_0^T \int\_{\mathbb{R}^d} (G - G\_\epsilon) w \varpi\_\epsilon \to 0 $$


Let me do the critical case for convenience; the argument should be similar for the subcritical cases with some adjustment of the exponents.


Noting that $p-1 = \frac{4}{d-2}$, by Strichartz inequality, $G$ belongs to space-time norm $L^1([0,T]; L^q\_x)$ for any $q\in [\frac{d}{2},d]$. And the mollification will converge in that norm. Since $w$ is energy class you have that it is uniformly bounded in $L^\infty\_t L^{2d/(d-2)}\_x$.


The Strichartz estimates together with Gronwell's inequality can be used to show that if $G\_\epsilon \in L^1\_t L^d\_x$, then


$$ \|\varpi\_\epsilon(t)\|\_{\dot{H}^1} \lesssim e^{C(T-t) \|G\_\epsilon\|\_{L^1\_t L^d\_x}} \|f\|\_{L^2} $$


This shows that $\|\varpi\_\epsilon(t)\|\_{L^\infty\_t L^{2d/(d-2)}}$ is uniformly bounded on the time interval $[0,T]$. And hence the fact that $G-G\_\epsilon$ converges to zero in $L^1\_t L^{d/2}$ gives that $|\int w(T,x) f(x) ~dx | = 0$ after taking the limit.


