[proofplan]
We rewrite the semilinear [wave equation](/page/Wave%20Equation) as a Duhamel fixed point problem in the energy space $X_T$. The key estimate is that the nonlinearity $F(a)=|a|^{p-1}a$ maps $H^1(\mathbb{R}^n)$ locally Lipschitzly into $L^2(\mathbb{R}^n)$, because the hypothesis $p \leq n/(n-2)$ gives the Sobolev embedding $H^1(\mathbb{R}^n)\hookrightarrow L^{2p}(\mathbb{R}^n)$. The forced linear wave energy estimate then gives a factor of $T$ after integration in time, so the Duhamel map is a contraction on a closed ball for $T$ sufficiently small. The same contraction estimate gives uniqueness and Lipschitz continuous dependence on bounded sets of initial data.
[/proofplan]
[step:Declare the energy space and the linear wave estimates used in the fixed point argument]
Let
\begin{align*}
E:=H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)
\end{align*}
with norm
\begin{align*}
\|(f,g)\|_E:=\|f\|_{H^1(\mathbb{R}^n)}+\|g\|_{L^2(\mathbb{R}^n)}.
\end{align*}
For $T>0$, define the time interval $I_T:=[-T,T]$ and the [Banach space](/page/Banach%20Space)
\begin{align*}
X_T:=C(I_T;H^1(\mathbb{R}^n))\cap C^1(I_T;L^2(\mathbb{R}^n))
\end{align*}
with norm
\begin{align*}
\|u\|_{X_T}:=\sup_{t\in I_T}\bigl(\|u(t)\|_{H^1(\mathbb{R}^n)}+\|\partial_tu(t)\|_{L^2(\mathbb{R}^n)}\bigr).
\end{align*}
We use the standard energy estimate for the forced linear wave equation (citing a result not yet in the wiki: Linear Wave Energy Estimate). Precisely, there exists a constant $C_{\mathrm{w}}=C_{\mathrm{w}}(n)\geq 1$ such that whenever $a<b$, $G\in L^1([a,b];L^2(\mathbb{R}^n))$, $t_0\in [a,b]$, and $w\in C([a,b];H^1(\mathbb{R}^n))\cap C^1([a,b];L^2(\mathbb{R}^n))$ is an energy-class distributional solution of
\begin{align*}
\partial_t^2w-\Delta w=G
\end{align*}
in $\mathcal{D}'(\mathbb{R}^n\times(a,b))$ with Cauchy data $(w(t_0),\partial_tw(t_0))=(f,g)\in E$, then for every $t\in [a,b]$,
\begin{align*}
\|(w(t),\partial_tw(t))\|_E \leq C_{\mathrm{w}}\|(f,g)\|_E+C_{\mathrm{w}}\int_{\min\{t_0,t\}}^{\max\{t_0,t\}}\|G(s)\|_{L^2(\mathbb{R}^n)}\,d\mathcal{L}^1(s).
\end{align*}
The same constant $C_{\mathrm{w}}$ is used on every translated time interval because the linear wave equation is invariant under time translation. We also use the corresponding Duhamel representation for the wave equation (citing a result not yet in the wiki: Duhamel Formula for the Linear Wave Equation): for each $G\in L^1(I_T;L^2(\mathbb{R}^n))$ and $(f,g)\in E$, there is a unique $w\in X_T$ satisfying the preceding forced wave equation and initial data.
[/step]
[step:Prove that the nonlinearity is locally Lipschitz from $H^1(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$]
Define the scalar nonlinearity
\begin{align*}
F:\mathbb{R}\to\mathbb{R},\qquad F(a):=|a|^{p-1}a.
\end{align*}
Since $1<p\leq n/(n-2)$, we have
\begin{align*}
2p\leq \frac{2n}{n-2}.
\end{align*}
By the [Sobolev embedding theorem](/theorems/903) on $\mathbb{R}^n$ (citing a result not yet in the wiki: Sobolev Embedding Theorem), there is a constant $S_{n,p}>0$ such that every $h\in H^1(\mathbb{R}^n)$ satisfies
\begin{align*}
\|h\|_{L^{2p}(\mathbb{R}^n)}\leq S_{n,p}\|h\|_{H^1(\mathbb{R}^n)}.
\end{align*}
For [real numbers](/page/Real%20Numbers) $a,b\in\mathbb{R}$, the [mean value theorem](/theorems/186) applied to $F$ on the interval between $a$ and $b$ gives
\begin{align*}
|F(a)-F(b)|\leq C_p\bigl(|a|^{p-1}+|b|^{p-1}\bigr)|a-b|,
\end{align*}
where $C_p:=p\,2^{p-1}$. Therefore, for $u,v\in H^1(\mathbb{R}^n)$, Hölder's inequality with conjugate exponents $p/(p-1)$ and $p$ gives
\begin{align*}
\|F(u)-F(v)\|_{L^2(\mathbb{R}^n)} \leq C_p\bigl(\|u\|_{L^{2p}(\mathbb{R}^n)}^{p-1}+\|v\|_{L^{2p}(\mathbb{R}^n)}^{p-1}\bigr)\|u-v\|_{L^{2p}(\mathbb{R}^n)}.
\end{align*}
Using the Sobolev embedding on each factor yields
\begin{align*}
\|F(u)-F(v)\|_{L^2(\mathbb{R}^n)} \leq L_{n,p}\bigl(\|u\|_{H^1(\mathbb{R}^n)}^{p-1}+\|v\|_{H^1(\mathbb{R}^n)}^{p-1}\bigr)\|u-v\|_{H^1(\mathbb{R}^n)},
\end{align*}
where
\begin{align*}
L_{n,p}:=C_pS_{n,p}^p.
\end{align*}
[guided]
The purpose of this step is to show that the nonlinear forcing term has exactly the regularity required by the linear wave estimate. Since the forcing space is $L^1(I_T;L^2(\mathbb{R}^n))$, we need $F(u)=|u|^{p-1}u$ to lie in $L^2(\mathbb{R}^n)$ whenever $u\in H^1(\mathbb{R}^n)$, and we need a Lipschitz bound to make the fixed point map contractive.
Define
\begin{align*}
F:\mathbb{R}\to\mathbb{R},\qquad F(a):=|a|^{p-1}a.
\end{align*}
The derivative of $F$ satisfies $|F'(a)|=p|a|^{p-1}$ for $a\neq 0$, and the same growth bound is valid at $a=0$ in the mean-value estimate. Thus, for all $a,b\in\mathbb{R}$,
\begin{align*}
|F(a)-F(b)|\leq C_p\bigl(|a|^{p-1}+|b|^{p-1}\bigr)|a-b|,
\end{align*}
with $C_p:=p2^{p-1}$. This is the pointwise Lipschitz estimate.
Now let $u,v\in H^1(\mathbb{R}^n)$. Applying the pointwise estimate and then taking the $L^2(\mathbb{R}^n)$ norm gives
\begin{align*}
\|F(u)-F(v)\|_{L^2(\mathbb{R}^n)}\leq C_p\bigl\|\bigl(|u|^{p-1}+|v|^{p-1}\bigr)|u-v|\bigr\|_{L^2(\mathbb{R}^n)}.
\end{align*}
We estimate the product by Hölder's inequality. The factor $|u|^{p-1}$ belongs to $L^{2p/(p-1)}(\mathbb{R}^n)$ whenever $u\in L^{2p}(\mathbb{R}^n)$, and $u-v$ belongs to $L^{2p}(\mathbb{R}^n)$. These exponents are compatible because
\begin{align*}
\frac{p-1}{2p}+\frac{1}{2p}=\frac{1}{2}.
\end{align*}
Therefore Hölder's inequality gives
\begin{align*}
\||u|^{p-1}|u-v|\|_{L^2(\mathbb{R}^n)}\leq \|u\|_{L^{2p}(\mathbb{R}^n)}^{p-1}\|u-v\|_{L^{2p}(\mathbb{R}^n)}.
\end{align*}
The same computation with $v$ in place of $u$ gives
\begin{align*}
\||v|^{p-1}|u-v|\|_{L^2(\mathbb{R}^n)}\leq \|v\|_{L^{2p}(\mathbb{R}^n)}^{p-1}\|u-v\|_{L^{2p}(\mathbb{R}^n)}.
\end{align*}
Adding the two estimates, we obtain
\begin{align*}
\|F(u)-F(v)\|_{L^2(\mathbb{R}^n)} \leq C_p\bigl(\|u\|_{L^{2p}(\mathbb{R}^n)}^{p-1}+\|v\|_{L^{2p}(\mathbb{R}^n)}^{p-1}\bigr)\|u-v\|_{L^{2p}(\mathbb{R}^n)}.
\end{align*}
This is where the restriction $p\leq n/(n-2)$ is used. Since $n\geq 3$, the Sobolev embedding theorem gives
\begin{align*}
H^1(\mathbb{R}^n)\hookrightarrow L^q(\mathbb{R}^n)
\end{align*}
for every $q$ satisfying $2\leq q\leq 2n/(n-2)$. Our hypothesis implies $2p\leq 2n/(n-2)$, so there exists $S_{n,p}>0$ such that
\begin{align*}
\|h\|_{L^{2p}(\mathbb{R}^n)}\leq S_{n,p}\|h\|_{H^1(\mathbb{R}^n)}
\end{align*}
for every $h\in H^1(\mathbb{R}^n)$. Applying this to $u$, $v$, and $u-v$ gives
\begin{align*}
\|F(u)-F(v)\|_{L^2(\mathbb{R}^n)} \leq L_{n,p}\bigl(\|u\|_{H^1(\mathbb{R}^n)}^{p-1}+\|v\|_{H^1(\mathbb{R}^n)}^{p-1}\bigr)\|u-v\|_{H^1(\mathbb{R}^n)},
\end{align*}
where
\begin{align*}
L_{n,p}:=C_pS_{n,p}^p.
\end{align*}
This proves the required local Lipschitz estimate from $H^1(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$.
[/guided]
[/step]
[step:Construct the Duhamel map on a closed ball in $X_T$]
Fix initial data $(u_0,u_1)\in E$ and set
\begin{align*}
N:=\|(u_0,u_1)\|_E.
\end{align*}
Define the radius
\begin{align*}
A:=2C_{\mathrm{w}}(N+1).
\end{align*}
For $T>0$, let
\begin{align*}
B_{T,A}:=\{u\in X_T:\|u\|_{X_T}\leq A\}.
\end{align*}
This is a closed subset of the Banach space $X_T$, hence is complete with the metric induced by $\|\cdot\|_{X_T}$.
For $u\in B_{T,A}$, define $\Phi(u)\in X_T$ to be the unique solution of the forced linear wave equation
\begin{align*}
\partial_t^2\Phi(u)-\Delta\Phi(u)=-F(u)
\end{align*}
with initial data
\begin{align*}
\Phi(u)(0)=u_0,\qquad \partial_t\Phi(u)(0)=u_1.
\end{align*}
Since $u\in C(I_T;H^1(\mathbb{R}^n))$, the Lipschitz estimate with $v=0$ implies $F(u)\in C(I_T;L^2(\mathbb{R}^n))$, hence $F(u)\in L^1(I_T;L^2(\mathbb{R}^n))$. Thus the Duhamel map $\Phi:B_{T,A}\to X_T$ is well-defined.
For each $t\in I_T$, the forced wave estimate gives
\begin{align*}
\|(\Phi(u)(t),\partial_t\Phi(u)(t))\|_E\leq C_{\mathrm{w}}N+C_{\mathrm{w}}\int_{\min\{0,t\}}^{\max\{0,t\}}\|F(u(s))\|_{L^2(\mathbb{R}^n)}\,d\mathcal{L}^1(s).
\end{align*}
Using the nonlinear estimate with $v=0$, for every $s\in I_T$,
\begin{align*}
\|F(u(s))\|_{L^2(\mathbb{R}^n)}\leq L_{n,p}\|u(s)\|_{H^1(\mathbb{R}^n)}^p\leq L_{n,p}A^p.
\end{align*}
Therefore
\begin{align*}
\|\Phi(u)\|_{X_T}\leq C_{\mathrm{w}}N+C_{\mathrm{w}}TL_{n,p}A^p.
\end{align*}
If
\begin{align*}
T\leq \frac{N+1}{L_{n,p}A^p},
\end{align*}
then
\begin{align*}
\|\Phi(u)\|_{X_T}\leq C_{\mathrm{w}}N+C_{\mathrm{w}}(N+1)\leq A.
\end{align*}
Thus $\Phi$ maps $B_{T,A}$ into itself for such $T$.
[/step]
[step:Choose the lifespan so that the Duhamel map is a contraction]
Let $u,v\in B_{T,A}$. The difference $\Phi(u)-\Phi(v)$ solves
\begin{align*}
\partial_t^2(\Phi(u)-\Phi(v))-\Delta(\Phi(u)-\Phi(v))=-(F(u)-F(v))
\end{align*}
with zero initial data. The forced wave estimate gives
\begin{align*}
\|\Phi(u)-\Phi(v)\|_{X_T}\leq C_{\mathrm{w}}\int_{-T}^{T}\|F(u(s))-F(v(s))\|_{L^2(\mathbb{R}^n)}\,d\mathcal{L}^1(s).
\end{align*}
For every $s\in I_T$, the nonlinear Lipschitz estimate and the bounds $\|u\|_{X_T},\|v\|_{X_T}\leq A$ imply
\begin{align*}
\|F(u(s))-F(v(s))\|_{L^2(\mathbb{R}^n)}\leq 2L_{n,p}A^{p-1}\|u(s)-v(s)\|_{H^1(\mathbb{R}^n)}.
\end{align*}
Hence
\begin{align*}
\|\Phi(u)-\Phi(v)\|_{X_T}\leq 4C_{\mathrm{w}}TL_{n,p}A^{p-1}\|u-v\|_{X_T}.
\end{align*}
Choose $T>0$ so that
\begin{align*}
T\leq \frac{1}{8C_{\mathrm{w}}L_{n,p}A^{p-1}}
\end{align*}
and also
\begin{align*}
T\leq \frac{N+1}{L_{n,p}A^p}.
\end{align*}
Then $\Phi:B_{T,A}\to B_{T,A}$ is a contraction with Lipschitz constant at most $1/2$. By the [Banach fixed point theorem](/theorems/270) (citing a result not yet in the wiki: [Banach Fixed Point Theorem](/theorems/289)), there exists a unique $u\in B_{T,A}$ such that $\Phi(u)=u$.
[/step]
[step:Identify the fixed point as an energy solution of the semilinear wave equation]
Since $u=\Phi(u)$, the function $u\in X_T$ satisfies the forced linear wave equation
\begin{align*}
\partial_t^2u-\Delta u=-F(u)
\end{align*}
in $\mathcal{D}'(\mathbb{R}^n\times I_T)$ with initial data $(u(0),\partial_tu(0))=(u_0,u_1)$. Substituting the definition $F(u)=|u|^{p-1}u$, we obtain
\begin{align*}
\partial_t^2u-\Delta u+|u|^{p-1}u=0
\end{align*}
in $\mathcal{D}'(\mathbb{R}^n\times I_T)$. Since $u\in C(I_T;H^1(\mathbb{R}^n))$ and $\partial_tu\in C(I_T;L^2(\mathbb{R}^n))$, this is an energy solution on $[-T,T]$.
[/step]
[step:Prove uniqueness in the whole energy class on the same time interval]
Let $u,v\in X_T$ be two solutions of the semilinear wave equation with the same initial data. Since $u,v\in X_T$, define
\begin{align*}
M:=\|u\|_{X_T}+\|v\|_{X_T}.
\end{align*}
The difference $z:=u-v$ belongs to $X_T$, has zero initial data at $t=0$, and satisfies
\begin{align*}
\partial_t^2z-\Delta z=-(F(u)-F(v))
\end{align*}
in $\mathcal{D}'(\mathbb{R}^n\times I_T)$. The forcing term belongs to $L^1(I_T;L^2(\mathbb{R}^n))$ by the local Lipschitz estimate for $F$ and the bounds defining $M$, so the shifted energy estimate for energy-class distributional solutions applies to $z$ on every closed subinterval of $I_T$.
Let $J=[a,b]\subset I_T$ be a closed interval on which $z(a)=0$ and $\partial_tz(a)=0$ in $E$, and define
\begin{align*}
\|z\|_{J}:=\sup_{t\in J}\bigl(\|z(t)\|_{H^1(\mathbb{R}^n)}+\|\partial_tz(t)\|_{L^2(\mathbb{R}^n)}\bigr).
\end{align*}
For every $s\in J$, the nonlinear Lipschitz estimate gives
\begin{align*}
\|F(u(s))-F(v(s))\|_{L^2(\mathbb{R}^n)}\leq L_{n,p}M^{p-1}\|z(s)\|_{H^1(\mathbb{R}^n)}.
\end{align*}
Applying the shifted forced wave estimate with initial time $a$ and zero data gives
\begin{align*}
\|z\|_{J}\leq C_{\mathrm{w}}(b-a)L_{n,p}M^{p-1}\|z\|_{J}.
\end{align*}
Choose $\tau_0>0$ such that
\begin{align*}
C_{\mathrm{w}}\tau_0L_{n,p}M^{p-1}<1.
\end{align*}
Then the preceding estimate forces $\|z\|_{J}=0$ on every interval $J=[a,b]\subset I_T$ of length at most $\tau_0$ whose left endpoint has zero Cauchy data for $z$.
Starting with $a=0$, we obtain $z=0$ on $[0,\min\{\tau_0,T\}]$. At the right endpoint of this interval, continuity of $z\in C(I_T;H^1(\mathbb{R}^n))$ and $\partial_tz\in C(I_T;L^2(\mathbb{R}^n))$ gives zero Cauchy data again, so the same estimate applies to the next interval of length at most $\tau_0$. Since finitely many intervals of length at most $\tau_0$ cover $[0,T]$, induction gives $z=0$ on $[0,T]$.
The same argument run backward in time, using intervals $[b,a]$ with $b<a$ and the shifted estimate integrated from $a$ down to $b$, gives $z=0$ on $[-T,0]$. Hence $z=0$ on all of $[-T,T]$, so the solution is unique in $X_T$.
[/step]
[step:Choose a lifespan on bounded sets and prove continuous dependence on the initial data]
Let $R>0$ and define
\begin{align*}
A_R:=2C_{\mathrm{w}}(R+1).
\end{align*}
Choose $T_R>0$ such that
\begin{align*}
T_R\leq \frac{R+1}{L_{n,p}A_R^p}
\end{align*}
and
\begin{align*}
T_R\leq \frac{1}{8C_{\mathrm{w}}L_{n,p}A_R^{p-1}}.
\end{align*}
Then the preceding construction applies to every datum $(u_0,u_1)\in E$ with $\|(u_0,u_1)\|_E\leq R$, and the corresponding fixed point lies in the ball of radius $A_R$ in $X_{T_R}$.
Let $(u_0,u_1)$ and $(v_0,v_1)$ be two initial data in the closed radius-$R$ ball of $E$, and let $u,v\in X_{T_R}$ be their corresponding solutions. The difference $u-v$ solves
\begin{align*}
\partial_t^2(u-v)-\Delta(u-v)=-(F(u)-F(v))
\end{align*}
with initial data
\begin{align*}
(u-v)(0)=u_0-v_0,\qquad \partial_t(u-v)(0)=u_1-v_1.
\end{align*}
The forced wave estimate and the nonlinear Lipschitz estimate on the radius-$A_R$ ball give
\begin{align*}
\|u-v\|_{X_{T_R}}\leq C_{\mathrm{w}}\|(u_0-v_0,u_1-v_1)\|_E+4C_{\mathrm{w}}T_RL_{n,p}A_R^{p-1}\|u-v\|_{X_{T_R}}.
\end{align*}
By the choice of $T_R$, the second coefficient is at most $1/2$. Absorbing this term into the left-hand side gives
\begin{align*}
\|u-v\|_{X_{T_R}}\leq 2C_{\mathrm{w}}\|(u_0-v_0,u_1-v_1)\|_E.
\end{align*}
Thus the data-to-solution map is Lipschitz, hence continuous, on the radius-$R$ ball of $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$ with values in $X_{T_R}$. This also shows that the lifespan may be chosen depending only on $R,n,p$, and the proof is complete.
[/step]
