Recovery of a Cubic Non-linearity in the Wave Equation in the Weakly Non-linear Regime

Abstract

We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-\Delta u+ \alpha (x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength h and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits ${{\,\mathrm{supp}\,}}\alpha$. We show that one can extract the Radon transform of $\alpha$ from the phase shift of such waves, and then one can recover $\alpha$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.

Appendices

Appendix A. Solving the Transport Equations

We start with the leading order equations (5.4). If

$$\begin{aligned} \mathsf {a}(s) = (\ldots , \mathsf {a}_{-k}(s), \ldots , \mathsf {a}_{-2}(s), \mathsf {a}_{-1}(s), \mathsf {a}_0(s), \mathsf {a}_{1}(s), \mathsf {a}_{2}(s),\ldots , \mathsf {a}_{k}(s), \ldots ), \end{aligned}$$

we say that

$$\begin{aligned} \begin{aligned} \mathsf {a}(s)&\in l^p, \;\ p\in [1,\infty ),&\text { if } \Vert \mathsf {a}(s)\Vert _{l^p}^p&:= \sum _{k=-\infty }^\infty |\mathsf {a}_k|^p<\infty , \\ \mathsf {a}(s)&\in l^\infty , \;\&\text { if } \Vert \mathsf {a}(s)\Vert _{l^\infty }&:= \sup _k |\mathsf {a}_k|<\infty , \\ \mathsf {a}(s)&\in {\mathsf {h}}^m, \;\ m \in [0,\infty ),&\text { if } \Vert \mathsf {a}(s)\Vert _{\mathsf {h}^m}^2&:= \sum _{k=-\infty }^\infty |k|^{2m} |\mathsf {a}_k|^2<\infty . \end{aligned} \end{aligned}$$

When $k=0$, the weight in the definition of $\Vert \mathsf {a}(s)\Vert _{\mathsf {h}^m}^2$ vanishes (and it is undefined when $m=0$) which is not the standard choice but our sequences will have non-zero terms when the index is odd only. If $I\subset \mathbb {R}$ is an interval, we say that $\mathsf {a}\in C(I, l^p)$ or $\mathsf {a}\in C(I, {\mathsf {h}}^m)$ if $\mathsf {a}(s)$ is continuous and

$$\begin{aligned} \begin{aligned} \Vert \mathsf {a}\Vert _{C(I, l^p)}= \sup _{s\in I} \Vert \mathsf {a}(s)\Vert _{l^p}<\infty , \text { or } \Vert \mathsf {a}\Vert _{C(I, {\mathsf {h}}^m)}= \sup _{s\in I} \Vert \mathsf {a}(s)\Vert _{{\mathsf {h}}^m} <\infty . \end{aligned} \end{aligned}$$

(A.1)

The discrete convolution $\mathsf {a}* \mathsf {b}$ is defined to be the sequence such that

$$\begin{aligned} (\mathsf {a}* \mathsf {b})_k = \sum _{k_1+k_2=k} \mathsf {a}_{k_1} \mathsf {b}_{k_2}, \end{aligned}$$

and in the case of three terms $\mathsf {a}* \mathsf {b}*\mathsf {c}$ is defined such that

$$\begin{aligned}\ (\mathsf {a}* \mathsf {b}* \mathsf {c})_k = \sum _{k_1+k_2+k_3=k} \mathsf {a}_{k_1}\mathsf {b}_{k_2} \mathsf {c}_{k_3}. \end{aligned}$$

Young’s inequality holds for discrete convolutions, since it holds for certain groups including $\mathbb {Z}$, but the standard proof which is an application of Hölder’s inequality, also works in this case, and

$$\begin{aligned} \begin{aligned}&\Vert \mathsf {a}*\mathsf {b}\Vert _{l^q}\le \Vert {\mathsf {a}}\Vert _{l^{p_1}} \Vert {\mathsf {b}}\Vert _{l^{p_2}}, \text { for } 1+\frac{1}{q}= \frac{1}{p_1}+\frac{1}{p_2}, \;\ p_1, p_2, q \in [1,\infty ], \\ \Vert \mathsf {a}*\mathsf {b}*\mathsf {c}\Vert _{l^q}&\le \Vert {\mathsf {a}}\Vert _{l^{p_1}} \Vert {\mathsf {b}}\Vert _{l^{p_2}} \Vert {\mathsf {c}}\Vert _{l^{p_3}}, \text { for } 1+\frac{1}{q}= \frac{1}{p_1}+\frac{1}{p_2} +\frac{1}{p_3}, \\&\quad p_1, p_2, p_3, q \in [1,\infty ]. \\ \end{aligned} \end{aligned}$$

(A.2)

In particular, if we choose $p_1=p_2=p_3=q=2$ in (A.2), we have the following inequality

$$\begin{aligned} \Vert \mathsf {a}*\mathsf {a}*\mathsf {a}\Vert _{l^2} \le \Vert \mathsf {a}\Vert _{l^2}^3. \end{aligned}$$

(A.3)

This defines the map

$$\begin{aligned} T: l^2 \longrightarrow l^2, \quad T(\mathsf {a})= \mathsf {a}*\mathsf {a}*\mathsf {a}. \end{aligned}$$

We can write (5.4), (5.5) as

$$\begin{aligned} \ \mathsf {a}_{k}(s) = \mathsf {a}_{k}(0)- \frac{\mathrm {i}}{2 k} \int _0^s\alpha (\sigma ) (\mathsf {a}*\mathsf {a}*\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma , \quad k=\pm 1, \pm 2, \dots , \end{aligned}$$

(A.4)

with $\mathsf {a}_{k}(0)=Ae_{-1}/2+Ae_1/2$. Since only odd values of k will be present, we have $k\not =0$.

We want to use Picard iteration and solve (A.4) locally, if the initial data is in $l^2,$ and in view of the conservation law (5.6), show that the solution is in fact global if the initial data is in ${\mathsf {h}}^{\frac{1}{2}}$. We define the map

$$\begin{aligned} (\Phi (\mathsf {a}))_k(s)= \mathsf {a}_k(0)- \frac{\mathrm {i}}{2k} \int _0^s\alpha (\sigma ) (\mathsf {a}*\mathsf {a}*\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma , \quad k=\pm 1, \pm 2, \dots . \end{aligned}$$

In view of (A.3), on the interval $[0,s_0]$, we have

$$\begin{aligned} \Vert \Phi (\mathsf {a})- \mathsf {a}(0)\Vert _{C([0,s_0],l^2)}\le \frac{s_0}{2} \Vert \alpha \Vert _{L^\infty } \Vert \mathsf {a}\Vert _{C([0,s_0],l^2)}^3. \end{aligned}$$

If $\mathcal {H}(\mathsf {a}(0),M)$ denotes the closed ball of radius M centered at $\mathsf {a}(0)$ in the space $C([0,s_0]; l^2)$ equipped with the norm (A.1), then as long as $s_0$ is such that

$$\begin{aligned} \frac{1}{2}\Vert \alpha \Vert _{L^\infty } (\Vert \mathsf {a}(0)\Vert _{l^2}+ M)^3 s_0 \le M, \end{aligned}$$

(A.5)

then $\Vert \Phi (\mathsf {a})- \mathsf {a}(0)\Vert \le M$ and so

$$\begin{aligned} \Phi : \mathcal {H}(\mathsf {a}(0),M) \longmapsto \mathcal {H}(\mathsf {a}(0),M). \end{aligned}$$

On the other hand, since

$$\begin{aligned} \mathsf {a}*\mathsf {a}*\mathsf {a}- \mathsf {b}*\mathsf {b}*\mathsf {b}= (\mathsf {a}-\mathsf {b})*\mathsf {a}*\mathsf {a}+ (\mathsf {a}-\mathsf {b})*\mathsf {b}*\mathsf {a}+(\mathsf {a}-\mathsf {b})*\mathsf {b}*\mathsf {b}, \end{aligned}$$

it follows from (A.3) that if $\mathsf {a}, \mathsf {b}\in C([0,s_0]; l^2)$; then

$$\begin{aligned} \begin{aligned}&\Vert \Phi (\mathsf {a})- \Phi (\mathsf {b})\Vert _{C([0,s_0],l^2)} \\&\quad \le \frac{s_0}{2} \Vert \alpha \Vert _{L^\infty } \Vert \mathsf {a}-\mathsf {b}\Vert _{C([0,s_0],l^2)}( \Vert \mathsf {a}\Vert _{C([0,s_0],l^2)}^2 + \Vert \mathsf {b}\Vert _{C([0,s_0],l^2)}^2\\&\qquad + \Vert \mathsf {a}\Vert _{C([0,s_0],l^2)}\Vert \mathsf {b}\Vert _{C([0,s_0],l^2)}). \end{aligned} \end{aligned}$$

And so, if $\mathsf {a}, \mathsf {b}\in \mathcal {H}(\mathsf {a}{}(0),M)$, and if $s_0$ is such that

$$\begin{aligned} \frac{3s_0}{2} \Vert \alpha \Vert _{L^\infty } (M+\Vert \mathsf {a}(0)\Vert _{l^2})^2 s_0<1, \end{aligned}$$

(A.6)

the map $\Phi$ is a contraction in $\mathcal {H}(\mathsf {a}(0),M)$, and therefore (A.4) has a unique solution $\mathsf {a}(s)\in \mathcal {H}(\mathsf {a}(0),M)\subset C([0,s_0]; l^2)$, provided M and $s_0$ satisfy (A.5) and (A.6).

If in addition we know that the initial data $\mathsf {a}(0)$ is in $\mathsf {h}^{\frac{1}{2}}$, we will show that we have a unique global solution $\mathsf {a}(s) \in C([0,s_0]; \mathsf {h}^{\frac{1}{2}})$, for any $s_0>0$. To prove this, notice that once we have a solution $\mathsf {a}(s) \in C([0,s_0]; l^2)$, then one can rewrite (A.4) as

$$\begin{aligned} \sqrt{|k|}\ \mathsf {a}_{k}(s)= & {} \sqrt{|k|}\ \mathsf {a}_{k}(0)- {{\,\mathrm{sign}\,}}(k) \frac{\mathrm {i}}{2 \sqrt{|k|}} \int _0^s\alpha (\sigma ) (\mathsf {a}*\mathsf {a}*\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma ,\\&\quad k=\pm 1, \pm 2, \dots , \end{aligned}$$

and in view of (A.3),

$$\begin{aligned} \Vert \mathsf {a}\Vert _{C([0,s_0],\mathsf {h}^{\frac{1}{2}})} \le \Vert \mathsf {a}(0)\Vert _{\mathsf {h}^{\frac{1}{2}}} + \Vert \mathsf {a}\Vert _{C([0,s_0],l^2)}^3; \end{aligned}$$

therefore, $\mathsf {a}\in C([0,s_0],h^{\frac{1}{2}})$, provided $s_0$ and M satisfy (A.5) and (A.6). Since

$$\begin{aligned} \Vert \mathsf {a}(s)\Vert _{l^2}\le \Vert \mathsf {a}(s)\Vert _{\mathsf {h}^{\frac{1}{2}}}, \end{aligned}$$

if we replace the conditions (A.5) and (A.6) with

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Vert \alpha \Vert _{L^\infty } (\Vert \mathsf {a}(0)\Vert _{l^2}+ M)^3 s_0&\le \frac{1}{2}\Vert \alpha \Vert _{L^\infty } (M+ \Vert \mathsf {a}(0)\Vert _{\mathsf {h}^{\frac{1}{2}}})^3 s_0 \le M, \\ \frac{3}{2} \Vert \alpha \Vert _{L^\infty } (M+\Vert \mathsf {a}(0)\Vert _{l^2})^2 s_0&\le \frac{3}{2} \Vert \alpha \Vert _{L^\infty } (M+\Vert \mathsf {a}(0)\Vert _{\mathsf {h}^{\frac{1}{2}}})^2 s_0<1, \end{aligned} \end{aligned}$$

(A.7)

then, in view of the conservation law (5.6),

$$\begin{aligned} \begin{aligned} \frac{1}{2}\Vert \alpha \Vert _{L^\infty } (\Vert \mathsf {a}(s_0)\Vert _{l^2}+ M)^3 s_0&\le \frac{1}{2}\Vert \alpha \Vert _{L^\infty } (M+ \Vert \mathsf {a}(s_0)\Vert _{\mathsf {h}^{\frac{1}{2}}})^3s_0 \le M, \\ \frac{3}{2} \Vert \alpha \Vert _{L^\infty } (M+\Vert \mathsf {a}(s_0)\Vert _{l^2})^2 s_0&\le \frac{3}{2} \Vert \alpha \Vert _{L^\infty } (M+\Vert \mathsf {a}(s_0)\Vert _{\mathsf {h}^{\frac{1}{2}}})^2 s_0<1. \end{aligned} \end{aligned}$$

This implies that we can solve equation (A.4) with initial data set at $s=s_0$ instead of $s=0$, and therefore this shows there exists a unique solution $\mathsf {a}(s)\in C([0,2s_0]; \mathsf {h}^{\frac{1}{2}})$ to (A.4), provided (A.7) is satisfied. This process can be repeated indefinitely and so we have proved the following:

Proposition A.1

If $\mathsf {a}(0)\in l^{2}$, then there exists a unique $\mathsf {a}(s)\in C([0,s_0]; l^2)$ which satisfies (A.4), as long as $s_0$ and M satisfy (A.5) and (A.6). If $\mathsf {a}(0)\in \mathsf {h}^{\frac{1}{2}}$, then there exists a unique $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{\frac{1}{2}})$ which satisfies (A.4).

We can use induction to obtain a similar result for initial data $\mathsf {a}(0)\in \mathsf {h}^{m},$ with $m\in \mathbb {N}$. Proposition A.1 guarantees that equation (A.4) has a unique solution $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{\frac{1}{2}})$ and in particular, $\mathsf {a}(s)\in C(\mathbb {R}; l^2)$. Moreover, by (A.3), the integrand in (A.4) is in the same space, therefore $\mathsf {a}_k(s)$ is $C^1$ in the s variable with values in $l^2$, i.e., it is a strong solution of (5.4).

Recast (5.4) as

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s} k\mathsf {a}_{k}(s) = -\frac{\mathrm {i}}{2 k } \alpha (s) k(\mathsf {a}*\mathsf {a}*\mathsf {a})_k(s) , \quad k=\pm 1, \pm 2, \dots , \end{aligned}$$

with some initial condition for $k\mathsf {a}_k(0)\in \mathsf {h}^{m-1}$. Notice that

$$\begin{aligned} k(\mathsf {a}*\mathsf {a}*\mathsf {a})= k\sum _{k_1+k_2+k_3=k} \mathsf {a}_{k_1} \mathsf {a}_{k_2} \mathsf {a}_{k_3}= 3 \sum _{k_1+k_2+k_3=k} \mathsf {a}_{k_1} \mathsf {a}_{k_2} k_3 \mathsf {a}_{k_3}= 3 (\mathsf {a}*\mathsf {a}* k \mathsf {a}). \end{aligned}$$

Therefore, the sequence ${\tilde{\mathsf {a}}}:=k \mathsf {a}_k$ satisfies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s} {{\tilde{\mathsf {a}}}}_{k}(s) = -\frac{3\mathrm {i}}{2 k} \alpha (\sigma ) (\mathsf {a}*\mathsf {a}*{{\tilde{\mathsf {a}}}} )_k(\sigma ) , \quad k=\pm 1, \pm 2, \dots \end{aligned}$$

(A.8)

This is a linear ODE with a generator

$$\begin{aligned} (A(s) \mathsf {b} )_k(s)= - \frac{3\mathrm {i}}{2k} \alpha (s) (\mathsf {a}*\mathsf {a}*\mathsf {b} )_k(s) , \quad k=\pm 1, \pm 2, \dots . \end{aligned}$$

(A.9)

By (A.3), A(s) is bounded in $l^2$ with a uniformly bounded norm. Then the solution to (A.9) is given by a two-parameter solution group U(t, s) applied to the initial condition $k\mathsf {a}_k(0)\in l^2$, and solves (A.9) in strong sense, see [21, Theorem X.96]. That group is obtained by successive iterations of the integrated equation (the Dyson expansion), similarly to what we did above for the non-linear equation.

So we have shown that if $\mathsf {a}(0)\in \mathsf {h}^{1},$ then there exists a unique $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{1})$ which satisfies (A.4).

When $m=2,$ we have

$$\begin{aligned} \begin{aligned} k^2 (\mathsf {a}*\mathsf {a}*\mathsf {a})_k&= k^2\sum _{k_1+k_2+k_3=k} a_{k_1} a_{k_2} a_{k_3}= 3k \sum _{k_1+k_2+k_3=k} a_{k_1} a_{k_2} k_3 a_{k_3} \\&= 6 \sum _{k_1+k_2+k_3=k} a_{k_1} k_2a_{k_2} k_3a_{k_3}+ 3 \sum _{k_1+k_2+k_3=k} a_{k_1} a_{k_2} k_3^2 a_{k_3}. \end{aligned} \end{aligned}$$

Note that this equality can be interpreted as taking second derivative w.r.t. the dual variable of k, when the convolution is a product; and this is how we got (5.4) in the first place, see (5.2).

Therefore, the ODE for $k^2\mathsf {a}_k$ is

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s} k^2 \mathsf {a}_{k}(s) =- \frac{3\mathrm {i}}{2 k} \alpha (\sigma ) (\mathsf {a}*\mathsf {a}*k^2\mathsf {a})_k(s) - \frac{6\mathrm {i}}{2 k} (\mathsf {a}*k \mathsf {a}*k \mathsf {a})_k(s) , \quad k=\pm 1, \pm 2, \dots .\nonumber \\ \end{aligned}$$

(A.10)

This is a linear non-homogeneous ODE, similar to (A.8) (which is homogeneous) and can be solved in $l^2$ used the solution group U(t, s) and Duhamel’s principle. The generator is the same as above, and the source terms is continuous in s with values in $l^2$ by the previous step.

We already know that $\mathsf {a}\in C(\mathbb {R}, \mathsf {h}^{1})$ and so the right hand side of (A.10) is in $C(\mathbb {R};l^2),$ and so the same contraction mapping argument, now applied to the non-homogeneous equation, shows that if $\mathsf {a}(0)\in \mathsf {h}^{2},$ then there exists a unique $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{2})$ which satisfies (A.4).

In general if m is a positive integer, $k^m \mathsf {a}$ satisfies

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s} k^m \mathsf {a}_{k}(s)&=- \frac{3\mathrm {i}}{2 k} \alpha (\sigma ) (\mathsf {a}*\mathsf {a}*|k|^m\mathsf {a})_k(\sigma ) \\&\qquad + \frac{\mathrm {i}}{ 2 k} \sum _{{\mathop {m_1>0}\limits ^{m_1+m_2+m_3=m}}} C_{m_1,m_2,m_3} \alpha (\sigma ) (|k|^{m_1}\mathsf {a}*|k|^{m_2}\mathsf {a}*|k|^{m_3}\mathsf {a})_k(\sigma ) , \end{aligned}\nonumber \\ \end{aligned}$$

(A.11)

$k=\pm 1, \pm 2, \dots$. By induction, the right hand side is in $C(\mathbb {R}, l^2)$ and the argument used above proves the following.

Proposition A.2

If m is a positive integer and $\mathsf {a}(0)\in \mathsf {h}^{m},$ then there exists a unique $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{m})$ which satisfies (5.4) with that initial condition.

Next we analyze the higher order transport equations. We return to the notation $a_0^{(k)}$ instead of $\mathsf {a}_k$. By (5.3), the next transport equation takes the form

$$\begin{aligned} 2k \frac{\mathrm {d}}{\mathrm {d}s} a_1^{(k)} +3\mathrm {i}(a_0* a_0* a_1)^{(k)}= -\mathrm {i}\Box a_0^{(k)} \end{aligned}$$

(A.12)

with zero initial conditions. Note that the D’Alembertian in the r.h.s. is written in the original (t, x) coordinates instead in the characteristic coordinates $(s,y)=( t,x-t\omega )$ but we can always convert it to the latter ones. This is a linear homogeneous system of ODEs, but before we can solve it we need to show that the right hand side is well defined. We can differentiate (A.4) to find

$$\begin{aligned} \ \partial _{y_j}\mathsf {a}_{k}(s) = \partial _{y_j}\mathsf {a}_{k}(0)- \frac{3\mathrm {i}}{2 k} \int _0^s\alpha (\sigma ) (\mathsf {a}*\mathsf {a}*\partial _{y_j}\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma , \quad k=\pm 1, \pm 2, \dots , \end{aligned}$$

and the argument used to prove Proposition A.2 shows that ${\partial }_{y_j}\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{m}),$ provided ${\partial }_{y_j}\mathsf {a}(0)\in \mathsf {h}^{m},$ and m is a positive derivative. For the second order derivatives, we have

$$\begin{aligned} \begin{aligned} \ \partial _{y_j}{\partial }_{y_r}\mathsf {a}_{k}(s)&= \partial _{y_j}{\partial }_{y_r}\mathsf {a}_{k}(0)- \frac{3\mathrm {i}}{2 k} \int _0^s\alpha (\sigma ) (\mathsf {a}*\mathsf {a}*{\partial }_{y_j}\partial _{y_r}\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma \\&\qquad -\frac{9\mathrm {i}}{2 k} \int _0^s\alpha (\sigma ) (\mathsf {a}*{\partial }_{y_m}\mathsf {a}*\partial _{y_j}\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma , \quad k=\pm 1, \pm 2, \dots . \end{aligned} \end{aligned}$$

(A.13)

We have already established that the first order derivatives satisfy

$$\begin{aligned} \int _0^s\alpha (\sigma ) (\mathsf {a}*{\partial }_{y_m}\mathsf {a}*\partial _{y_j}\mathsf {a})_k(\sigma )\,\mathrm {d}\sigma \in C(\mathbb {R}; \mathsf {h}^m), \end{aligned}$$

and again we apply the argument used in the proof of Proposition A.2 to the non-homogeneous system (A.13) and we find that ${\partial }_{y_j}\partial _{y_r}\mathsf {a}\in C(\mathbb {R}; \mathsf {h}^m)$. We can treat derivatives involving $\partial _s$ in a similar way, differentiating (A.4) w.r.t. s. Once we obtain the result for second order derivatives, the same argument proves the result for third order derivatives and so by induction we obtain the following.

Proposition A.3

If m is a positive integer and $(\partial _s, \partial _y)^\alpha \mathsf {a}(0)\in \mathsf {h}^{m},$ for $|\alpha |\le M,$ then there exists a unique $\mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{m})$ such that $(\partial _s, \partial _y)^\alpha \mathsf {a}(s)\in C(\mathbb {R}; \mathsf {h}^{m})$ for $|\alpha |\le M,$ which satisfies (A.4).

In particular, if $(\partial _s,\partial _y)^\alpha \mathsf {a}(0)\in \mathsf {h}^{m},$ with $|\alpha |\le 2$ it follows that $\Box a_0^{(k)} \in C(\mathbb {R}; \mathsf {h}^m),$ and we can once again apply the argument used in the proof of Proposition A.2 to show the following.

Proposition A.4

If m is a positive integer and $(\partial _s,\partial _y)^\alpha \mathsf {a}(0)\in \mathsf {h}^{m},$ with $|\alpha |\le 2,$ then there exists a unique $a_1 (s)\in C(\mathbb {R}; \mathsf {h}^{m})$ which satisfies (A.12).

The higher order equations can be treated similarly.

Appendix B. Global Existence of Solutions and Well-Posedness

We formulate global existence and well-posedness results for the semilinear wave equation (1.1) when $n=2,3$, with initial conditions

$$\begin{aligned} u|_{t=0} = f_1, \quad u_t|_{t=0} = f_2. \end{aligned}$$

(B.1)

Related results can be found in [3, 6, 9, 10, 14, 24]. We follow [21, section X.13], where even more general non-linearities are considered. The theorems below follow from the theorems there, see more specifically pp. 303–310, when $n=3$. We will show that they hold when $n=2$ as well. We will modify the energy space a bit. We are interested in solutions with initial data belonging to the energy space locally only propagating over time interval [0, T] with $R>0$, $T>0$ fixed. By [21, Theorem X.77], the speed of propagation does not exceed one. Then it is enough to study initial conditions supported in the ball B(0, R), see the paragraphs following Theorem B.1. The support of the solution would not expand beyond $B(0,R+T)$. We can just work in the latter ball by imposing zero boundary conditions on its boundary. The solutions we are interested in would never reflect from the boundary. In what follows, we replace $R+T$ by R. The energy space then becomes $\mathcal {H} := H_0^1(\Omega )\times L^2(\Omega )$, where $\Omega =B(0,R)$. Then $-\Delta$ is essentially self-adjoint on $C_0^\infty (\Omega )$, extending the Dirichlet Laplacian $-\Delta _D$ on $\Omega$ as a self-adjoint one, having a positive minimal eigenvalue. Then $B=(-\Delta _D)^{1/2}$ is a well defined positive operator on $L^2(\Omega )$. Moreover, $D(B)=H_0^1(\Omega ),$ and for every $f\in D(B)$, we have $\Vert Bf\Vert =\Vert \nabla f\Vert$.

In [21, section X.13], there is the Klein-Gordon term $m^2$ added to the Laplacian with $m>0$, then $B=(-\Delta +m^2)^{1/2}$ in $L^2(\mathbf {R}^3)$. All the proofs apply to our situation as well. Another way to make the mass $m=0$ is outlined in Problem 76 there: add $m^2$ to $-\Delta$ and subtract it from the non-linearity $\alpha |u|^2u$. The space dimension is $n=3$ there however.

The well-posedness for $n=2,3$ also follows from [10] in a similar way. They consider more general non-linearities as well.

1.1 B.1. Existence and uniqueness

We view (1.1) as on ODE in the energy space $\dot{H}^1(\Omega ) \times L^2(\Omega )$, as it is usually done:

$$\begin{aligned} \mathbf {u}_t = \begin{pmatrix} 0&{}\mathrm{Id}\\ \Delta &{}0 \end{pmatrix}\mathbf {u} -\begin{pmatrix} 0\\ alpha |u|^2u \end{pmatrix}, \quad \mathbf {u}(0)=\mathbf {f}:= \begin{pmatrix} f_1\\ f_2 \end{pmatrix}. \end{aligned}$$

(B.2)

The space $\dot{H}^1(\Omega )$ is defined as the completion of $C_0^\infty (\Omega )$ under the norm $\Vert f\Vert _{\dot{H}^1}=\Vert \nabla f\Vert _{L^2}$. If $\Omega$ is a bounded domain, then $\dot{H}^1(\Omega )$ is topologically equivalent to $H_0^1(\Omega )$. We denote by A the matrix operator above. Its domain is $D(A) = H^2(\Omega ) \cap H_0^1(\Omega )\times H_0^1(\Omega )$.

One uses the Picard iteration to solve it. We convert it to an integral equation, i.e., we are seeking the weak solution now:

$$\begin{aligned} \mathbf {u}(t) = U_0(t)\begin{pmatrix} f_1\\ f_2 \end{pmatrix} - \int _0^t U_0(t-s) \begin{pmatrix} 0\\ \alpha |u(s)|^2u(s)\end{pmatrix}\mathrm {d}s, \end{aligned}$$

(B.3)

where $U_0$ is the solution group of the linear equation and we suppressed the dependence on x in u. Then we replace u in the non-linearity on the right with $\mathbf {u}_0(t) :=U_0(t){\mathbf {f}}$, compute the first iteration by that formula, then iterate, and take the limit. For this to work at least locally, we need the non-linearity to map $\dot{H}^1$ to $L^2$ continuously and be Lipschitz there. That would give us a (weak) solution in the energy space only. For a strong solution, one needs the Cauchy data to be in the domain D(A) of the matrix operator in (B.2), and wants to prove that the solution exists in that space as well. The analysis is similar but in a new space. To prove existence of a global solution, the energy preservation (1.2) plays a crucial role. This is the strategy in [21, section X.13], as well as in the papers cited above.

We assume $n=2$ or $n=3$ below. We will show that the sequence of lemmas in [21, section X.13] which imply the desired theorem hold when $n=2$ as well but we also allow $n=3$ below. All norms are in $\Omega \subset B(0,R)$.

The first lemma shows that the non-linearity is a continuous operator in the energy space, see also Lemma B.2 below.

Lemma B.1

For every $u\in C_0^\infty (\Omega )$, we have

$$\begin{aligned} \Vert u\Vert _{L^6} \le C \Vert \nabla u\Vert _{L^2}. \end{aligned}$$

(B.4)

Proof

By the Sobolev embedding inequality,

$$\begin{aligned} \Vert u\Vert _{L^q}\le C(n,p)\Vert \nabla u\Vert _{L^p}, \quad 1\le p< n, \quad 1/q =1/p-1/n. \end{aligned}$$

(B.5)

Set $n=3$, $p=2$ in (B.5), then $q=6$; hence

$$\begin{aligned} \Vert u\Vert _{L^6}\le C \Vert \nabla u\Vert _{L^2}, \quad n=3. \end{aligned}$$

(B.6)

In [10] one can find a refined argument which covers $n=2$ as well, and is also useful to prove the Lipschitz property below for $n=2,3$. Writing $|u|^6= |u|^2 |u|^4$, we apply Hölder’s inequality first

$$\begin{aligned} \int \left| \alpha |u|^2u\right| ^2 \,\mathrm {d}x \le \Vert u\Vert ^4_{L^{4q_1}}\Vert u\Vert ^2_{L^{2q_2}} ,\quad 1/q_1+1/q_2=1,\quad q_1, q_2>0. \end{aligned}$$

Take $4q_1=2q_2$; then $q_1=3/2$, $q_2=3$. We apply (B.5) with $n=2$, $q=4q_1=2q_2=6$. For the corresponding p, we get $p_1=p_2=6n/(6+n)$, which does not exceed 2 when $n=2,3$. Another application of Hölder’s inequality to the pair of functions u and 1 implies that (B.6) still holds for $n=2$ as well. Note that the constant C in (B.4) is independent of R when $n=3$ but it depends on it when $n=2$. $\square$

Next lemma is a refinement of the previous one.

Lemma B.2

For every $u\in H_0^1(\Omega )$, we have

$$\begin{aligned} \Vert u_1u_2u_3\Vert _{L^2} \le C \Vert \nabla u_1\Vert _{L^2}\Vert \nabla u_2\Vert _{L^2}\Vert \nabla u_3\Vert _{L^2}. \end{aligned}$$

The proof is as in [21, Lemma 3, X.13]; in particular, Lemma B.1 above is used.

The next lemma says that non-linearity is a continuous operator in the energy space as it follows directly from Lemma B.1, and that it is Lipschitz there.

Lemma B.3

For every $u_1, u_2\in H_0^1(\Omega )$, we have

$$\begin{aligned} \begin{aligned} \left\| \alpha |u_1|^2 u_1\right\| _{L^2}&\le C \Vert \nabla u_1\Vert ^3_{L^2},\\ \left\| \alpha |u_1|^2 u_1 - \alpha |u_2|^2 u_2\right\| _{L^2}&\le C(u_1,u_2)\Vert \nabla (u_1-u_2)\Vert _{L^2} \end{aligned} \end{aligned}$$

(B.7)

with $C(u_1,u_2) = C_0\left( \Vert \nabla u_1\Vert ^2 +\Vert \nabla u_1\Vert \Vert \nabla u_2\Vert +\Vert \nabla u_2\Vert ^2\right)$ and $C_0>0$ depending on R only.

The proof is as in [21, Lemma 4, X.13] and it is based on the previous lemmas. In particular, it works for $n=2$ as well as in all lemmas so far.

Next lemma is an analogue of Lemma B.3 but the smoothness requirements are one degree higher, so are the conclusions. It corresponds to [21, Lemma 5, X.13].

Lemma B.4

For every $u_1, u_2\in H^2(\Omega )\cap H_0^1(\Omega )$, we have

$$\begin{aligned} \begin{aligned} \left\| \nabla (\alpha |u_1|^2 u_1)\right\| _{L^2}&\le C \Vert \nabla u_1\Vert _{L^2}^2 \Vert \Delta u_1\Vert _{L^2},\\ \left\| \nabla \left( \alpha |u_1|^2 u_1 - \alpha |u_2|^2 u_2\right) \right\| _{L^2}&\le C(\nabla u_1,\nabla u_2, \Delta u_1,\Delta u_2 )\Vert \Delta (u_1-u_2)\Vert _{L^2} \end{aligned} \end{aligned}$$

with C above some continuous function, increasing in each of its arguments.

Sketch of the proof

The proof is the same as that of [21, Lemma 5, X.13] with one caveat. The latter uses the Fourier transform since $\Omega =\mathbf {R}^3$ there. We can adapt this to the current setup however. In the proof of [21, Lemma 5, X.13], one needs to estimate $\Vert \nabla u_{x_j}\Vert$. For every u as in the lemma, we have

$$\begin{aligned} \Vert \nabla u_{x_j}\Vert _{L^2}\le C\Vert \Delta u\Vert _{L^2} \end{aligned}$$

by standard elliptic estimates about the solution to $\Delta u=f$, $u|_{\partial \Omega }=0$. This is also the estimate first established in the proof of [21, Lemma 5, X.13] using the Fourier transform. Another inequality used there is $\Vert \nabla u\Vert _{L^2}\le C \Vert \Delta u\Vert _{L^2}$ for u as in the lemma, which follows again from standard elliptic estimates. The rest of the proof is the same as in [21, Lemma 5, X.13]. $\square$

The next lemma states the energy preservation property (1.2) for solutions with regularity as in Lemma B.4 above. Of course, assuming enough smoothness, (1.2) is immediate.

Lemma B.5

Let u be a solution of (1.1), (B.1) on [0, T) with $\mathbf {f} = (f_1,f_2) \in D(A)$. Then the energy (1.2) is independent of t.

Let $E_0(\mathbf {u}(t))$ be the “free” energy, defined as $E(\mathbf {u}(t))$ in (1.2) but with $\alpha =0$, i.e., $E_0(\mathbf {u}(t))=\frac{1}{2}\Vert \mathbf {u} (t)\Vert ^2_\mathcal {H}$. In the next lemma, $\Omega =\mathbf {R}^n$ instead of being bounded because this case is of its own interest as T grows and the support expands. The proof applies when $\Omega =B(0,R)$ as well; then (B.8) still holds, and the rest is unchanged.

Lemma B.6

Assume that $\mathbf {f} = (f_1,f_2)\in D(A)$ and is supported in the ball B(0, R), and let $\mathbf {u} =(u,u_t)$ solve (1.1), (B.1). Then $E_0(\mathbf {u}(t))$ remains bounded with a bound dependent on R but independent of T.

Proof

Recall Poincaré’s inequality

$$\begin{aligned} \Vert f\Vert _{L^2}\le CR \Vert \nabla f\Vert _{L^2}, \quad {{\,\mathrm{supp}\,}}f\subset B(0,R). \end{aligned}$$

(B.8)

In particular,

$$\begin{aligned} \Vert f\Vert _{H^1}\le C(1+R)\Vert \nabla f\Vert _{L^2}, \quad {{\,\mathrm{supp}\,}}f\subset B(0,R). \end{aligned}$$

By the Hölder inequality, if v is supported in a bounded domain $\Omega \subset B(0,R)$,

$$\begin{aligned} \begin{aligned} \int _\Omega |v|^p\,\mathrm {d}x&\le \Big (\int _\Omega |v|^{p q'}\,\mathrm {d}x\Big )^\frac{1}{q'} \Big (\int _\Omega \,\mathrm {d}x\Big )^\frac{1}{q''} \\&\le CR^{n/q''} \Big (\int _\Omega |v|^{p q'}\,\mathrm {d}x\Big )^\frac{1}{q'}, \qquad \frac{1}{q'}+\frac{1}{q''}=1,\quad q'>1, q''>1, \end{aligned} \end{aligned}$$

thus

$$\begin{aligned} \Vert v\Vert _{L^p}\le C R^{n/p-n/q} \Vert v\Vert _{L^q}\quad \text {as long as }p\le q. \end{aligned}$$

(B.9)

We apply the Sobolev embedding inequality (B.5) with $q=4$. Then for the non-quadratic term in the definition (1.2) of $E(\mathbf {u}(t))$ we have

$$\begin{aligned} \int \alpha |u|^4\,\mathrm {d}x \le C \Vert \nabla u\Vert ^4_{L^p}, \quad 1/4=1/p-1/n\quad \Longrightarrow \quad p=\frac{4n}{n+4}. \end{aligned}$$

When $n\le 4$, we have $p\le 2$. Therefore, for every $\mathbf {u}(0)$ supported in B(0, R), we have, for $t=0$,

$$\begin{aligned} \int \alpha |u|^4\,\mathrm {d}x \le C R^{4} \Vert \nabla u\Vert ^4_{L^2}\le C R^{4} E_0^2 (\mathbf {u}(0)), \end{aligned}$$

where we used (B.9). In fact, the first inequality is a known generalized version of the Poincaré inequality. Hence,

$$\begin{aligned} E(\mathbf {u}(0))\le C\left( R^{4} E_0^2(\mathbf {u}(0)) + E_0(\mathbf {u}(0)) \right) . \end{aligned}$$

Since the energy is preserved,

$$\begin{aligned} E_0(\mathbf {u}(t))\le E(\mathbf {u}(t))= E(\mathbf {u}(0)) \le C\big ( E_0(\mathbf {u}(0)) + R^{4} E_0^2(\mathbf {u}(0)) \big ). \end{aligned}$$

$\square$

The analysis in [21, section X.13], see Theorem X.75 there, yields the following.

Theorem B.1

Let $n=2$ or $n=3$. Let $\alpha \in C^2(\mathbf {R}^n)$. Assume that $(f_1,f_2)\in H^2(\mathbf {R}^n)\cap \dot{H}^1(\mathbf {R}^n)\times \dot{H}^1(\mathbf {R}^n)$ is compactly supported. Then the PDE (1.1) with Cauchy data (B.1) has a unique solution in $\mathbf {R}_t\times \mathbf {R}^n_x$ so that $u\in C^j(\mathbf {R}_t;\; H^{2-j}(\mathbf {R}^n) )$, $j=0,1,2$.

In fact, we first prove the theorem with $\mathbf {R}^n$ replaced by a bounded domain $\Omega$. Then using the finite speed of propagation, we reduce the $\Omega =\mathbf {R}^n$ case to this one, as explained at the beginning of this section.

Remark B.1

Assume that $\alpha$ has compact support. Then we can remove the requirement that $\mathbf {u}(0)=(u_1,u_2)$ has compact support and the latter needs to belong to the indicated space locally only. We can localize $\mathbf {u}(0)$ in a large ball with some smooth cutoff $\chi$ so that signals supported outside it do not reach ${{\,\mathrm{supp}\,}}\alpha$ for time $T>0$ fixed (they solve the linear wave equation there). Write $\mathbf {u}(0)=\chi \mathbf {u}(0)+ (1-\chi )\mathbf {u}(0)$. Apply the theorem to solutions with initial data the first term; and solve the linear problem with initial data the second one getting a solution with finite local energy. Then the sum solves the non-linear problem for $|t|\le T$.

1.2 B.2. Well-posedness

The following theorem and its proof correspond to [21, Theorem X.75]. Problem 80 there shows that one can increase the Sobolev norms in which the estimates are made, i.e., work in $D(A^k)$ with $k\ge 2$.

Theorem B.2

Let $n=2$ or $n=3$ and $\alpha \in C^2(\Omega )$. Let $u^{(1)}$, $u^{(2)}$ solve

$$\begin{aligned} \begin{aligned} u_{tt}^{(j)}-\Delta u^{(j)} + \alpha (x) |u^{(j)}|^2u^{(j)}&=0,\\ u^{(j)}|_{t=0}&=f_1^{(j)}, \\ u_t^{(j)}|_{t=0}&=f_2^{(j)}, \end{aligned} \end{aligned}$$

$j=1,2$. Assume $\Vert \mathbf {f}^{(j)}\Vert _{\mathcal {H}}\le C_0$, $j=1,2$, where $\mathbf {f}^{(j)}= (f_1^{(j)}, f_2^{(j)})$. Then

$$\begin{aligned} \Vert \mathbf {u}^{(1)}(t)- \mathbf {u}^{(2)}(t)\Vert _{D(A)}\le e^{C(C_0)t} \Vert \mathbf {f}^{(1)} -\mathbf {f}^{(2)} \Vert _{D(A)}. \end{aligned}$$

(B.10)

Proof

Dropping the superscripts, we have, similarly to (B.3),

$$\begin{aligned} \mathbf {u}(t) = U_0(t)\mathbf {u}_0 - \int _0^t U_0(t-s) \begin{pmatrix} 0\\ \alpha |u(s)|^2u(s)\end{pmatrix}\mathrm {d}s . \end{aligned}$$

(B.11)

Subtract those identities for $j=1,2$ and use Lemma B.3 to get

$$\begin{aligned} \begin{aligned} \Vert \mathbf {u}^{(1)}(t)- \mathbf {u}^{(2)}(t)\Vert _{\mathcal {H}}&\le \Vert \mathbf {f}^{(1)} -\mathbf {f}^{(2)} \Vert _{\mathcal {H}} + C(C_0) \int _0^t\Vert \mathbf {u}^{(1)}(s)- \mathbf {u}^{(2)}(s) \Vert _\mathcal {H}, \end{aligned} \end{aligned}$$

where $C(C_0)$ is the constant in the second inequality in (B.7) which depends on $C_0$ only since the free energy $E_0(\mathbf {u}(t))$ remains bounded by Lemma B.6. Then (B.10), with the norms there in $\mathcal {H}$ instead of D(A), follows by Gronwall’s inequality.

To prove (B.10) with the norms as stated, we need an a priori bound for $E_0(A\mathbf {u}(t)) =\frac{1}{2}\Vert A\mathbf {u}(t)\Vert ^2_\mathcal {H}$. This follows from [21, Lemma 1], the needed conditions for it to hold in our situation are guaranteed by Lemma B.3. Then we apply A to (B.11) and argue as above. $\square$

Remark B.3

As we mentioned above, Problem 80 in [21] outlines a way to prove even higher order estimates. For that, one needs to prove higher order versions of Lemmas B.3–B.5, see [21, Theorem X.74].

The next theorem shows that the formal asymptotic solution (parametrix) is close to an actual one; thus justifying the parametrix construction.

Theorem B.3

Let $n=2$ or $n=3$ and $\alpha \in C^2(\Omega )$. Let u solve the unperturbed equation (1.1) with initial conditions (B.1), where $\Vert \mathbf {f}\Vert _{\mathcal {H}}\le C_0$. Let $u^\sharp$ solve

$$\begin{aligned} \begin{aligned} u_{tt}^\sharp -\Delta u^\sharp + \alpha (x) |u^\sharp |^2u^\sharp&=r(t,x),\\ u ^\sharp |_{t=0}&=f_1^\sharp , \\ u_t^\sharp |_{t=0}&=f_2^\sharp , \end{aligned} \end{aligned}$$

with $\Vert \mathbf {u}^\sharp (t)\Vert _{D(A)}\le C^\sharp \Vert \mathbf {f}\Vert _{D(A)}$ for $t\in [0,T]$. Then

$$\begin{aligned} \Vert \mathbf {u}(t)- \mathbf {u}^\sharp (t)\Vert _{D(A)}\le e^{C(C_0, C^\sharp )t}\left( \int _0^t\Vert r(s,\cdot ) \Vert _{H^1}\, \mathrm {d}s + \Vert \mathbf {f} -\mathbf {f}^\sharp \Vert _{D(A)}\right) \end{aligned}$$

for $t\in [0,T]$ assuming that r is such that the norm in the r.h.s. above for each one of them is finite.

Proof

We argue as above. Subtracting the two solutions, we get

$$\begin{aligned} \begin{aligned} \Vert \mathbf {u} (t)- \mathbf {u}^\sharp (t)\Vert _{\mathcal {H}}&\le \Vert \mathbf {f} -\mathbf {f}^\sharp \Vert _{\mathcal {H}} + C(C_0, C^\sharp ) \int _0^t\Vert \mathbf {u}(s)- \mathbf {u}^\sharp (s) \Vert _\mathcal {H}\, \mathrm {d}s \\&\qquad + \int _0^t\Vert r(s, \cdot ) \Vert _{L^2}\,\mathrm {d}s. \end{aligned} \end{aligned}$$

Now we apply A to the difference and estimate in $\mathcal {H}$ again. Note that the needed a priori estimate for $\Vert \mathbf {u}\Vert _{D(A)}$ is guaranteed by the argument in the previous proof while that for $\Vert \mathbf {u}^\sharp \Vert _{D(A)}$ is postulated but it naturally holds for the parametrix we constructed. $\square$

Remark B.3

By Sobolev embedding, since $n=2$ of $n=3$, for every $\mathbf {u}=(u,u_t)\in D(A)$, we have $u\in H^2$ for every t, therefore, $u\in C^0$ as well, with a continuous dependence on t. The estimates above hold in $C^0$ as well for $u(t,\cdot )$.