Existence of Classical Solution to $\square u + u = 0$ in unbounded domains

Question

Let $\Omega \subseteq \mathbb{R}^n$ be a domain, not necessarily bounded. Let's also assume for simplicity that $\Omega$ has smooth boundary. Fix two functions $$ \alpha, \beta : \bar{\Omega} \to \mathbb{R}, $$ each assumed to be smooth in $\Omega$, continuous up to the boundary, with $\alpha \equiv \beta \equiv 0$ on $\partial \Omega$. In the case that $\Omega$ is unbounded, we will assume in addition that $$ \alpha,\beta \to 0 \quad \text{as } {|x|} \to \infty. $$ Can we ensure the existence of a smooth solution $u$ to the Cauchy problem \begin{align} \square u + u \equiv 0 \quad \text{in }\Omega \times \mathbb{R},\\ u(x,0) \equiv \alpha(x) \quad \text{in } \Omega,\\ u_t(x,0) \equiv \beta(x) \quad \text{in } \Omega. \end{align}

Naturally, I wanted to obtain a weak solution to this problem and to then invoke regularity results to lift it to a classical solution. However, $\alpha$ and $\beta$ need not belong to $L^p(\Omega)$ for any $1 < p < \infty$. Consequently, I cannot see how to formulate this problem in a "weak sense".

Is this approach correct? Or am I missing something entirely?

Answer 1

I assume the structure of the wave operator $\square$ to be considered is the following one: $$ \square =\frac{\partial^2}{\partial t^2}-c^2\Delta\quad c>0,\;(x,t)\text{ in }\;\Bbb R^{n+1}\equiv \Bbb R^n\times \Bbb R \label{w}\tag{W} $$ That said, it is possible to prove the existence and uniqueness of a solution for the posed problem, which could be called homogeneous Cauchy problem for the perturbed wave equation, and construct it in a fairly explicit way: also, depending on the properties of the initial data $\alpha$ and $\beta$, it is possible to determine the smoothness of the solution and thus if it is a classical solution or not.

Existence, uniqueness and explicit construction of the solution
Let's start by reformulationg the problem $$ \left\{ \begin{align} \square u(x,t)+u(x,t)&=0 &\\ u(x,0)&=\alpha(x) & x\in\Omega\\ u_t(x,0)&=\beta(x) & x\in\Omega \end{align} \right.\label{pw}\tag{PW} $$ By using techniques from Fourier analysis and distribution theory, precisely by applying to problem \eqref{pw} the partial Fourier transform respect to the $x$ variable $\mathscr{F}_{x\to\xi}$, we can transform it to the following Cauchy problem for a linear constant coefficient ODE: $$ \left\{ \begin{align} \frac{\partial^2 \hat{u}(\xi,t)}{\partial t^2}+(c^2|\xi|^2+1)\hat{u}(\xi,t)&=0 &\\ \hat{u}(\xi,0)=\hat{\alpha}(\xi) & &\\ \hat{u}_t(\xi,0)=\hat{\beta}(\xi) & & \end{align} \right.\label{1}\tag{1} $$ Having applied the Fourier transform, we implicitly seek the solutions of \eqref{pw} as a distribution of slow growth, and the solutions of \eqref{pw} are the Fourier transforms of this distributions. Now, since from the theory of ODE we know that problem \eqref{1} is uniquely solvable and since the Fourier transform is an isomorphism in $\mathscr{S}^\prime(\Bbb R^n)$ (see [1], chapter VII, §7.1, theorem 7.1.10, p. 164), we have solved the existence and uniqueness \eqref{pw}: furthermore, the explicit solution to \eqref{1} is $$ \hat{u}(\xi,t)=\hat{\alpha}(\xi)\frac{\cos\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}} + \hat{\beta}(\xi)\frac{\sin\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}}\label{2'}\tag{2'} $$ thus the solution of \eqref{pw} has the form $$ u(x,t)=\alpha\ast\mathrm{cp}(x,t)+\beta\ast\mathrm{sp}(x,t)\in\mathscr{S}^\prime(\Bbb R^n)\times C^\infty(\Bbb R) \label{2}\tag{2} $$ where $$ \begin{split} \mathrm{sp}(x,t)&=\mathscr{F}_{\xi\to x}^{-1}\left[\tfrac{\sin\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}}\right](x)=\frac{1}{(2\pi)^n}\int\limits_{\Bbb R^n} \tfrac{\sin\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}}e^{i\langle \xi,x\rangle}\mathrm{d}\xi\\ \mathrm{cp}(x,t)&=\mathscr{F}_{\xi\to x}^{-1}\left[\tfrac{\cos\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}}\right](x)=\frac{1}{(2\pi)^n}\int\limits_{\Bbb R^n} \tfrac{\cos\left(t\sqrt{c^2|\xi|^2+1}\right)}{\sqrt{c^2|\xi|^2+1}}e^{i\langle \xi,x\rangle}\mathrm{d}\xi \end{split}\label{3}\tag{3}$$

The regularity of the solution $u(x,t)$ and conditions for it to be a classical solution
The first thing to note is that the sine and cosine term in \eqref{2'} are bounded oscillating $O\big(|\xi|^{-1}\big)$ for $|\xi|\to\infty$ functions, thus their inverse Fourier transform is a function (of slow growth) (i.e. the Fourier integrals in \eqref{3} exist in classical sense). Then

• If $\Omega$ is bounded, $\alpha(x)|_{x\in\partial\Omega}=\beta(x)|_{x\in\partial\Omega}=0$ and $\alpha,\beta\in C^k(\Omega)$, $k\in\Bbb N$, by standard properties of distributions (see for example [2], chapter 3, §3.4 pp. 48-50) $u(x,\cdot)\in C^k(\Bbb R^n)$ for every fixed $t\in\Bbb R$. In particular, if $k=2$, $u(x,t)$ is a classical solution
• If $\Omega$ is unbounded, the regularity of $u(x,t)$ depends not only on the smoothness of initial data, but also on their behavior at infinity. Precisely, if apart from being $\alpha,\beta\in C^2(\Omega)$, we have also that their Fourier transform satisfies the following relation $$ |\xi|^2\hat{\alpha}(\xi),|\xi|^2\hat{\beta}(\xi)\in O\big(|\xi|^{-\varepsilon}\big)\iff\hat{\alpha},\hat{\beta}\in O\big(|\xi|^{-(2+\varepsilon)}\big)\label{g}\tag{G} $$ for $|\xi|\to\infty$ and $\varepsilon >0$, $u(x,t)$ is again a classical solution: and condition \eqref{g} is surely satisfied if $\alpha,\beta\to 0$ as $|x|\to \infty$ sufficiently fast.

Note on the structure of Fourier integrals \eqref{3}: differently from what happens for the standard wave equation I was not able to find an explicit expression of the inverse Fourier transforms which define $\mathrm{sp}(x)$ and $\mathrm{cp}(x)$, despite they are not distributions due the the above considerations: perhaps this could be an interesting question to ask.

Bibliography

[1] Lars Hörmander (1990), The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft, 256 (2nd ed.), Berlin-Heidelberg-New York: Springer-Verlag, ISBN 0-387-52343-X/ 3-540-52343-X, MR1065136, Zbl 0712.35001.

[2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.