# Nonuniqueness of solutions to the wave equation for infinite energy

Given initial data lying in $$\dot{H}^1(\mathbb{R}^n)$$, one can prove uniqueness of solutions to the wave equation $$\Box u =0$$ through conservation of energy

i.e. $$E'(t)=0$$ where $$E(t)=\frac{1}{2}\int_{\mathbb{R}^n} ( |\nabla u (t) |^2 + |u_t (t)|^2 ) dx$$,

which implies that the only solution for zero initial data is the trivial one, and hence if two solutions $$u$$,$$\tilde{u}$$ have the same initial data, as their difference $$u-\tilde{u}$$ is a solution with zero initial data, it follows that $$u-\tilde{u}=0$$.

I have been told that uniqueness of solutions fails when the initial energy is allowed to be infinite, but cannot find an example of this. Can anyone give an example (or reference) for failure of uniqueness of solutions?

$$u(x,t) = \sum_{n=0}^\infty \frac{g^{(n)}(t)}{n!}x^n,$$
where $g$ is the bump function
$$g(t) = \begin{cases} e^{-\frac{1}{t^2}},&\text{if } t>0\\ 0,&\text{otherwise.}\end{cases}$$
Then $u(x,0)=0$ and $u_t(x,0)=0$, but $u(x,t)$ has infinite energy for $t>0$. Basically it starts with zero energy, and an infinite amount of energy arrives from $\pm\infty$ in finite time.
• I have recently been having some problems with this solution, can we prove that $u(x,t)$ is a smooth solution? Can we even prove that it is continuous? Jul 8, 2019 at 17:01