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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?

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Consider a solution of the form

$$u(x,t) = \sum_{n=0}^\infty \frac{g^{(n)}(t)}{n!}x^n,$$

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.

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  • $\begingroup$ Great answer. I realise ive been thinking about these things naively.. of course conservation of energy only applies given an a-priori bound on the entire solution. It hadn't really occurred to me that it's not enough just to require finite energy initial data, but that without imposing the requirement of finite energy in the very definition of a solution you are not guaranteed uniqueness even if your initial data is regular (e.g. even for 0 initial data). I also see how this trick extends to the heat equation now. Thanks :) $\endgroup$ – Aerinmund Fagelson Sep 10 '16 at 16:45
  • $\begingroup$ 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? $\endgroup$ – Aerinmund Fagelson Jul 8 at 17:01

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