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Consider an infinite string with the initial condition $$u_{tt}=c^2 u_{xx},~~x\in R,~~t\geq 0$$ $$u(x,0)=f(x), ~~x\in R$$ $$u_t(x,0)=g(x),~~x\in R.$$ The solution is given by $$u(x,t)=\frac{1}{2}(f(x-ct)+f(x+ct))+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds.$$ This solution is unique if $f$ is twice continuously differentiable function and $g$ is continuously differentiable function.

Question: I want an example such that $u(x,t)$ is not unique, that is, an example with two different solution satisfying the given wave equation and its initial conditions.

Attempt: Choose $g=0$ and $f(x)=|x|$, which is non-differentiable function. Suppose my wave equation is $$u_{tt}=u_{xx}.$$ So, the solution is given by $$u(x,t)=\frac{1}{2}(|x-t|+|x+t|).$$ This is one of the solution. Now, what is the another solution in this case?

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There is only one (unique) solution for the example you have taken as it is a well posed problem with well defined initial data. In fact, for the general problem that you have considered, the solution when it exists, is unique. It is a direct consequence of the energy method. See this discussion for a proof.

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