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I am trying to show the ill-posedness of the problem $\partial_{t}u=i\Delta\bar{u}$, with $u(x,0)=u_{0}(x)$. The sugestion of my reference is differentiate the equation with respect to the variable $t$, and use the conjugate of the equation. Thus, we can conclude that if $u$ solve the problem above then $u$ is a solution of the problem $\partial_{t}^{2}u=\Delta^{2}u$, with the same initial condition. Under some assumptions, this last problem is ill-posed. However, there is not a equivalence between the two problems, because we can find a plane wave that is a solution for the last, but not solve the first problem. Thus, if someone can help me with this problem, i will be grateful.

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    What is the meaning of $\bar{u}$ in this question?2012-12-28

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