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I'm studying differential operators. For example, in the evolution equation

\begin{align} u_t&=(1-\partial_x^2)u\\ u(0)&=u_0 \end{align}

Question 1. Does this problem have any name in the literature?

the symbol of the operator $(1-\partial_x^2)$ is $(1+\xi^2)$. Now, the operator $(1+x^2)(1-\partial_x^2)$ has the symbol $(1+x^2)(1+\xi^2)$.

Question 2. Is there some problem as \begin{align} u_t&=(1+x^2)(1-\partial_x^2)u\\ u(0)&=u_0 \end{align} or this has non sense?

Thanks

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  • $\begingroup$ 1a. It is a backwards heat equation with source term, which is ill-posed, and so of limited use. 1b. I don’t think it makes sense to have a Fourier(?) symbol of an operator that has variable coefficients. 2. It is a variable coefficient backward heat equation that is ill-posed and thus of limited use. $\endgroup$
    – messenger
    Commented Jun 29 at 8:34

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