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I am trying to solve a equation that looks like $$u_t=f(x)u_x+Du_{xx}$$ any insight to whether an solution might exist would be very much appreciated!

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    $\begingroup$ for an advection-diffusion equation you need $[f(x)u]_x$, not $f(x)u_x$; an overview of existence conditions is given, for example, in doi.org/10.1155/2014/450417 $\endgroup$
    – Carlo Beenakker
    Commented Jan 12 at 17:59

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