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Such an equation would represent an ill-posed problem for all positive time (i.e. the solution is not defined). The irreversibility of diffusive processes is closely related to the second law of thermodynamics.

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Q: How do you solve a diffusion equation with a negative diffusion coefficient?
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How do you solve a diffusion equation with a negative diffusion coefficient numerically It is unstable at all. Solving it with finite difference is best?

One common approach is using an implicit method (such as the Crank-Nicolson scheme) for numerical integration, as it is unconditionally stable. Another option is to use the exponential finite difference method, which can handle negative diffusion coefficients while ensuring stability. Additionally, modifying the equation to transform the negative diffusion coefficient into a positive one can also be effective for numerical stability.


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