## Porous medium equation and fast diffusion equation as gradient systems.(English)Zbl 1363.35083

Summary: We show that the porous medium equation and the fast diffusion equation, $$\dot u-\Delta u^m=f$$, with $$m\in (0,\infty)$$, can be modeled as a gradient system in the Hilbert space $$H^{-1}(\Omega)$$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $$\Omega \subseteq \mathbb{R}^n$$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

### MSC:

 35G25 Initial value problems for nonlinear higher-order PDEs 47J35 Nonlinear evolution equations 47H99 Nonlinear operators and their properties 34G20 Nonlinear differential equations in abstract spaces
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### References:

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