The porous medium equation with variable exponent revisited

Abstract

The author presents a simplified proof for the local continuity of the weak solutions to the porous medium equation with variable positive bounded exponent \(\gamma (x,t)\)

$$\begin{aligned} u_t-\nabla \cdot \left( |u|^{\gamma (x,t)} \, \nabla u\right) =0 , \quad \mathrm {in} \quad \Omega _T =\Omega \times (0,T] , 0<T<\infty . \end{aligned}$$

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Correspondence to Eurica Henriques.

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The research of the author was partially financed by Portuguese Funds through FCT (Fundao para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020.

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Henriques, E. The porous medium equation with variable exponent revisited. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00632-8

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Keywords

  • Porous medium equation
  • Singular/degenerate PDE
  • Regularity theory
  • Intrinsic scaling

Mathematics Subject Classification

  • 35B65
  • 35K55
  • 35K65