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The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions

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Abstract

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign-changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles—the well-known solutions of the classical porous medium equation.

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Author information

Correspondence to Cyril Imbert.

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Communicated by L. Saint-Raymond

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Biler, P., Imbert, C. & Karch, G. The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions. Arch Rational Mech Anal 215, 497–529 (2015). https://doi.org/10.1007/s00205-014-0786-1

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Keywords

  • Weak Solution
  • Mild Solution
  • Decay Estimate
  • Differential Inequality
  • Fourier Multiplier