Semigroup Forum

, Volume 88, Issue 2, pp 479–511 | Cite as

Diffusion processes on an interval under linear moment conditions

  • Delio MugnoloEmail author
  • Serge Nicaise


We discuss a class of linear and nonlinear diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.


Nonlocal conditions for PDEs Porous medium equation Heat equation Subdifferentials 


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut für AnalysisUniversität UlmUlmGermany
  2. 2.LAMAV, FR CNRS 2956, ISTVUniversité de Valenciennes et du Hainaut CambrésisValenciennes Cedex 9France

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