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Nonlocal Heat Content

  • José M. Mazón
  • Julio Daniel Rossi
  • J. Julián Toledo
Chapter
Part of the Frontiers in Mathematics book series (FM)

Abstract

The heat content of a Borel measurable set \(D \subset \mathbb {R}^N\) at time t is defined by M. van der Berg in [69] (see also [70]) as:
$$\displaystyle \mathbb {H}_D(t) = \int _D T(t) {\chi }_D (x) dx, $$
with (T(t))t≥0 being the heat semigroup in \(L^2(\mathbb {R}^N)\). Therefore, the heat content represents the amount of heat in D at time t if in D the initial temperature is 1 and in \(\mathbb {R}^N \setminus D\) the initial temperature is 0.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • José M. Mazón
    • 1
  • Julio Daniel Rossi
    • 2
  • J. Julián Toledo
    • 3
  1. 1.Departamento de Análisis MatemáticoUniversitat de ValènciaValenciaSpain
  2. 2.Departamento de MatemáticasUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Departamento de Análisis MatemáticoUniversitat de ValènciaValènciaSpain

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