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RESEARCH PRODUCT

Nonlocal Heat Content

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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.