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RESEARCH PRODUCT

The Heat Content for Nonlocal Diffusion with Non-singular Kernels

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Abstract We study the behavior of the heat content for a nonlocal evolution problem.We obtain an asymptotic expansion for the heat content of a set D, defined as ℍ D J ⁢ ( t ) := ∫ D u ⁢ ( x , t ) ⁢ 𝑑 x ${\mathbb{H}_{D}^{J}(t):=\int_{D}u(x,t)\,dx}$ , with u being the solution to u t = J ∗ u - u ${u_{t}=J\ast u-u}$ withinitial condition u 0 = χ D ${u_{0}=\chi_{D}}$ . This expansion is given in terms of geometric values of D. As a consequence, we obtain that ℍ D J ⁢ ( t ) = | D | - P J ⁢ ( D ) ⁢ t + o ⁢ ( t ) ${\mathbb{H}^{J}_{D}(t)=\lvert D\rvert-P_{J}(D)t+o(t)}$ as t ↓ 0 ${t\downarrow 0}$ .We also recover the usual heat content for the heat equation when we rescale the kernel J in an appropriate way.Finally, we also find an asymptotic expansion for the nonlocal analogous to the spectral heat content that is defined as before but considering u ⁢ ( x , t ) ${u(x,t)}$ a solution to the equation u t = J ∗ u - u ${u_{t}=J\ast u-u}$ inside D with u = 0 ${u=0}$ in ℝ N ∖ D ${\mathbb{R}^{N}\setminus D}$ andinitial condition u 0 = χ D ${u_{0}=\chi_{D}}$ .