The Heat Content for Nonlocal Diffusion with Non-singular Kernels
José M. Mazón, Julio Daniel Rossi, Julián Toledo · Advanced Nonlinear Studies · 2017
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 )