Reduced problems are elliptic problems with a large parameter (as the spectral parameter) given by the Laplace transform of time dependent problems. In this paper, asymptotic behavior of the solutions of the reduced problem for the classical heat equation in bounded domains with the inhomogeneous Robin type conditions is discussed. The boundary of the domain consists of two disjoint surfaces, outside one and inside one. When there are inhomogeneous Robin type data at both boundaries, it is shown that asymptotics of the value of the solution with respect to the large parameter at a given point inside the domain is closely connected to the distance from the point to the both boundaries. It is also shown that if the inside boundary is strictly convex and the data therein vanish, then the asymptotics is different from the previous one. The method for the proof employs a representation of the solution via single layer potentials. It is based on some non trivial estimates on the integral kernels of related integral equations which are previously established and used in studying an inverse problem for the heat equation via the enclosure method.
"Asymptotic behavior of the solutions for the Laplace equation with a large spectral parameter and the inhomogeneous Robin type conditions." Osaka J. Math. 55 (1) 117 - 163, January 2018.