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Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase‐field model
Author(s) -
Colombo Fabrizio,
Guidetti Davide,
Vespri Vincenzo
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.658
Subject(s) - mathematics , uniqueness , field (mathematics) , mathematical analysis , inverse problem , inverse , heat equation , banach space , phase space , boundary (topology) , parabolic partial differential equation , phase (matter) , principal part , partial differential equation , pure mathematics , geometry , physics , quantum mechanics , thermodynamics
Abstract In this paper we consider a system of two integro‐differential evolution equations coming from a conservative phase‐field model in which the principal part of the elliptic operators, involved in the two evolution equations, have different orders. The inverse problem consists in finding the evolution of: the temperature, the phase‐field, the two memory kernels and the time dependence of the heat source when we suppose to know additional measurements of the temperature on some part of the body Ω. Our results are set within the framework of Hölder continuous function spaces with values in a Banach space X . We prove that the inverse problem admits a local in time solution, but we are also able to prove a global in time uniqueness result. Our setting, when we choose for example X = C ( $\bar{\Omega}$ ), allows us to make additional measurements of the temperature on the boundary of the body Ω. Copyright © 2005 John Wiley & Sons, Ltd.