Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics

Michele Colturato

doi:10.3934/eect.2018011

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2018, Volume 7, Issue 2: 217-245. Doi: 10.3934/eect.2018011

This issue Previous Article The recovery of a parabolic equation from measurements at a single point Next Article Robust Stackelberg controllability for linear and semilinear heat equations

Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics

Michele Colturato^,

Dipartimento di Matematica, Universitá di Pavia, Via Ferrata 5, Pavia, PV 27100, Italy

* Corresponding author: Michele Colturato
* Corresponding author: Michele Colturato

Received: November 2017

Revised: December 2017

Published: May 2018

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Abstract

We consider a singular phase field system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic nonlinearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.

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Mathematics Subject Classification: Primary: 35K61, 35K20, 35D30; Secondary: 80A22.

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