Global Existence for The Conserved Phase Field Model with Memory and Quadratic Nonlinearity

P. Colli, G. Gilardi, M. Grasselli and G. Schimperna

Abstract: A nonlinear system for the heat diffusion inside a material subject to phase changes is considered. A thermal memory effect is assumed in the heat conduction law; moreover, on account of thermodynamical considerations, a linear growth is allowed for the latent heat density. The resulting problem couples a second order integrodifferential equation, derived from the balance of energy, with a fourth order parabolic inclusion which rules the evolution of an order parameter $\chi$. Homogeneous Neumann boundary conditions guarantee that the space average of $\chi$ is conserved in time. Global existence of solutions is proved in a variational setting.

Keywords: Conserved phase-field model; Integrodifferential evolution system; Heat transfer with memory; Maximal monotone graph; Bootstrap argument.

Classification (MSC2000): 35R99, 45K05, 80A22.

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