ABSTRACT
In the framework of a recent joint research we have proposed and compared some phase change models in materials with memory (see [5-10] and references therein) by studying questions like existence, uniqueness, and asymptotic properties for the solutions of the resulting initial and boundary value problems. One of these problems, which could be termed as parabolic Stefan problem with memory, consists in finding a couple of functions, the (relative) temperature d and the phase variable χving the integrodifferentia! equatibn https://www.w3.org/1998/Math/MathML"> ∂ t ( α 0 ϑ + β 0 χ + α * ϑ + β * χ ) - Δ ( k 0 ϑ + k * ϑ ) = f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1g.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and satisfying the graph relationship https://www.w3.org/1998/Math/MathML"> χ = 0 if ϑ < 0 , χ ∈ [ 0 , 1 ] if ϑ = 0 , χ = if ϑ > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1h.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in the cylindrical domain https://www.w3.org/1998/Math/MathML"> Ω × ] 0 , T [ , Ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math210.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> being a smooth bounded open set in R3 and T > 0 standing for a final time. Regarding (1.1), we should specify that https://www.w3.org/1998/Math/MathML"> ∂ t ≔ ∂ / ∂ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math211.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the symbol “ * ” denotes the usual convolution product with respect to time over [0,t] (t ∈ [0, T]), and △ is nothing but the Laplacian acting on the space variable. Moreover, the data α0, β0, k0 are positive constants, α, β, k: ]0, T[→ R represent memory functions, and the right hand side https://www.w3.org/1998/Math/MathML"> f : Ω × ] 0 , T [ → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math212.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> accounts for the heat source and for the past histories of temperature and phase proportions up to t = 0. The system (1.1-2) is complemented by the initial condition https://www.w3.org/1998/Math/MathML"> ( α 0 ϑ + β 0 χ ) ( ⋅ , 0 ) = η 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_3a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where η0: Ω → R provides a sort of initial energy, and by a suitable boundary condition. For instance, letting {Г0, ГV} be a partition of the boundary https://www.w3.org/1998/Math/MathML"> ∂ Ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math213.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , into two measurable subsets, one can take https://www.w3.org/1998/Math/MathML"> ϑ = 0 on Γ 0 × ] 0 , T [ , ∂ ν ( k 0 ϑ + k * ϑ ) = g o n Γ v × ] 0 , T [ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_4a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 83with g: https://www.w3.org/1998/Math/MathML"> Γ ν × ] 0 , T [ → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math214.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> known function depending on the normal heat flux (cf, [5]). Here ∂ v indicates the outward normal derivative on ∂Ω and Г v is allowed to coincide with ∂Ω. Thus, the problem (1.1-4) describes a phase transition process influenced by what has already occurred during the evolution. This memory effect is governed by the relaxation kernels α, β, k. Note that if α = β = k ≡, then (1.1-4) reduces to a usual two-phase Stefan problem (see, e.g., [11]).
