ABSTRACT
We consider the following nonlinear system: https://www.w3.org/1998/Math/MathML"> ∂ ρ ( u ) ∂ t + ∂ w ∂ t - Δ u = f ( t , x ) in Q ≔ ( 0 , + ∞ , ) × Ω , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_1b.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ν ∂ w ∂ t + β ( w ) + g ( w ) ∋ u in Q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_2c.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with lateral boundary condition: https://www.w3.org/1998/Math/MathML"> ∂ u ∂ n + α N ( x ) u = h N ( t , x ) on Σ ≔ ( 0 , + ∞ ) × Γ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_3d.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and initial conditions: https://www.w3.org/1998/Math/MathML"> u ( 0 , ⋅ ) = u 0 , w ( 0 , ⋅ ) = w 0 i n Ω . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math1_4d.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Here Ω is a bounded domain in RN (N ≥ 1) with smooth boundary https://www.w3.org/1998/Math/MathML"> Γ ≔ ∂ 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-math706.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ρ is a monotone increasing and bi-Lipschitz continuous function on R; v is a positive constant; β is a maximal monotone graph in R × R; g is a smooth function defined on R; α N is a non-negative, bounded and measurable function on Γ such that α N > 0 on a subset of Γ with positive measure; f, hN ,u 0 and w 0 are given data.
