ABSTRACT
In this paper we consider the following parabolic-elliptic interface problem: Let Ω ∈ ℝ N, N ≥ 2, be a simply connected, bounded domain with boundary ∂Ω ∈ C 1, and I = [0, T]. For given nonnegative g = g(x,t,u): Ω × I × ℝ → ℝ we define Ω ℰ = Ω \ supp x ∈ Ω g ( x , t , u ) , Ω P = Ω \ Ω ℰ ¯ , Γ = ∂ Ω ℰ ∩ ∂ Ω P , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter_22_265_1.tif"/> and assume that Ω ε, Ω P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter22_265_1.tif"/> are independent of t and u. Then we look for a weak solution of g ( x , t , u ) u t + A ( t ) u = f ( x , t , u ) in Q T : = Ω × I , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqn_chapter_22_1.tif"/> u ℰ ( x , t ) = u P ( x , t ) , ∂ ν A u ℰ ( x , t ) = ∂ ν A u P ( x , t ) on Γ T : = Γ × I , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqn_chapter_22_2.tif"/> u ( x , t ) = 0 on B T : = ∂ Ω × I , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqn_chapter_22_3.tif"/> u ( x , 0 ) = U 0 ( x ) x ∈ Ω P , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqn_chapter_22_4.tif"/> where u P = u | Ω P , u ℰ = u | Ω ℰ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter22_265_2.tif"/> , and A ( t ) u = − ∑ i , k = 1 N ∂ ∂ x k ( a i k ( x , t ) ∂ u ∂ x i ) + a 0 ( x , t ) u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter22_265_3.tif"/> . Since g = 0 for x ∈ Ω ε the problem is elliptic on Ω ε × I and parabolic on Ω P × I https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter22_265_4.tif"/> .
