Global attractor of a contact parabolic problem in a thin two-layer domain

Journal Article · Sat Feb 28 04:00:00 EST 2004 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2004V195N01ABEH000795; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21260457

Rekalo, A M; Chueshov, I D ^[1]

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1. V.N. Karazin Kharkiv National University, Kharkiv (Ukraine)

A semilinear parabolic equation is considered in the union of two bounded thin cylindrical domains {omega}{sub 1,{epsilon}}={gamma}x(0,{epsilon}) and {omega}{sub 2,{epsilon}}={gamma}x(-{epsilon},0) adjoining along their bases, where {gamma} is a domain in R{sup d}, d{<=}3. The unknown functions are related by means of an interface condition on the common base {gamma}. This problem can serve as a reaction-diffusion model describing the behaviour of a system of two components interacting at the boundary. The intensity of the reaction is assumed to depend on {epsilon} and the thickness of the domains, and to be of order {epsilon}{sup {alpha}}. Under investigation are the limiting properties of the evolution semigroup S{sub {alpha}}{sub ,{epsilon}}(t), generated by the original problem as {epsilon}{yields}0 (that is, as the domain becomes ever thinner). These properties are shown to depend essentially on the exponent {alpha}. Depending on whether {alpha} is equal to, greater than, or smaller than 1, the original system can have three distinct systems of equations on {gamma} as its asymptotic limit. The continuity properties of the global attractor of the semigroup S{sub {alpha}}{sub ,{epsilon}}(t) as {epsilon}{yields}0 are established under natural assumptions.

• OSTI ID: 21260457
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 1 Vol. 195; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

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ASYMPTOTIC SOLUTIONS
ATTRACTORS
CYLINDRICAL CONFIGURATION
EQUATIONS
FUNCTIONS
GROUP THEORY
MATHEMATICAL EVOLUTION