Global attractor of a contact parabolic problem in a thin two-layer domain
Andrey M. Rekalo, Igor Dmitrievich Chueshov · Sbornik Mathematics · 2004
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.