Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Abstract

We consider a semi-linear parabolic problem in a thick junction Ωε, which is the union of a domain Ω0 and a lot of joined thin trees situated ε-periodically along some manifold on the boundary of Ω0. The trees have finite number of branching levels. The following nonlinear Robin boundary condition ∂νvε+εαiμ(t,x2,vε)=εβgε is given on the boundaries of the branches from the i-th branching layer; {αi} and β are real parameters. The asymptotic analysis of this problem is made as ε→0, i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution vε and prove the corresponding asymptotic estimate in the space C([0,T];L2(Ωε))∩L2(0,T;H1(Ωε)), which shows the influence of the parameters {αi} and β on the asymptotic behavior of the solution.