Semilinear Elliptic Equations on Star Graph with Small Edges

Konyrkulzhayeva M.N. Borisov D.I.
December 2025 Pleiades Publishing

Russian Journal of Mathematical Physics
2025 #32 Issue 4 634 - 661 pp.

Abstract: We consider a second order semilinear elliptic equation on a star graph with finitely many edges, some of which are supposed to have small lengths proportional to a small parameter At the common vertex, a general boundary condition is imposed, while at the boundary vertices, we impose the Dirichlet or Robin or Neumann condition. The nonlinearity in the equation depends only on the unknown function and variable and is supposed to satisfy a Lipschitz condition as well as a certain monotonicity condition. In addition, the nonlinearity on the small edges is multiplied by The main results of the paper establish the unique solvability of the considered problem and describe the convergence and asymptotic expansions for the solution as goes to zero. The limiting problem is considered on the graph without small edges and involves a certain limiting boundary condition at the common vertex. This limiting boundary condition depends in a nontrivial nonlinear way on the nonlinearities in the original equation on the small edges. The convergence is established uniformly in -norm of the right-hand side in the equation. We also construct the asymptotic expansion for the solution of perturbed problem up to an arbitrary power of and estimate the remainders in the asymptotics uniformly in -norm of the right-hand side in the equation.

Text of the article Перейти на текст статьи

International Information Technology University, Almaty, Kazakhstan
Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russian Federation
Peoples Friendship University of Russia (RUDN University), Moscow, Russian Federation
Bashkir State Pedagogical University named after M. Akhmulla, Ufa, Russian Federation

International Information Technology University
Institute of Mathematics
Peoples Friendship University of Russia (RUDN University)
Bashkir State Pedagogical University named after M. Akhmulla

10 лет помогаем публиковать статьи Международный издатель

Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026