On the generalized boundary-value problem for linear Sobolev type equations on the geometric graph

On the geometric graph, where in addition to the continuity conditions and balance flow, condition of immobility is first introduced into the vertices of the graph, which is converted to a Dirichlet condition when the graph has one edge with two vertices. To solve this problem we first consider the corresponding Sturm–Liouville problem, and the results are then used to solve the Cauchy problem for two linear models, defined on the graph: Hoff equation and Barenblatt–Zheltov–Kochina equation. A feature of the work is the fact that on each edge of the graph given by the equation with different coefficients, which coupled with the introduction of vertices, is fixed for the first time in this problem.
Both models relate to Sobolev type equations, the study of which is experiencing an era of its heyday. Reduction of these equations to an abstract Sobolev type equation makes it possible to apply the method of degenerate semigroups of operators. The phase space of solutions is determined by the phase space method, which consists in reducing the singular equation to a regular equation defined on some subspace of the original space. The obtained results of theorems can be used in consideration of inverse problems, optimal control problems, the initial-end and multipoint problems, and also in consideration of stochastic equations for the models set in a geometric graph.