: We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

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In this paper, we consider the inverse problem for semilinear ultraparabolic equation. The equation has two unknown functions of different arguments in its minor coefficient and in right-hand side function. The sufficient conditions of the existence and the uniqueness of solution on some interval $[0,T],$ where $T$ depends on the coefficients of the equation, are obtained.