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- Title
GLOBAL SOLVABILITY OF A MIXED PROBLEM FOR A SINGULAR SEMILINEAR HYPERBOLIC 1D SYSTEM.
- Authors
KYRYLYCH, V. M.; PELIUSHKEVYCH, O. V.
- Abstract
Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle Π = {(x, t): 0 < x < l < ∞, 0 < t < T < ∞} for the first order hyperbolic system of semi-linear equations of the form... for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characteristics and with nonlinear boundary conditions, where Λ(x, t) = diag(λ1(x, t),. . ., λk(x, t)), u = (u1, . . ., uk), v = (v1, . . ., vm), w = (w1, . . ., wn), f = (f1, . . ., fk), g = (g1, . . ., gm), h = (h1, . . ., hn) and besides sign λi(0, t) = const ̸= 0, sign λi(l, t) = const ̸= 0 for all t ∈ [0, T] and for all i ∈ {1, . . ., k}. The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems.
- Subjects
HYPERBOLIC differential equations; FIXED point theory; HYPERBOLIC functions; UNIQUENESS (Mathematics); NONLINEAR boundary value problems; CONTINUOUS functions; ORTHOGONAL systems
- Publication
Matematychni Studii, 2024, Vol 61, Issue 2, p188
- ISSN
1027-4634
- Publication type
Article
- DOI
10.30970/ms.61.2.188-194