Classical solution to the first mixed problem for a mildly quasilinear wave equation: a fixed-point approach
https://doi.org/10.67268/1812-5093-2026-34-1-85-95
EDN: DCQCNW
Abstract
For a one-dimensional mildly quasilinear wave equation given in the first quadrant, we consider a mixed problem in which Cauchy conditions are specified on the spatial semi-axis and a Dirichlet condition is specified on the temporal semi-axis. The nonlinearity contains independent variables, the unknown function, and its derivatives. We construct the solution in implicit analytical form as the solution of some integro-differential equations. We prove the solvability of these integro-differential equations using a generalization of the Banach fixed-point theorem. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists
Keywords
mildly quasilinear wave equation,
mixed problem,
classical solution,
fixed-point principle,
matching conditions.
Supporting agencies: The work was supported by state program for scientific research “Interdisciplinary and Synergetic Research” (“Convergence-2030”), subprogram “Modern Mathematical Methods and Their Applications”, task 1.05 “Classical Solutions, Development of New Methods for Studying Problems of the Theory of Partial Differential Equations”, R&D 1.05.1 “Classical Methods for Solving and Proving the Hadamard Well-Posedness of Problems for Partial Differential Equations”.
About the Author
J. V. Rudzko
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
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For citations:
Rudzko J.V. Classical solution to the first mixed problem for a mildly quasilinear wave equation: a fixed-point approach. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2026;34(1):85-95. https://doi.org/10.67268/1812-5093-2026-34-1-85-95. EDN: DCQCNW
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