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Generalized Solutions to Hyperbolic Systems with Nonlinear Conditions and Strongly Singular Data

Irina Kmit

TL;DR

The paper develops global generalized solvability results for semilinear hyperbolic systems with nonlinear nonlocal boundary conditions in the presence of strongly singular data, using the Colombeau algebra ${\cal G}(\overline\Pi)$. It first establishes existence-uniqueness for Lipshitz-type nonlinearities within this framework, leveraging characteristic-based integral formulations and a priori estimates to lift smooth solutions to Colombeau representatives. It then extends the results to non-Lipshitz nonlinearities by introducing growth-control assumptions and a refined moderateness concept, enabling solvability even with highly singular initial data and boundaries. A key achievement is handling strongly singular coefficients and nonlinear boundary terms without requiring global boundedness of the coefficients, broadening the applicability to physical models with singular data. Overall, the work provides a robust method for proving global solvability of hyperbolic systems under severe singularities in both the equations and the boundary data.

Abstract

Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions. Our analysis covers the case of non-Lipshitz nonlinearities both in the differential equations and in the boundary conditions.

Generalized Solutions to Hyperbolic Systems with Nonlinear Conditions and Strongly Singular Data

TL;DR

The paper develops global generalized solvability results for semilinear hyperbolic systems with nonlinear nonlocal boundary conditions in the presence of strongly singular data, using the Colombeau algebra . It first establishes existence-uniqueness for Lipshitz-type nonlinearities within this framework, leveraging characteristic-based integral formulations and a priori estimates to lift smooth solutions to Colombeau representatives. It then extends the results to non-Lipshitz nonlinearities by introducing growth-control assumptions and a refined moderateness concept, enabling solvability even with highly singular initial data and boundaries. A key achievement is handling strongly singular coefficients and nonlinear boundary terms without requiring global boundedness of the coefficients, broadening the applicability to physical models with singular data. Overall, the work provides a robust method for proving global solvability of hyperbolic systems under severe singularities in both the equations and the boundary data.

Abstract

Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions. Our analysis covers the case of non-Lipshitz nonlinearities both in the differential equations and in the boundary conditions.

Paper Structure

This paper contains 5 sections, 3 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence-uniqueness results in the Colombeau algebra of generalized functions
  4. Colombeau Lipshitz nonlinearities
  5. Non-Lipshitz nonlinearities

Key Result

Theorem 1

Let U\in{\cal G}(\Omega) (U\in{\cal G}(\overline\Omega)). Then the following two conditions are equivalent: (i) U is invertible in {\cal G}(\Omega) (in {\cal G}(\overline\Omega)), i.e., there exists V\in{\cal G}(\Omega) (V\in{\cal G}(\overline\Omega)) such that UV=1 in {\cal G}(\Omega) (in {\cal G}(

Theorems & Definitions (6)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Example 5
  • Remark 6