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Inverse problems with integral conditions for the generalized Korteweg-de Vries equation

Oleg S. Balashov, Andrei V. Faminskii

TL;DR

The paper studies inverse problems for the generalized Korteweg-de Vries equation on a bounded interval with integral overdetermination, recovering unknown forcing and/or boundary data without growth restrictions on the nonlinearity by leveraging time-regular function spaces. It first develops solvability for linear inverse problems and then proves nonlinear well-posedness via fixed-point arguments under small data or short-time conditions. Key contributions include existence, uniqueness, and Lipschitz continuous dependence of the solution and recovered quantities, along with explicit reconstruction operators. The work extends inverse-problem theory for KdV-type equations on bounded domains and provides a framework that avoids growth restrictions on the nonlinearity through temporal regularity.

Abstract

Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the right-hand side of equation or the boundary data, or both are chosen as controls. The considered solutions are regular with respect to the time variable. Assumptions on smallness of the input data or smallness of a time interval are required.

Inverse problems with integral conditions for the generalized Korteweg-de Vries equation

TL;DR

The paper studies inverse problems for the generalized Korteweg-de Vries equation on a bounded interval with integral overdetermination, recovering unknown forcing and/or boundary data without growth restrictions on the nonlinearity by leveraging time-regular function spaces. It first develops solvability for linear inverse problems and then proves nonlinear well-posedness via fixed-point arguments under small data or short-time conditions. Key contributions include existence, uniqueness, and Lipschitz continuous dependence of the solution and recovered quantities, along with explicit reconstruction operators. The work extends inverse-problem theory for KdV-type equations on bounded domains and provides a framework that avoids growth restrictions on the nonlinearity through temporal regularity.

Abstract

Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the right-hand side of equation or the boundary data, or both are chosen as controls. The considered solutions are regular with respect to the time variable. Assumptions on smallness of the input data or smallness of a time interval are required.

Paper Structure

This paper contains 5 sections, 17 theorems, 160 equations.

Table of Contents

  1. Introduction. Notation. Description of main results
  2. Auxiliary nonlinear estimates
  3. Auxiliary linear problem
  4. Linear inverse problems
  5. Nonlinear results

Key Result

Theorem 1.1

Let Denote Then either for the fixed $\delta>0$ there exists $T_0>0$ or for the fixed $T_0>0$ there exists $\delta>0$ such that in both cases if $c_1\leq \delta$ and $T\in (0,T_0]$, there exist a unique pair of functions $(F, \nu_1)$, satisfying $F\in \widetilde{H}^k(0,T)$, $\nu_1 \in H^k(0,T)$, $\nu_1^{(m)}(0 is Lipschitz continuous on the corresponding subset of the closed ball of the radius $

Theorems & Definitions (41)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • ...and 31 more