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Inverse Source Problem for a Nonlinear Parabolic Equation via Paralinearization

Hu Xirui

TL;DR

This work studies an inverse source problem for a nonlinear parabolic equation with a quadratic reaction and a Burgers-type convection term, using partial boundary measurements of $\partial_t u$ and $\nabla\partial_t u$ and an interior snapshot $u(\cdot,t_0)$. The main approach is to paralinearize the nonlinear term via Besov/Littlewood–Paley techniques on a short time window, turning the nonlinearity into small bounded coefficients, and then differentiate in time to obtain a linearized equation for $z=\partial_t u$ that fits a Carleman-estimate framework. The key contribution is showing that the nonlinear inverse source problem inherits the same conditional Hölder–logarithmic stability as the corresponding linear problem under the same observation geometry and a smallness/short-time regime. This provides a robust template for handling semilinear/parabolic inverse problems by combining paralinearization with Carleman techniques, potentially extendable to broader nonlinearities and quasilinear settings.

Abstract

This thesis investigates an inverse source problem for a semilinear parabolic equation that includes both a quadratic reaction and a convection-type nonlinearity depending on the unknown function and its gradient. Based on partial boundary observations of the time derivative and its spatial gradient on a nonempty portion of the boundary, together with interior data at a fixed time, the goal is to recover an unknown spatial source term. On a short time interval, the nonlinear part is treated by Bony paralinearization technique, which transforms the original nonlinear problem into a linear parabolic equation with small, bounded coefficients and a negligible remainder term. By differentiating the equation with respect to time, a linearized model for the time derivative of the solution is obtained, which is suitable for the application of Carleman estimates. Using these tools, the work establishes conditional stability of Holder-logarithmic type for the recovery of the source term under the same observation geometry as in the corresponding linear case. The analysis combines techniques from Littlewood-Paley theory, Besov space estimates, and Carleman inequalities, showing that the stability structure of the linear inverse source problem extends naturally to the nonlinear setting under smallness and short-time assumptions.

Inverse Source Problem for a Nonlinear Parabolic Equation via Paralinearization

TL;DR

This work studies an inverse source problem for a nonlinear parabolic equation with a quadratic reaction and a Burgers-type convection term, using partial boundary measurements of and and an interior snapshot . The main approach is to paralinearize the nonlinear term via Besov/Littlewood–Paley techniques on a short time window, turning the nonlinearity into small bounded coefficients, and then differentiate in time to obtain a linearized equation for that fits a Carleman-estimate framework. The key contribution is showing that the nonlinear inverse source problem inherits the same conditional Hölder–logarithmic stability as the corresponding linear problem under the same observation geometry and a smallness/short-time regime. This provides a robust template for handling semilinear/parabolic inverse problems by combining paralinearization with Carleman techniques, potentially extendable to broader nonlinearities and quasilinear settings.

Abstract

This thesis investigates an inverse source problem for a semilinear parabolic equation that includes both a quadratic reaction and a convection-type nonlinearity depending on the unknown function and its gradient. Based on partial boundary observations of the time derivative and its spatial gradient on a nonempty portion of the boundary, together with interior data at a fixed time, the goal is to recover an unknown spatial source term. On a short time interval, the nonlinear part is treated by Bony paralinearization technique, which transforms the original nonlinear problem into a linear parabolic equation with small, bounded coefficients and a negligible remainder term. By differentiating the equation with respect to time, a linearized model for the time derivative of the solution is obtained, which is suitable for the application of Carleman estimates. Using these tools, the work establishes conditional stability of Holder-logarithmic type for the recovery of the source term under the same observation geometry as in the corresponding linear case. The analysis combines techniques from Littlewood-Paley theory, Besov space estimates, and Carleman inequalities, showing that the stability structure of the linear inverse source problem extends naturally to the nonlinear setting under smallness and short-time assumptions.

Paper Structure

This paper contains 27 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction and main results
  2. Mathematical model
  3. Previous works on related parabolic inverse source problems
  4. Linear models.
  5. Semilinear or quasilinear variants.
  6. Analytical tools.
  7. Motivation and novelty
  8. Main results
  9. Organization of the paper
  10. Analytical framework
  11. Littlewood--Paley blocks and Besov norms
  12. Bernstein inequalities
  13. Bony decomposition and paraproduct bounds
  14. Proof of the paraproduct bound.
  15. Paralinearization of the nonlinear term
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $u$ satisfy eq:P--eq:bc--eq:NL-concrete with the regularity eq:Besov-reg, and assume the observation geometry described above. Then there exist constants $C>0$, $\kappa\in(0,1)$, and a small $\delta>0$ such that for $I=(t_0-\delta,t_0+\delta)$, for any $\Omega_0\Subset\Omega\cup\Gamma$ with $\partial\Omega_0\cap\partial\Omega\subset\Gamma$. The constants depend only on $\Omega$, $\Gamma$, the

Theorems & Definitions (7)

  • Theorem 1.1: Stability for the nonlinear inverse source problem
  • Remark 1.2
  • Proposition 3.1: Parabolic energy on a short window
  • proof : Proof
  • Corollary 3.2: Closed a priori bound
  • Lemma 5.1: Carleman inequality with small perturbations
  • proof