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Recovering Initial States in Certain Quasilinear Parabolic Problems from Time Averages

Bogdan-Vasile Matioc, Christoph Walker

TL;DR

The paper addresses the inverse problem of recovering the initial state of a quasilinear parabolic system from time-averaged data by developing a fixed-point framework centered on the evolution-operator $U_A$ and the initial-encoding operator $Φ_A$. It proves that $Φ_A$ is invertible (under Fredholm index zero) and that the map $A\mapsto Φ_A$ is Lipschitz, enabling a contraction mapping in time-weighted interpolation spaces to recover $u(0)$ from $M$ for small $M$ on any fixed $T>0$. The results yield a unique solution with precise regularity and demonstrate applicability to chemotaxis models and quasilinear reaction-diffusion systems with nonlocal coefficients. This furnishes a practical method for reconstructing initial states from time-averaged measurements in nonlinear parabolic PDEs with nonlocal data, with concrete instances in biology and nonlinear diffusion.

Abstract

The inverse problem of reconstructing the initial state in quasilinear parabolic equations from time averages is investigated. Under suitable regularity assumptions on the quasilinear structure and a superlinear growth condition near zero for the semilinear part, it is shown that the initial state can be uniquely recovered from small time averages taken over an arbitrary time period. The applicability of the result is demonstrated for certain chemotaxis models and reaction-diffusion systems.

Recovering Initial States in Certain Quasilinear Parabolic Problems from Time Averages

TL;DR

The paper addresses the inverse problem of recovering the initial state of a quasilinear parabolic system from time-averaged data by developing a fixed-point framework centered on the evolution-operator and the initial-encoding operator . It proves that is invertible (under Fredholm index zero) and that the map is Lipschitz, enabling a contraction mapping in time-weighted interpolation spaces to recover from for small on any fixed . The results yield a unique solution with precise regularity and demonstrate applicability to chemotaxis models and quasilinear reaction-diffusion systems with nonlocal coefficients. This furnishes a practical method for reconstructing initial states from time-averaged measurements in nonlinear parabolic PDEs with nonlocal data, with concrete instances in biology and nonlinear diffusion.

Abstract

The inverse problem of reconstructing the initial state in quasilinear parabolic equations from time averages is investigated. Under suitable regularity assumptions on the quasilinear structure and a superlinear growth condition near zero for the semilinear part, it is shown that the initial state can be uniquely recovered from small time averages taken over an arbitrary time period. The applicability of the result is demonstrated for certain chemotaxis models and reaction-diffusion systems.
Paper Structure (11 sections, 16 theorems, 252 equations)

This paper contains 11 sections, 16 theorems, 252 equations.

Table of Contents

  1. Introduction and Main Result
  2. The Operator $\Phi$: Linear Theory
  3. Invertibility of $\Phi_A$
  4. Lipschitz Continuity of $\Phi_A$ with respect to $A$
  5. Proof of Theorem \ref{['MT1']}
  6. Preliminaries
  7. Proof of Theorem \ref{['MT1']}
  8. Applications
  9. A Chemotaxis System with Local Sensing
  10. Quasilinear Reaction-Diffusion Systems with Nonlocal Coefficients
  11. Quasilinear Equations with Other Nonlocal Conditions

Key Result

Theorem 1.1

Let $T>0$ and assume EQ:A and AA. Then, there exists $m_0>0$ such that EE has for each $M\in E_{1+\alpha}$ with $\|M\|_{E_{1+\alpha}}\le m_0$ a unique solution such that $\lim_{t\to 0^+}t^\mu\|u(t)\|_{E_\xi}=0$ forGiven $x\in\mathbb R$, we set $x_+:=\max\{0,x\}$.$(\xi-\alpha)_+<\mu< (\ell+1)^{-1}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 24 more