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Observation estimates for a semilinear heat equation in \mathbb{R}^n

Guojie Zheng, Xin Yu

TL;DR

The paper addresses state observation for the semilinear heat equation $\partial_t y-\Delta y+f(y)=0$ on $\mathbb{R}^n$, establishing a global interpolation inequality that links final-time energy to initial data and partial observations in a thick region $\omega$. It introduces a local frequency-function framework and logarithmic convexity arguments to derive a local-to-global estimate, enabling reconstruction of the full state from partial measurements and yielding a unique continuation result. The authors prove forward-problem well-posedness under varying data regularity, and extend the observation results to a time-shifted setting to obtain a robust conditional stability bound, with proofs grounded in energy estimates and fixed-point techniques. Overall, the work provides a rigorous observability and stability framework for nonlinear parabolic dynamics driven by semilinear sources, with implications for control and inverse problems in unbounded domains.

Abstract

This paper studies the state observation problems for the semilinear heat equation in R^n. We derive observation estimates for the equation using the logarithmic convexity property of the frequency function (see [12]). As an application, we show that if two solutions coincide on a nonempty open subset ω\subsetΩat some time T>0, then they must be identical.

Observation estimates for a semilinear heat equation in \mathbb{R}^n

TL;DR

The paper addresses state observation for the semilinear heat equation on , establishing a global interpolation inequality that links final-time energy to initial data and partial observations in a thick region . It introduces a local frequency-function framework and logarithmic convexity arguments to derive a local-to-global estimate, enabling reconstruction of the full state from partial measurements and yielding a unique continuation result. The authors prove forward-problem well-posedness under varying data regularity, and extend the observation results to a time-shifted setting to obtain a robust conditional stability bound, with proofs grounded in energy estimates and fixed-point techniques. Overall, the work provides a rigorous observability and stability framework for nonlinear parabolic dynamics driven by semilinear sources, with implications for control and inverse problems in unbounded domains.

Abstract

This paper studies the state observation problems for the semilinear heat equation in R^n. We derive observation estimates for the equation using the logarithmic convexity property of the frequency function (see [12]). As an application, we show that if two solutions coincide on a nonempty open subset ω\subsetΩat some time T>0, then they must be identical.

Paper Structure

This paper contains 10 sections, 12 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of the system (\ref{['1.1']})
  3. A local interpolation inequality
  4. Proof of the main results
  5. Proof of Theorem \ref{['theorem1']}
  6. Proof of Theorem \ref{['theorem2']}
  7. Appendix A
  8. Proof of Theorem \ref{['theorem3.1']}
  9. Proof of Lemma \ref{['lemma2.3']}
  10. Proof of Theorem \ref{['theorem3.2']}

Key Result

Theorem 1.1

Suppose that $(A_1)$ and $(A_2)$ hold. Let $y_i^0\in L^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$$(i=1,2)$ be the initial data and let $y_i \in L^\infty(0,T;L^\infty(\mathbb{R}^n))$$(i=1,2)$ be the solution to the system (1.1) with the initial value $y_i^0$. Let $L_{M}>0$ be the Lipschitz constant Then, we obtain the following estimates: $(i)$ There exist two positive numbers $\beta=\beta(L, r)\

Theorems & Definitions (26)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.2
  • Theorem 3.1
  • ...and 16 more