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Strongly Singular Nonlocal Kirchhoff-Type Equations with Variable Exponents: Existence, Regularity, and Renormalized Solutions

M. H. M. Rashid

Abstract

This work resolves the open problem of strong singularity ($α(z)> 1$) in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with weighted Sobolev spaces and existence via truncation, we develop comparison principles, optimal regularity results, and when classical solutions cease to exist, the construction of renormalized solutions. Building upon these foundations, we establish three advanced results: optimal convergence of truncated sequences to renormalized solutions, refined energy estimates characterizing asymptotic behavior as the truncation parameter vanishes, and a quantitative comparison principle yielding sharp pointwise bounds. Subsequently, we derive sharp two-sided pointwise estimates, a uniqueness theorem with quantitative stability, and Lipschitz continuous dependence of solutions on parameters and boundary data. Each theorem is supported by rigorous proofs employing nonlinear analysis, variational methods, and elliptic regularity theory. A computational illustration visualizes the solution behavior near the boundary and demonstrates convergence of truncated approximations.

Strongly Singular Nonlocal Kirchhoff-Type Equations with Variable Exponents: Existence, Regularity, and Renormalized Solutions

Abstract

This work resolves the open problem of strong singularity () in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with weighted Sobolev spaces and existence via truncation, we develop comparison principles, optimal regularity results, and when classical solutions cease to exist, the construction of renormalized solutions. Building upon these foundations, we establish three advanced results: optimal convergence of truncated sequences to renormalized solutions, refined energy estimates characterizing asymptotic behavior as the truncation parameter vanishes, and a quantitative comparison principle yielding sharp pointwise bounds. Subsequently, we derive sharp two-sided pointwise estimates, a uniqueness theorem with quantitative stability, and Lipschitz continuous dependence of solutions on parameters and boundary data. Each theorem is supported by rigorous proofs employing nonlinear analysis, variational methods, and elliptic regularity theory. A computational illustration visualizes the solution behavior near the boundary and demonstrates convergence of truncated approximations.

Paper Structure

This paper contains 19 sections, 17 theorems, 106 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Weighted Sobolev Spaces and Analytical Tools
  3. From Truncation to Renormalized Solutions: A Unified Theory
  4. Qualitative Theory: Comparison, Regularity, and Stability
  5. Regularity Theory: From Local Estimates to Boundary Asymptotics
  6. Renormalized Solutions: Existence, Uniqueness, and Asymptotic Behavior
  7. Problem Setup and Numerical Method
  8. Model Problem
  9. Numerical Method
  10. Numerical Results
  11. Convergence of Truncated Solutions
  12. Convergence Rate Analysis
  13. Sharp Two-Sided Pointwise Estimates
  14. Energy Decay
  15. Effect of Variable Exponent
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $\phi \in W^{1,p(z)}_0(\Omega)$ be a nonnegative function, and suppose $\alpha(z) \ge 1$. Then there exists a constant $C > 0$, depending only on $\Omega$, $p^+$, and the embedding constants of $W^{1,p(z)}_0(\Omega) \hookrightarrow L^{\gamma}(\Omega)$, such that where $|\Omega|$ denotes the Lebesgue measure of $\Omega$.

Figures (6)

  • Figure 1: Convergence of truncated solutions for decreasing $\varepsilon$. The solutions approach a limiting profile as $\varepsilon \to 0^+$, consistent with the theoretical convergence results.
  • Figure 2: Convergence rate of truncated solutions. The numerical error decays as $\varepsilon^\gamma$ with $\gamma \approx 0.16$, matching the theoretical prediction $\gamma = 1/6$.
  • Figure 3: Boundary behavior near $x=0$. The solution exhibits linear growth, consistent with the lower bound estimate.
  • Figure 4: Convergence of the energy functional. The energy difference decays as $\varepsilon \to 0^+$.
  • Figure 5: Comparison between variable and constant exponent solutions. The variable exponent introduces asymmetry and modifies the solution profile.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Definition 2.1: Weighted Sobolev Space
  • Definition 2.2: Truncation Operator
  • Lemma 2.3: Key Estimate
  • proof
  • Theorem 3.1: Existence of Truncated Solutions
  • proof
  • Theorem 3.2: Convergence to Renormalized Solutions
  • proof
  • Theorem 3.3: Sharp Energy Estimates and Asymptotics
  • proof
  • ...and 26 more