In this paper, we studied the continuous dependence and stability of solutions for a class of fractional partial differential equations with multiple spatially varying coefficient parameters. The nonlocal operator was defined by a symmetric kernel, yielding a self-adjoint structure essential to the analysis. Using variational methods and the Minty–Browder theorem, we established the existence and uniqueness of weak solutions in the energy space $ X_0 $ for each admissible parameter vector $ w $. We extended single-parameter stability to a multi-parameter framework by proving that the solution operator $ S_f $ is continuous with respect to $ w $ in the product space $ \prod_i L^{q_i}(\Omega) $. Moreover, we derived an explicit global Lipschitz estimate for $ S_f $ and showed its Gâteaux differentiability under mild regularity assumptions on $ f(x, u, w) $. Numerical simulations confirmed continuity, Lipschitz stability, and differentiability of $ S_f $ with respect to all parameters. These results provided rigorous guarantees for inverse problems and uncertainty quantification in multi-parameter fractional PDE models.
Citation: Jia Zheng, Xiuling Li, Yanni Pang, Hongying Wang, Tongchao Wang, Jiaxuan Sun. Continuous dependence and stability for a class of fractional partial differential equations with multiple parameters[J]. AIMS Mathematics, 2025, 10(11): 27837-27861. doi: 10.3934/math.20251223
In this paper, we studied the continuous dependence and stability of solutions for a class of fractional partial differential equations with multiple spatially varying coefficient parameters. The nonlocal operator was defined by a symmetric kernel, yielding a self-adjoint structure essential to the analysis. Using variational methods and the Minty–Browder theorem, we established the existence and uniqueness of weak solutions in the energy space $ X_0 $ for each admissible parameter vector $ w $. We extended single-parameter stability to a multi-parameter framework by proving that the solution operator $ S_f $ is continuous with respect to $ w $ in the product space $ \prod_i L^{q_i}(\Omega) $. Moreover, we derived an explicit global Lipschitz estimate for $ S_f $ and showed its Gâteaux differentiability under mild regularity assumptions on $ f(x, u, w) $. Numerical simulations confirmed continuity, Lipschitz stability, and differentiability of $ S_f $ with respect to all parameters. These results provided rigorous guarantees for inverse problems and uncertainty quantification in multi-parameter fractional PDE models.
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Jia Zheng, Xiuling Li, Yanni Pang, Hongying Wang, Tongchao Wang, Jiaxuan Sun. Continuous dependence and stability for a class of fractional partial differential equations with multiple parameters[J]. AIMS Mathematics, 2025, 10(11): 27837-27861. doi: 10.3934/math.20251223
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