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Well-posedness of initial-boundary value problem for time-fractional diffusion-wave equation with time-dependent coefficients

Xinchi Huang, Masahiro Yamamoto

TL;DR

This work proves well-posedness for the initial-boundary value problem of a time-fractional diffusion-wave equation with time-dependent coefficients in the regime $1<\alpha\le 2$. The authors develop a Galerkin framework that reduces the PDE to a system of time-fractional ODEs, for which they establish unique solvability and estimates, then pass to the limit to obtain a unique weak solution with precise regularity in $H^1(0,T;L^2(\Omega))$, $L^{\infty}(0,T;H^1_0(\Omega))$, and $u-a_0-ta_1\in H_{\alpha}(0,T;H^{-1}(\Omega))$, plus $\partial_t^{\alpha-1}(u-a_0)\in L^{\infty}(0,T;L^2(\Omega))$. Under stronger data compatibility ($F-Aa_0\in H_1(0,T;L^2(\Omega))$), they obtain improved spatial-temporal regularity, including $u\in L^{\infty}(0,T;H^2(\Omega))$ and $\|\partial_t^{\alpha}(u-a_0-ta_1)\|_{L^{\infty}(0,T;L^2(\Omega))}$ bounds. The results are supported by a rigorous development of the fractional Sobolev spaces $H_{\gamma}(0,T;X)$, coercivity inequalities for fractional derivatives, and a generalized Grönwall framework, providing a solid foundation for anomalous-diffusion/wave models with time-varying coefficients. These findings advance the mathematical understanding of fractional PDEs with nonstationary coefficients and enable further quantitative regularity analyses in related settings.

Abstract

We consider the well-posedness of the initial-boundary value problem for a time-fractional partial differential equation with the fractional order lying in (1,2]. For the case of time-dependent coefficients, it is difficult to give an explicit solution formula by the eigenfunction expansion method. In order to deal with the case of time-varying coefficients, we first show the unique existence and regularity of solution to a system of time-fractional ordinary differential equations. Then the unique existence of the weak solution to the time-fractional partial differential equation and improved regularity are derived by using the Galerkin method.

Well-posedness of initial-boundary value problem for time-fractional diffusion-wave equation with time-dependent coefficients

TL;DR

This work proves well-posedness for the initial-boundary value problem of a time-fractional diffusion-wave equation with time-dependent coefficients in the regime . The authors develop a Galerkin framework that reduces the PDE to a system of time-fractional ODEs, for which they establish unique solvability and estimates, then pass to the limit to obtain a unique weak solution with precise regularity in , , and , plus . Under stronger data compatibility (), they obtain improved spatial-temporal regularity, including and bounds. The results are supported by a rigorous development of the fractional Sobolev spaces , coercivity inequalities for fractional derivatives, and a generalized Grönwall framework, providing a solid foundation for anomalous-diffusion/wave models with time-varying coefficients. These findings advance the mathematical understanding of fractional PDEs with nonstationary coefficients and enable further quantitative regularity analyses in related settings.

Abstract

We consider the well-posedness of the initial-boundary value problem for a time-fractional partial differential equation with the fractional order lying in (1,2]. For the case of time-dependent coefficients, it is difficult to give an explicit solution formula by the eigenfunction expansion method. In order to deal with the case of time-varying coefficients, we first show the unique existence and regularity of solution to a system of time-fractional ordinary differential equations. Then the unique existence of the weak solution to the time-fractional partial differential equation and improved regularity are derived by using the Galerkin method.
Paper Structure (12 sections, 16 theorems, 171 equations)

This paper contains 12 sections, 16 theorems, 171 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Time-fractional derivatives in function spaces
  4. Statement of the main results
  5. System of time-fractional ordinary differential equations
  6. Proof of Theorem \ref{['thm:pde1']}
  7. Proof of Theorem \ref{['thm:pde2']}
  8. Appendix
  9. Characterization of $H_{\gamma}(0,T;\mathbb{R})$ in terms of Sobolev-Slobodeckij spaces
  10. Proof of Lemma \ref{['lem:dense']}
  11. Coercivity inequalities of the time-fractional derivative
  12. Generalized Grönwall's inequality

Key Result

Lemma 1.1

Let $\gamma>0$. $\mathrm{(i)}$ The operator $J^{\gamma}: L^2(0,T;X) \longrightarrow L^2(0,T;X)$ is bounded. $\mathrm{(ii)}$ The operator $J^{\gamma}: L^2(0,T;X) \longrightarrow L^2(0,T;X)$ is injective. $\mathrm{(iii)}$$J^{\alpha+\beta} = J^{\alpha}J^{\beta}$ in $L^2(0,T;X)$ for $\alpha, \beta>0$.

Theorems & Definitions (26)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Proposition 1.1
  • proof
  • ...and 16 more