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Fractional diffusion-wave equations with critical nonlinearities in Lebesgue spaces

Masterson Costa, Claudio Cuevas, Bruno de Andrade

TL;DR

The paper addresses the existence, regularity, and long-time behavior of semilinear fractional diffusion-wave equations with Caputo time derivative $\partial_t^\alpha$ ($\alpha\in(1,2)$) in a bounded domain, under critical nonlinearity with growth $|f(r)-f(s)|\le c(|r|^{\rho-1}+|s|^{\rho-1})|r-s|$ and initial data in the critical Lebesgue space $L^q(\Omega)$ with $q=\frac{N(\rho-1)}{2}$. The authors formulate the problem in the abstract scale $\{X_q^\gamma\}\,$ using a sectorial operator $\mathcal{A}_q$ and Mittag-Leffler operators $E_\alpha,S_\alpha,R_\alpha$ to define $\varepsilon$-regular mild solutions, proving local well-posedness via a fixed-point in a weighted ball and establishing smoothing estimates that enable a contraction argument. They then obtain global well-posedness for small data and analyze asymptotic behavior, showing that the weighted norm $t^{\alpha\varepsilon}\|u(t)\|_{X_q^{1+\varepsilon}}$ remains bounded globally and that the difference between two global solutions decays in time, with explicit examples illustrating the long-time dynamics. A key, novel finding is that the critical Lebesgue exponent $q=\frac{N(\rho-1)}{2}$ is independent of the fractional order $\alpha$, unifying superdiffusive, diffusive, and subdiffusive regimes in this setting. The results advance the theory of critical nonlinear fractional PDEs by introducing the $\varepsilon$-regular mild solution framework and providing precise smoothing and asymptotic tools for fractional diffusion-wave equations.

Abstract

This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the continuous dependence of the solutions on initial data. Secondly, we establish the existence of global mild solutions and investigate their asymptotic behavior.

Fractional diffusion-wave equations with critical nonlinearities in Lebesgue spaces

TL;DR

The paper addresses the existence, regularity, and long-time behavior of semilinear fractional diffusion-wave equations with Caputo time derivative () in a bounded domain, under critical nonlinearity with growth and initial data in the critical Lebesgue space with . The authors formulate the problem in the abstract scale using a sectorial operator and Mittag-Leffler operators to define -regular mild solutions, proving local well-posedness via a fixed-point in a weighted ball and establishing smoothing estimates that enable a contraction argument. They then obtain global well-posedness for small data and analyze asymptotic behavior, showing that the weighted norm remains bounded globally and that the difference between two global solutions decays in time, with explicit examples illustrating the long-time dynamics. A key, novel finding is that the critical Lebesgue exponent is independent of the fractional order , unifying superdiffusive, diffusive, and subdiffusive regimes in this setting. The results advance the theory of critical nonlinear fractional PDEs by introducing the -regular mild solution framework and providing precise smoothing and asymptotic tools for fractional diffusion-wave equations.

Abstract

This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the continuous dependence of the solutions on initial data. Secondly, we establish the existence of global mild solutions and investigate their asymptotic behavior.
Paper Structure (5 sections, 5 theorems, 107 equations)

This paper contains 5 sections, 5 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Abstract setting of the problem and known results
  3. Proof of main results
  4. Local well-posedness
  5. Global well-posedness and asymptotic behavior

Key Result

Theorem 1.1

Let $1<\alpha<\frac{2\phi_q}{\pi}$, $1<\rho, q<\infty$, such that $q=\frac{N(\rho-1)}{2}$, and $0<\varepsilon<\frac{N}{N+2q}$, with $\alpha\rho\varepsilon<1$. Then, for all $v_0 \in L^q(\Omega)$, there exist $r=r(v_0)>0$ and $\tau_0=\tau_0(v_0)>0$ such that the problem diffwavep has an $\varepsilon$

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2: deAS
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['local']}
  • Remark 3.1
  • ...and 2 more