Fractional diffusion-wave equations with critical nonlinearities in Lebesgue spaces

Technical Summary: Fractional Diffusion-Wave Equations with Critical Nonlinearities in Lebesgue Spaces

Problem Statement
This paper investigates the semilinear fractional diffusion-wave equation defined on a smooth bounded domain $\Omega \subset \mathbb{R}^N$:
$$\begin{cases} \partial_t^\alpha u = \Delta u + f(u), & \text{in } [0, \infty) \times \Omega, \\ u = 0, & \text{on } [0, \infty) \times \partial\Omega, \\ u(0, x) = u_0(x), \quad u'(0, x) = u_1(x), & \text{in } \Omega, \end{cases}$$
where $\alpha \in (1, 2)$, $\partial_t^\alpha$ denotes the Caputo fractional derivative, and the nonlinearity $f: \mathbb{R} \to \mathbb{R}$ satisfies a growth condition $|f(r) - f(s)| \leq c(|r|^{\rho-1} + |s|^{\rho-1})|r-s|$ with $\rho > 1$.

The primary focus is the analysis of critical nonlinearities where the initial data $(u_0, u_1)$ belongs to the Lebesgue space $L^q(\Omega)$ with $1 < q < \infty$. Specifically, the authors address the critical case defined by the relation $q = \frac{N(\rho-1)}{2}$ (or equivalently $\rho = 1 + \frac{2q}{N}$). Previous works (e.g., de Andrade & Santos [11]) established well-posedness for subcritical cases ($\rho < 1 + \frac{2q}{N}$) but left the critical case as an open problem due to the insufficiency of standard technical tools when the regularity parameter $\beta$ approaches zero in the fractional power scale.

Methodology
The authors employ an abstract functional analytic framework based on the scale of fractional power spaces associated with the Laplace operator.

1. Abstract Formulation: The problem is reformulated in the space $X^0_q = L^q(\Omega)$ using the operator $A_q = -\Delta$ (with Dirichlet boundary conditions). The equation is rewritten as an abstract evolution equation: $D_t^\alpha u = A_q u + f(u)$.
2. Solution Concept: The authors utilize the notion of $\epsilon$-regular mild solutions. A function $u$ is an $\epsilon$-regular mild solution if it belongs to $C((0, \tau]; X^{1+\epsilon}_q)$ and satisfies the variation of constants formula involving Mittag-Leffler operators ($E_\alpha, S_\alpha, R_\alpha$) generated by $A_q$.
3. Key Technical Tools:
   • Smoothing Estimates: The analysis relies heavily on the smoothing properties of the Mittag-Leffler operators, which map initial data from lower regularity spaces to higher regularity spaces with specific time-decay rates (e.g., $\|E_\alpha(tA_q)x\|_{X^{1+\theta}_q} \leq M t^{-\alpha(1+\theta-\beta)} \|x\|_{X^\beta_q}$).
   • $\epsilon$-Regular Maps: The authors prove that the nonlinearity $f$ induces an $\epsilon$-regular map from $X^{1+\epsilon}_q$ to $X^{\rho\epsilon}_q$. This property is crucial for handling the critical exponent where standard Lipschitz arguments in $L^q$ fail.
   • Fixed Point Arguments: Local and global existence are established using the Banach Fixed Point Theorem on carefully constructed complete metric spaces equipped with weighted norms (e.g., $\sup_{t} t^{\alpha\epsilon} \|u(t)\|_{X^{1+\epsilon}_q}$).
   • Singular Gronwall Inequality: This is used to prove uniqueness and continuous dependence on initial data.

Key Contributions and Results

1. Resolution of the Critical Case: The paper provides a rigorous proof of local well-posedness for the critical exponent $\rho = 1 + \frac{2q}{N}$. This resolves an open problem left by previous studies where the standard contraction mapping arguments collapsed when the regularity parameter $\beta$ was zero.

   • Theorem 1.1 establishes the existence, uniqueness, and continuous dependence of local $\epsilon$-regular mild solutions for initial data in $L^q(\Omega)$. It further demonstrates immediate spatial regularization (solutions enter $X^{1+\theta}_q$ for $t>0$) and provides blow-up criteria for maximal existence times.

2. Independence of Critical Exponent from Time-Fractional Order: A significant theoretical finding is that the critical Lebesgue space for existence and uniqueness is independent of the order of the time-fractional derivative $\alpha$. The paper demonstrates that the critical exponent for the superdiffusive case ($1 < \alpha < 2$), the standard diffusive heat equation ($\alpha = 1$), and the subdiffusive case ($0 < \alpha < 1$) are identical when initial data is taken in $L^q(\Omega)$.

3. Global Existence for Small Data:

   • Theorem 1.2 proves the existence of global $\epsilon$-regular mild solutions for initial data with sufficiently small norms in $L^q(\Omega)$ and $X^{1-1/\alpha}_q$. The solutions satisfy uniform bounds in the weighted norm $t^{\alpha\epsilon} \|u(t)\|_{X^{1+\epsilon}_q}$.

4. Asymptotic Stability:

   • Theorem 1.3 addresses the asymptotic behavior of global solutions. It establishes that the stability of the difference between two solutions (in the weighted norm) is equivalent to the stability of the difference between their corresponding linear evolution terms (involving the Mittag-Leffler operators). This provides a characterization of the basin of attraction for these solutions.

Significance and Claims
The authors claim that this work is the first to treat the fractional diffusion-wave equation with critical nonlinearities in the context of $L^q(\Omega)$ initial data using the framework of $\epsilon$-regular maps. The significance lies in:

• Closing a Gap: Successfully addressing the critical case $\rho = 1 + \frac{2q}{N}$, which was previously an open problem due to technical limitations in existing literature.
• Unifying Theory: Demonstrating that the critical exponent for existence and uniqueness is invariant across different types of fractional evolution equations (subdiffusive, diffusive, and superdiffusive) when formulated in Lebesgue spaces.
• Refined Regularity: Providing a detailed analysis of the spatial regularity and asymptotic decay of solutions, showing that solutions possess greater regularity than the initial data immediately after $t=0$.

The paper does not propose new experimental applications or future directions beyond the mathematical analysis of well-posedness and stability for the specified class of equations.