Fujita Phenomenon for a Mixed Local-Nonlocal Hardy-Hénon Equation with Regularly Varying Time Weights

1. Problem Statement

The authors investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local–nonlocal diffusion operator with a Hardy–Hénon nonlinearity and a time-dependent forcing term. The equation is given by:

$$\begin{cases} \partial_t u - (\Delta - (-\Delta)^s)u = h(t)|x|^{-b}|u|^p + t^\varrho w(x), & (x, t) \in \mathbb{R}^N \times (0, \infty), \\ u(x, 0) = u_0(x), & x \in \mathbb{R}^N, \end{cases}$$

Key Parameters and Components:

• Operator: $L = \Delta - (-\Delta)^s$, where $\Delta$ is the classical Laplacian (local diffusion) and $(-\Delta)^s$ ($s \in (0,1)$) is the fractional Laplacian (nonlocal diffusion). This models competition between short-range and long-range interactions.
• Nonlinearity: $|x|^{-b}|u|^p$ with $b \ge 0$ and $p > 1$. The term $|x|^{-b}$ introduces a spatial singularity (Hardy potential).
• Time-Dependent Coefficient: $h(t)$ belongs to the generalized class $M(\gamma)$ of functions with logarithmic growth rate $\gamma > -1$. This class extends the classical regularly varying functions to include oscillatory and non-regular behaviors (e.g., $t^\gamma \sin(\log t)$).
• Forcing Term: $t^\varrho w(x)$ with $\varrho > -1$ and a prescribed spatial source $w(x)$.

The study focuses on two main regimes:

1. Unforced Case ($w \equiv 0$): Determining the critical Fujita exponent separating global existence from finite-time blow-up.
2. Forced Case ($w \not\equiv 0$): Establishing conditions for the nonexistence of global weak solutions and deriving smallness conditions for global mild solutions.

2. Methodology

The analysis combines several advanced mathematical techniques:

• Semigroup Theory: The authors utilize the semigroup $\{e^{tL}\}_{t \ge 0}$ generated by the mixed operator $L$. They establish smoothing estimates (decay rates) for the kernel of this semigroup, leveraging the fact that the fundamental solution is a convolution of the classical heat kernel and the fractional heat kernel.
• Regularly Varying Functions (Karamata Theory): A core novelty is the use of the class $M(\gamma)$. The authors employ asymptotic properties of these functions (specifically Karamata's theorems on integrals and uniform convergence) to handle the time-dependent coefficient $h(t)$, which may not be strictly regularly varying but shares similar growth rates.
• Test Function Method: For proving nonexistence (blow-up) results, the authors construct specific cut-off test functions $\psi_R(x,t)$ involving spatial and temporal scaling. They apply this to the weak formulation of the equation to derive integral inequalities that lead to contradictions for global solutions.
• Fixed-Point Argument (Banach Contraction): For proving global existence, the problem is reformulated as a Duhamel integral equation. The authors define a suitable weighted Banach space (incorporating time-decay weights) and prove that the associated nonlinear mapping is a strict contraction for small initial data and forcing terms.
• Scaling Arguments: Precise scaling identities are used to determine the critical exponents and to handle the singular weight $|x|^{-b}$ in the nonlinear term.

3. Key Contributions and Results

A. The Unforced Problem (Fujita Phenomenon)

The paper extends previous results (which assumed $h(t) \sim t^\gamma$) to the broader class $h \in M(\gamma)$.

• Critical Fujita Exponent: The authors establish that the critical exponent $p_F$ remains:
  $$p_F = 1 + \frac{2s(\gamma + 1)}{N}$$
  • Blow-up: If $1 < p < p_F$, every nonnegative nontrivial solution blows up in finite time.
  • Global Existence: If $p > p_F$, global mild solutions exist for sufficiently small initial data.
• Significance: This result demonstrates the robustness of the Fujita phenomenon. Even if $h(t)$ exhibits oscillations or irregular behavior (as long as its logarithmic growth rate is $\gamma$), the threshold for blow-up remains unchanged. The local diffusion term $\Delta$ does not alter the critical exponent, which is governed solely by the fractional part $s$.

B. The Forced Problem (Nonexistence of Global Solutions)

For the equation with a source term $t^\varrho w(x)$, the authors derive conditions under which no global weak solutions exist, regardless of the size of the initial data.

• Conditions: Assuming $w \in C_0 \cap L^1$ with $\int w > 0$:
  • If $\varrho \le 0$, $0 \le \frac{b}{1+\gamma} < 2s < N$, and $1 < p < p^*$, where:
    $$p^* = \frac{N - b - 2s(\varrho - \gamma)}{N - 2s(\varrho + 1)}$$
  • If $\varrho > 0$ and $b, \gamma \ge 0$, nonexistence holds for all $p > 1$.
• Method: The proof relies on the test function method, carefully balancing the decay of the operator kernel against the growth of the time weights $h(t)$ and $t^\varrho$.

C. The Forced Problem (Global Existence)

The authors provide sufficient conditions for the existence of global mild solutions when the parameters allow for it ($p > p^*$).

• Smallness Condition: There exists $\epsilon > 0$ such that if the initial data $u_0$ and source $w$ satisfy:
  $$\|u_0\|_{L^{p_c}} + \|w\|_{L^{q_c}} < \epsilon$$
  where $p_c$ and $q_c$ are specific critical exponents derived from the scaling analysis, then a unique global mild solution exists.
• Functional Framework: The proof utilizes a weighted space with norms involving time-decay factors $t^\mu$, ensuring the solution remains bounded globally.

4. Significance and Novelty

1. Generalization of Time Weights: This is the first study to address blow-up phenomena for nonlinear diffusion equations with coefficients in the class $M(\gamma)$. Previous works were restricted to power-law coefficients ($t^\gamma$). This generalization covers oscillatory and irregular time dependencies, making the results applicable to a wider range of physical models.
2. Mixed Operator Analysis: The work rigorously analyzes the interplay between local ($\Delta$) and nonlocal ($(-\Delta)^s$) diffusion. It confirms that the critical Fujita exponent is determined by the fractional component, a non-trivial result given the competition between the two operators.
3. Hardy-Hénon with Forcing: The inclusion of the spatial singularity $|x|^{-b}$ combined with a time-dependent forcing term $t^\varrho w(x)$ creates a complex system. The paper provides a complete classification of existence vs. nonexistence regimes based on the interplay of $N, s, b, p, \gamma, \varrho$.
4. Unified Framework: The results recover several known cases (e.g., pure fractional, pure local, constant coefficients) as special instances, providing a unified theoretical framework for mixed diffusion equations.

5. Conclusion

Ben Belgacem and Majdoub successfully characterize the blow-up and global existence behavior for a complex class of mixed diffusion equations. By integrating semigroup estimates with the asymptotic theory of regularly varying functions, they extend the classical Fujita theory to include irregular time weights and external forcing. The findings highlight that while the specific form of the time coefficient can be highly irregular, its asymptotic growth rate is the sole determinant of the critical threshold for blow-up. Future work suggested by the authors includes analyzing long-time asymptotics, gradient nonlinearities, and probabilistic interpretations via Lévy processes.