A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces

Abstract: In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}<em>γ\left(|u|^{{p_1}\right)|u|^{p_2},} p_1, p_2&gt;0, γ\in[0, n)$, with initial data taken in the homogeneous Besov spaces $\dot{B}</em>{2, \infty}<sup>{-β}$, where $β\in\left[0, \frac{n}{2}\right)$. Our approach is based on deriving decay estimates for solutions to the associated linear damped wave equation with initial data belonging to $\dot{B}<em>{2, \infty}<sup>{-β}$, combined with refined tools from Harmonic Analysis. As a consequence, we identify a new critical exponent given by $$p_1+p_2:=p</em>{\mathrm{Fuji}}\left(\tfrac{n+2β}{2+γ}\right):=1+\tfrac{4+2γ}{n+2β} \quad \text{ for } β\in\left[0, \tfrac{n}{2}\right) \text{ and } γ\in [0, n).$$ More precisely, we establish the global (in time) existence of small data solutions in the supercritical and critical regimes $$p_1+p_2 \geq p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$$. In contrast, we prove finite-time blow-up of weak solutions, even for arbitrarily small initial data, in the subcritical range $2&lt;p_1+p_2&lt;p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$.

• The paper introduces a new critical exponent that delineates the boundary between global existence and finite-time blow-up for solutions with low-regularity initial data in homogeneous Besov spaces.
• It utilizes advanced harmonic analysis techniques including Fourier analysis, fractional inequalities, and fixed point arguments to derive precise decay estimates.
• The study highlights the impact of nonlocal Hartree-type nonlinearities on solution dynamics, offering insights relevant to both theoretical analysis and practical applications.

Critical Exponent Analysis for Semilinear Damped Wave Equations with Hartree-Type Nonlinearity and Besov Initial Data

Introduction

This work addresses the Cauchy problem for a semilinear damped wave equation incorporating Hartree-type nonlinearity: $$\partial_t^2 u - \Delta u + \partial_t u = \mathcal{I}_\gamma (|u|^{p_1})|u|^{p_2},$$ with initial data in the homogeneous Besov space $\dot{B}_{2, \infty}^{-\beta}$, for $\beta \in [0, n/2)$. Here, $\mathcal{I}_\gamma$ is the Riesz potential operator of order $\gamma \in [0, n)$. The primary result is the identification of a new critical exponent for the existence and blow-up dichotomy for solutions with such low regularity initial data.

Mathematical Setting and Background

The analysis builds upon the extensive tradition of critical exponent theory for damped wave and parabolic equations. Historically, the Fujita exponent $p_{\text{Fuj}}(n) = 1 + 2/n$ separated global existence from blow-up for semilinear heat-type and damped wave equations with $L^1$ initial data. Extensions to $L^m$ and more refined function spaces showed that this threshold shifts with the regularity of the data, a phenomenon here further explored in the context of homogeneous Besov spaces.

A distinguishing feature of this work is the consideration of nonlocal Hartree-type nonlinearities and convolution operators, rather than local powers of $u$. The Hartree-type term, $\mathcal{I}_\gamma(|u|^{p_1})|u|^{p_2}$, models spatially nonlocal nonlinear effects relevant in mathematical physics and introduces analytic subtleties, especially under weak regularity assumptions for the initial data.

Main Results

New Critical Exponent

The main theorem asserts that the critical exponent for global existence versus blow-up is shifted to

$\dot{B}_{2, \infty}^{-\beta}$0

This encompasses the usual Fujita exponent as a limiting case ($\dot{B}_{2, \infty}^{-\beta}$1, $\dot{B}_{2, \infty}^{-\beta}$2), but allows for arbitrary low-regularity data from $\dot{B}_{2, \infty}^{-\beta}$3. The regime analysis is as follows:

• Supercritical ($\dot{B}_{2, \infty}^{-\beta}$4): Global (in time) existence of small data solutions is achieved.
• Critical ($\dot{B}_{2, \infty}^{-\beta}$5): Global existence persists at the threshold, with explicit polynomial decay of norms over time for solutions and their derivatives.
• Subcritical ($\dot{B}_{2, \infty}^{-\beta}$6): Any nontrivial solution blows up in finite time, regardless of data size, under mild positivity and decay assumptions.

Precise Decay Estimates

For solutions in the supercritical/critical regimes, the paper establishes sharp decay rates in $\dot{B}_{2, \infty}^{-\beta}$7, homogeneous Sobolev, and Besov spaces: $\dot{B}_{2, \infty}^{-\beta}$8 for any $\dot{B}_{2, \infty}^{-\beta}$9, reflecting the effect of the low regularity of the initial data and the dissipative mechanism.

Blow-Up Criterion

The blow-up proof employs a nontrivial adaptation of the test function and energy method, leveraging the convolution structure of the nonlinearity and fine properties of Besov-regular data. The lower bound condition required for the initial data is shown to be nonempty, ensuring that the solution space is analytically significant.

Methodological Innovations

The analysis crucially hinges on new $\beta \in [0, n/2)$0 and $\beta \in [0, n/2)$1 decay estimates for the linear damped wave equation, obtained via explicit Fourier analysis and careful frequency decomposition. The authors exploit the heat kernel characterization of negative-order Besov spaces and combine this with advanced harmonic analysis, specifically the fractional Gagliardo–Nirenberg and Hardy–Littlewood–Sobolev inequalities, to control the nonlocal nonlinearities.

A fixed point argument (in a tailored function space $\beta \in [0, n/2)$2 capturing the combined decay rates) delivers the existence results. The blow-up regime is characterized rigorously using a duality approach for weak solutions.

Implications and Future Directions

Theoretical Ramifications

This work generalizes the framework for critical exponent analysis in several directions:

• It incorporates spatially nonlocal (Hartree-type) nonlinearities with convolutions, a setting directly relevant for models in quantum mechanics and nonlinear optics.
• It systematically accounts for weak regularity, covering cases of highly singular initial data.
• The interplay between the order of the Riesz potential, the regularity parameter of the Besov space, and the nonlinearity exponents is fully elucidated, highlighting the sensitivity of global existence thresholds to these parameters.

Practical Impact

Though primarily mathematical, these results inform the modeling of physical systems—especially where data may lack smoothness or compactness (as in turbulent or random media), and where nonlocal self-interactions are dominant.

Prospective Research

Several avenues naturally emerge. Extending the analysis to even more general functional frameworks (Triebel–Lizorkin or Morrey spaces), or to equations with variable coefficients, remains open. The methods might be adapted for quasilinear evolution problems or to treatment of equations with memory terms or nonlocal-in-time damping.

Numerical verification and exploration of the sharpness in concrete settings would also enhance practical understanding, as would deriving explicit lifespan bounds in the intermediate regime between global existence and blow-up.

Conclusion

This study rigorously determines a new critical exponent for a semilinear damped wave equation with Hartree-type nonlinearity and low regularity initial data in homogeneous Besov spaces. The main results deliver a sharp dichotomy between global existence and finite-time blow-up, accompanied by explicit decay estimates. The work broadens the theoretical foundation for nonlocal nonlinear evolution equations and showcases the rich interplay between initial data regularity and nonlinear dynamics (2604.23764).