Global and local existence of solutions for a novel type of parabolic Kirchhoff system with singular term

TL;DR

Addresses global and local existence for a fractional Kirchhoff parabolic system with a logarithmic coupling between two unknowns. A weak solution is constructed via Faedo–Galerkin under structural assumptions on the Kirchhoff function, and a potential-well framework with an energy functional $\varphi$ and a Nehari functional $\psi$ is developed to distinguish global existence from finite-time blow-up. The depth of the potential well is shown positive, and stabilization of global solutions is obtained through Komornik’s integral inequality, yielding explicit decay rates for the energy in various parameter regimes. The work advances the theory of nonlocal, Kirchhoff-type parabolic systems with logarithmic nonlinearities and provides sharp blow-up criteria and time estimates.

Abstract

In this paper, we investigate solutions for a fractional system involving a novel class of Kirchhoff functions and logarithmic nonlinearity: \begin{equation*} \left\{\begin{array}{lll} \displaystyle \mathfrak{u}_{t}+\mathcal{K}\left([\mathfrak{u}]_p^s\right) \mathscr{L}_p^s u=\vert \mathfrak{v} \vert^{σ}\vert \mathfrak{u} \vert^{σ-2} u \log | \mathfrak{u} \mathfrak{v}|, \, \, & \mbox{in}\quad &\mathcal{U} \times[0, T),\\ \mathfrak{v}_t+\mathcal{K}\left([\mathfrak{v}]_q^s\right) \mathscr{L}_q^s \mathfrak{v}=\vert \mathfrak{u} \vert^{σ}|\mathfrak{v}|^{σ-2} \mathfrak{v} \log | \mathfrak{u} \mathfrak{v}|, & \text { in } & \mathcal{U} \times[0, T), \\ \mathfrak{u}(\mathrm{x}, t)=\mathfrak{v}(\mathrm{x}, t)=0, & \text { in } & \partial \mathcal{U} \times[0, T), \\ \mathfrak{u}(\mathrm{x}, 0)=\mathfrak{u}_0(\mathrm{x}), \mathfrak{v}(\mathrm{x}, 0)=\mathfrak{v}_0(\mathrm{x}), & \text { in } & \mathcal{U}, \end{array}% \right. \end{equation*} where $\mathcal{K}$ is Kirchhoff function, and $\mathscr{L}_{p}^{s}$ is the fractional $p-$ Laplacian operator. We prove the existence of a weak solution using the Faedo-Galerkin method under suitable assumptions on the Kirchhoff function. We investigate the finite-time blow-up and global existence of solutions based on critical, subcritical, and supercritical initial energy levels. Subsequently, we establish the stabilization of the solution with positive initial energy by applying Komornik's integral inequality.

Theorems & Definitions (26)

• Example 1.1
• Theorem 2.1
• Theorem 2.2
• Lemma 2.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2