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

TL;DR

This paper establishes the existence, blow-up, and stabilization of solutions for a fractional Kirchhoff system with a logarithmic nonlinearity, involving novel functions and fractional operators, under various initial energy conditions.

Contribution

It introduces a new class of fractional Kirchhoff systems with logarithmic nonlinearities and proves existence, blow-up, and stabilization results using the Faedo-Galerkin method and integral inequalities.

Findings

Proved existence of weak solutions under certain conditions.

Analyzed finite-time blow-up for specific initial energies.

Demonstrated stabilization of solutions with positive initial energy.

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^{\sigma }\vert \mathfrak{u} \vert^{\sigma-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^{\sigma }|\mathfrak{v}|^{\sigma-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.