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

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.