Some existence results for a Kirchhoff-type equation involving fractional p-Laplacian with logarithmic nonlinearity

In this paper, we study the global existence of weak solutions for parabolic Kirchhoff-type problems of the following form:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t+ M(\Vert u\Vert _{W_0}^p)(-\Delta )_p^s u+\pi _{p\theta }(u)=\pi _{p\theta }(u)\log (\vert u\vert ) & \text{ in } \Omega , \quad t>0,\\ u(x, 0)=u_0(x) & \text{ in } \Omega \\ u=0 & \text{ in } (\mathbb {R}^n \backslash \Omega ),\quad t>0, \end{array}\right. } \end{aligned}$$

where \(\Omega \subset \mathbb {R}^n\), \(\pi _{p\theta }(x)=\vert x\vert ^{p\theta -2}x\), \(1\le \theta <\frac{p_s^*}{p}\), \(2< p<\frac{n}{s}\), \(0<s<1\), \(M: \mathbb {R}^+\rightarrow \mathbb {R}^+\) is a continuous function defined by \(M(r)=r^{\theta -1}\) and \((-\Delta )_p^s\) is the fractional p-Laplacian operator. Based on the potential well method combined with the theory of Young measures and the Galerkin method, we obtain the existence of global solution.