We study the Dirichlet problem for a class of Kirchhoff-type evolution equations involving the p-Laplace operator \( u_{t}-a\left( \left\| \nabla u\right\| _{L^p(\Omega )}^{p}\right) \Delta _p u=\ln \left( \Vert u\Vert _{L^2(\Omega )}^2\right) |u|^{q(x,t)-2}u,\quad (x,t)\in \Omega \times (0,T), \) where the coefficient of the diffusion and the source terms nonlocally depend on the sought solution. We assume that the coefficient \(a:[0,\infty )\rightarrow [0,\infty ) \) is a non-decreasing function, and \(a(s)\rightarrow 0\) as \(s\rightarrow 0^+\) ; therefore, the equation degenerates as \(\Vert \nabla u(t)\Vert _{p}\rightarrow 0\) . Sufficient conditions for local and global in time solvability of the problem are found. The phenomena of blow-up or vanishing of solutions in a finite time are studied, and the upper bound for the blow-up moment is found.
