On a Class of Kirchhoff Type <i>p</i>-Laplacian Evolution Equation with Nonlocal Logarithmic Nonlinearity

We study the Dirichlet problem for a class of Kirchhoff-type evolution equations involving the p-Laplace operator ut-a del uLp(Omega)p Delta pu=ln & Vert;u & Vert;L2(Omega)2|u|q(x,t)-2u,(x,t)is an element of Omega x(0,T),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{t}-a\left( \left\| \nabla u\right\| _{L<^>p(\Omega )}<^>{p}\right) \Delta _p u=\ln \left( \Vert u\Vert _{L<^>2(\Omega )}<^>2\right) |u|<^>{q(x,t)-2}u,\quad (x,t)\in \Omega \times (0,T), $$\end{document}where the coefficient of the diffusion and the source terms nonlocally depend on the sought solution. We assume that the coefficient a:[0,infinity)->[0,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a:[0,\infty )\rightarrow [0,\infty ) $$\end{document} is a non-decreasing function, and a(s)-> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(s)\rightarrow 0$$\end{document} as s -> 0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\rightarrow 0<^>+$$\end{document}; therefore, the equation degenerates as & Vert;del u(t)& Vert;p -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \nabla u(t)\Vert _{p}\rightarrow 0$$\end{document}. 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.