On a Class of Kirchhoff Type p-Laplacian Evolution Equation with Nonlocal Logarithmic Nonlinearity
Uğur Sert, Sergey Shmarev · Mediterranean Journal of Mathematics · 2025
Abstract We study the Dirichlet problem for a class of Kirchhoff-type evolution equations involving the p-Laplace operator $$ u_{t}-a\left( \left\| abla u\right\| _{L^p(\Omega )}^{p}\right) \Delta _p u=\ln \left( \Vert u\Vert _{L^2(\Omega )}^2\right) |u|^{q(x,t)-2}u,\quad (x,t)\in \Omega \times (0,T), $$ u t - a ∇ u L p ( Ω ) p Δ p u = ln ‖ u ‖ L 2 ( Ω ) 2 | u | q ( x , t ) - 2 u , ( x , t ) ∈ Ω × ( 0 , T ) , where the coefficient of the diffusion and the source terms nonlocally depend on the sought solution. We assume that the coefficient $$a:[0,\infty )\rightarrow [0,\infty ) $$ a : [ 0 , ∞ ) → [ 0 , ∞ ) is a non-decreasing function, and $$a(s)\rightarrow 0$$ a ( s ) → 0 as $$s\rightarrow 0^+$$ s → 0 + ; therefore, the equation degenerates as $$\Vert abla u(t)\Vert _{p}\rightarrow 0$$ ‖ ∇ u ( t ) ‖ p → 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.