Existence for a Nonlocal Porous Medium Equations of Kirchhoff Type with Logarithmic Nonlinearity

Authors : Uğur Sert

Pages : 247-257

Doi:10.47000/tjmcs.1260780

View : 79 | Download : 47

Publication Date : 2023-12-31

Article Type : Research Paper

Abstract :We study the Dirichlet problem for the nonlocal parabolic equation of the Kirchhoff type \\[ u_{t}-a\\left(\\|u\\|_{L^{p}(\\Omega)}^{p}\\right)\\sum\\limits_{i=1}^{n}D_{i}\\left( \\left\\vert u\\right\\vert ^{p-2}D_{i}u\\right) +b(x,t) \\left\\vert u \\right\\vert ^{\\alpha \\left( x,t\\right) -2}u\\log|u|=f\\left( x,t\\right) \\quad \\text{in $Q_T=\\Omega \\times (0,T)$}, \\] where $p\\geq2$, $T>0$, $\\Omega \\subset \\mathbb{R}^{n}$, $n\\geq 2$, is a smooth bounded domain. The coefficient $a(\\cdot)$ is real-valued function defined on $\\mathbb{R}_+$. It is shown that the problem has a weak solution under appropriate and general conditions on $a(\\cdot)$, $\\alpha(\\cdot,\\cdot)$ and $b(\\cdot)$.
Keywords : Kirchhoff type equation, nonlocal, existence, logarithmic nonlinearity

ORIGINAL ARTICLE URL