- Turkish Journal of Mathematics and Computer Science
- Cilt: 15 Sayı: 2
- Existence for a Nonlocal Porous Medium Equations of Kirchhoff Type with Logarithmic Nonlinearity
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 : 34 | Download : 39
Publication Date : 2023-12-31
Article Type : Research
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