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

Sert U.

Turkish Journal of Mathematics and Computer Science, cilt.15, sa.2, ss.247-257, 2023 (TRDizin)

Özet

We study the Dirichlet problem for the nonlocal parabolic equation of the Kirchhoff type
ut−a(‖u‖Lp(Ω)p)∑i=1nDi(|u|p−2Diu)+b(x,t)|u|α(x,t)−2ulog⁡|u|=f(x,t)in QT=Ω×(0,T)," role="presentation">ut−a(∥u∥pLp(Ω))n∑i=1Di(|u|p−2Diu)+b(x,t)|u|α(x,t)−2ulog|u|=f(x,t)in QT=Ω×(0,T),
where p≥2" role="presentation">p≥2, T>0" role="presentation">T>0, Ω⊂Rn" role="presentation">Ω⊂Rn, n≥2" role="presentation">n≥2, is a smooth bounded domain. The coefficient a(⋅)" role="presentation">a(⋅) is real-valued function defined on R+" role="presentation">R+. It is shown that the problem has a weak solution under appropriate and general conditions on a(⋅)" role="presentation">a(⋅), α(⋅,⋅)" role="presentation">α(⋅,⋅) and b(⋅)" role="presentation">b(⋅).