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- W4383893166 abstract "We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $ m = frac{2d}{d+2s} $ in such a way that the associated free energy is conformal invariant and there is a family of stationary solutions $ U(x) = cleft(frac{lambda}{lambda^2+|x-x_0|^2}right)^{frac{d+2s}{2}} $ for some constant $ c $ and $ lambda>0, x_0 in mathbb{R}^d$. We analyse under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of dynamical solutions by virtue of stationary solutions. Precisely, solutions exist globally in time if the $ L^m $ norm of the initial data $ |u_0|_{L^m(mathbb{R}^d)} $ is less than the $ L^m $ norm of stationary solutions $ |U(x)|_{L^m(mathbb{R}^d)} $. Whereas there are blowing-up solutions for $ |u_0|_{L^m(mathbb{R}^d)}>|U(x)|_{L^m(mathbb{R}^d)} $." @default.
- W4383893166 created "2023-07-12" @default.
- W4383893166 creator A5029974693 @default.
- W4383893166 date "2023-01-01" @default.
- W4383893166 modified "2023-10-16" @default.
- W4383893166 title "The aggregation-diffusion equation with energy critical exponent" @default.
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- W4383893166 doi "https://doi.org/10.3934/dcdsb.2023126" @default.
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