On the cauchy problem with degenerate diffusion and nonlocal nonlinear sources

TL;DR

This paper analyzes the behavior of solutions to a generalized nonlinear diffusion equation with nonlocal reaction, identifying conditions for global existence, bounds, and decay, supported by numerical simulations.

Contribution

It introduces a critical exponent for reaction terms that determines solution behavior, combining theoretical analysis with numerical verification.

Findings

Solutions exist globally for subcritical reaction exponents.

Decay properties are established for supercritical and critical cases.

Numerical simulations confirm theoretical predictions.

Abstract

This paper is devoted to the analysis of non-negative solutions for a generalisation of the parabolic equation with porous medium like nonlinear diffusion and nonlinear nonlocal reaction. We investigate under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and aggregation modelled by nonlinear reaction, occurs. Precisely, we exhibit that the qualitative behavior of solutions is decided by the nonlinear diffusion which is chosen in such a way that its scaling and the reaction term coincide, i.e. that there is a critical exponent $m+2/n$ for the reaction exponent $\alpha,$ solutions exist globally with uniformly upper bounds in the case of (i)$1\le \alpha<m+2/n$ for any initial data, (ii) $\alpha>m+2/n$ for small initial data and (iii) $\alpha=m+2/n$ for small mass capacity $M_0$. In the case of (ii) and (iii), the decay properties of the solution are also discussed. Moreover, numerical simulations are carried out to verify the theoretical analysis and explore other issues that lie beyond the scope of the analysis.