2022/09/11 by Isanka Garli Hevage, Hevage, Isanka Garli, Akif Ibragimov +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #35C06 #35K65 #35Q35 #35Q76 #76R50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2209.04950
openalex publication_date 2022/09/11 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28
The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality and the Ladyzhenskaya-Uraltseva iteration lemma.