2020/09/22 by Hai‐Yang Jin, Tian Xiang, Jin, Hai-Yang +1
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #35B44 #35B45 #35K51 #35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Slime Mold and Myxomycetes Research
paper · pdf · doi:10.48550/arxiv.2009.10423
openalex publication_date 2020/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotaxis on asymptotical behavior, we show that any global-in-time haptotaxis solution component vanishes exponentially as time approaches infinity, and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, the aforementioned critical mass blow-up phenomenon for the chemotaxis-only model is almost undestroyed even with arbitrary introduction of haptotaixs, showing negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and longtime behavior in the chemotaxis-haptotaxis model.