On the existence of solutions for a parabolic‐elliptic chemotaxis model with flux limitation and logistic source - Sastre‐Gomez, Silvia; Tello, Jose Ignacio (Mathematical methods in the applied sciences, 2023)
In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ u$$ u $$” and a chemical stimulus “ v$$ v $$” in a bounded and regular domain Ω$$ Ω $$ of ℝN$$ {\mathbb{R}}^N $$. The equation for u$$ u $$ is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as −χdiv(u|∇v|p−2∇v),$$ -χ \operatorname{div}\left(u{\left|\nabla v\right|}^{p-2}\nabla v\right), $$ for p>1$$ p>1 $$. The chemical substance distribution v$$ v $$ satisfies the elliptic equation −Δv+v=u.$$ -Δ v+v=u. $$ The evolution of u$$ u $$ is also determined by a logistic type growth term μu(1−u)$$ μ u\left(1-u\right) $$. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for p<3/2$$ p<3/2 $$ and any N≥2$$ N\ge 2 $$.