Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data

DOI10.1016/j.jde.2011.08.047zbMath1241.35118OpenAlexW2035611675MaRDI QIDQ657464

Sachiko Ishida, Tomomi Yokota

Publication date: 16 January 2012

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jde.2011.08.047

zbMATH Keywords

generalized Fujita's critical exponentsuper-critical case

Mathematics Subject Classification ID

Reaction-diffusion equations (35K57) Critical exponents in context of PDEs (35B33) Weak solutions to PDEs (35D30) Initial value problems for second-order parabolic systems (35K45) Quasilinear parabolic equations (35K59)

Related Items (34)

Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions ⋮ Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller-Segel system ⋮ Uniqueness theorem on weak solutions to the Keller-Segel system of degenerate and singular types ⋮ Ill-posedness of the two-dimensional Keller-Segel model in Triebel-Lizorkin spaces ⋮ Global existence and boundedness in a supercritical quasilinear degenerate Keller-Segel system under relaxed smallness conditions for initial data ⋮ A meshless technique based on the local radial basis functions collocation method for solving parabolic-parabolic Patlak-Keller-Segel chemotaxis model ⋮ On effects of the nonlinear signal production to the boundedness and finite-time blow-up in a flux-limited chemotaxis model ⋮ Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity ⋮ Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation ⋮ Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure ⋮ Boundedness through nonlocal dampening effects in a fully parabolic chemotaxis model with sub and superquadratic growth ⋮ Hölder estimates of weak solutions to degenerate chemotaxis systems with a source term ⋮ Existence of weak solutions to an anisotropic parabolic–parabolic chemotaxis system ⋮ Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion ⋮ Lower Bound of Mass in a Chemotactic Model with Advection and Absorbing Reaction ⋮ Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type ⋮ Stationary Ring and Concentric-Ring Solutions of the Keller--Segel Model with Quadratic Diffusion ⋮ Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains ⋮ Nonlinear diffusion in the Keller-Segel model of parabolic-parabolic type ⋮ Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion ⋮ Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models ⋮ Uniform \(L^{\infty}\) boundedness for a degenerate parabolic-parabolic Keller-Segel model ⋮ The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion ⋮ Possibility of the existence of blow‐up solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type ⋮ Keller-Segel chemotaxis models: a review ⋮ Properties of blow-up solutions and their initial data for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type ⋮ A refined criterion and lower bounds for the blow-up time in a parabolic-elliptic chemotaxis system with nonlinear diffusion ⋮ Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments ⋮ Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system ⋮ The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in ℝ^d,d ≥ 3 ⋮ Threshold of global behavior of solutions to a degenerate drift-diffusion system in between two critical exponents ⋮ Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues ⋮ Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system ⋮ An iterative approach to \(L^\infty\)-boundedness in quasilinear Keller-Segel systems

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems.
• Time global existence and asymptotic behavior of solutions to generate quasi-linear parabolic systems of chemotaxis.
• Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
• Initiation of slime mold aggregation viewed as an instability
• Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model
• A user's guide to PDE models for chemotaxis
• Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces
• Existence and nonexistence of global solutions to quasilinear parabolic equations with convection
• Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces
• Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term
• Finite-time blow-up in a quasilinear system of chemotaxis
• MATHEMATICAL MODELLING OF CANCER INVASION OF TISSUE: THE ROLE AND EFFECT OF NONLOCAL INTERACTIONS
• Does a ‘volume-filling effect’ always prevent chemotactic collapse?
• Heat kernels and maximal l^p—l^qestimates for parabolic evolution equations

This page was built for publication: Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data