NONLINEAR STABILITY OF LARGE AMPLITUDE VISCOUS SHOCK WAVES OF A GENERALIZED HYPERBOLIC–PARABOLIC SYSTEM ARISING IN CHEMOTAXIS

DOI10.1142/S0218202510004830zbMath1213.35081OpenAlexW2011487751MaRDI QIDQ3067102

Tong Li, Zhi-An Wang

Publication date: 20 January 2011

Published in: Mathematical Models and Methods in Applied Sciences (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1142/s0218202510004830

zbMATH Keywords

energy estimatesnumerical simulationsnonlinear kineticsShizuta-Kawashima conditionnondiffusive signals

Mathematics Subject Classification ID

Shocks and singularities for hyperbolic equations (35L67) Stability in context of PDEs (35B35) First-order nonlinear hyperbolic equations (35L60) Developmental biology, pattern formation (92C15) Initial value problems for first-order hyperbolic systems (35L45) General biology and biomathematics (92B05) Cell movement (chemotaxis, etc.) (92C17) Traveling wave solutions (35C07)

Related Items (58)

Riemann problem for a non-strictly hyperbolic system in chemotaxis ⋮ Optimal time-decay estimates in the critical framework for a chemotaxis model ⋮ Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity ⋮ Stability of Boundary Layers for a Viscous Hyperbolic System Arising from Chemotaxis: One-Dimensional Case ⋮ Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states ⋮ Stability of traveling waves of the Keller–Segel system with logarithmic sensitivity ⋮ Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations ⋮ Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption ⋮ Steadily propagating waves of a chemotaxis model ⋮ Global well-posedness and boundary layer effects of radially symmetric solutions for the singular Keller-Segel model ⋮ A note for global existence of a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant ⋮ Ill-posedness of a multidimensional chemotaxis system in the critical Besov spaces ⋮ Traveling bands for the Keller-Segel model with population growth ⋮ Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity ⋮ Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis ⋮ Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with nutrient-supported proliferation and signal consumption ⋮ Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces ⋮ The shock waves for a mixed-type system from chemotaxis ⋮ Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces ⋮ Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis ⋮ Nonlinear stability of strong traveling waves for a chemotaxis model with logarithmic sensitivity and periodic perturbations ⋮ Nonlinear stability of traveling waves to a parabolic-hyperbolic system modeling chemotaxis with periodic perturbations ⋮ Travelling waves in hyperbolic chemotaxis equations ⋮ Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions ⋮ Blow up criterion for a hyperbolic-parabolic system arising from chemotaxis ⋮ Development of traveling waves in an interacting two-species chemotaxis model ⋮ Boundary layers and stabilization of the singular Keller-Segel system ⋮ Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework ⋮ Asymptotic dynamics of a system of conservation laws from chemotaxis ⋮ Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis ⋮ Global solutions for a hyperbolic-parabolic system of chemotaxis ⋮ Global existence and asymptotic dynamics in a 3D fractional chemotaxis system with singular sensitivity ⋮ Well-posedness and ill-posedness of a multidimensional chemotaxis system in the critical Besov spaces ⋮ Exact traveling wave solutions of one-dimensional parabolic-parabolic models of chemotaxis ⋮ Nonlinear stability of strong traveling waves for the singular Keller-Segel system with large perturbations ⋮ Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis ⋮ Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane ⋮ Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate ⋮ Global Cauchy Problem of a System of Parabolic Conservation Laws Arising From a Keller--Segel Type Chemotaxis Model ⋮ Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity ⋮ On the fractional Fisher information with applications to a hyperbolic-parabolic system of chemotaxis ⋮ The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two ⋮ Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis ⋮ Boundary layer problem on a hyperbolic system arising from chemotaxis ⋮ On a parabolic-hyperbolic chemotaxis system with discontinuous data: well-posedness, stability and regularity ⋮ Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology ⋮ Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space ⋮ Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces ⋮ Large-time behavior in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with signal absorption ⋮ ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS ⋮ Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions ⋮ Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data ⋮ A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source ⋮ Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities ⋮ Asymptotic stability of a composite wave of two traveling waves to a hyperbolic-parabolic system modeling chemotaxis ⋮ Explicit decay rates for a generalized Boussinesq-Burgers system ⋮ A nonlocal model describing tumor angiogenesis ⋮ Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

Cites Work

• Traveling wave solutions from microscopic to macroscopic chemotaxis models
• A model of herd grazing as a travelling wave, chemotaxis and stability
• Initiation of slime mold aggregation viewed as an instability
• Model for chemotaxis
• Traveling bands of chemotactic bacteria: a theoretical analysis
• Global solutions of some chemotaxis and angiogenesis system in high space dimension
• Time-asymptotic behavior of wave propagation around a viscous shock profile
• Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation
• Nonlinear asymptotic stability of viscous shock profiles for conservation laws
• Traveling waves in a chemotactic model
• On the stability of steadily propagating bands of chemotactic bacteria
• Nonlinear stability of viscous shock waves
• Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems
• A chemotaxis model motivated by angiogenesis
• A constructive approach to traveling waves in chemotaxis
• Overview of mathematical approaches used to model bacterial chemotaxis. II: Bacterial popu\-lations
• Random walk with persistence and external bias
• Nonlinear Stability of Traveling Waves to a Hyperbolic-Parabolic System Modeling Chemotaxis
• Hyperbolic systems of conservation laws II
• Global existence of solutions to a hyperbolic-parabolic system
• Shock formation in a chemotaxis model
• Partial differential equations and non-diffusive structures
• Stability of Traveling Waves in Quasi-Linear Hyperbolic Systems with Relaxation and Diffusion
• On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas
• Stability of shock profiles in viscoelasticity with non‐convex constitutive relations
• Mathematical modeling of the onset of capillary formation initiating angiogenesis

This page was built for publication: NONLINEAR STABILITY OF LARGE AMPLITUDE VISCOUS SHOCK WAVES OF A GENERALIZED HYPERBOLIC–PARABOLIC SYSTEM ARISING IN CHEMOTAXIS