Journal of Differential Equations, ISSN 0022-0396, 10/2016, Volume 261, Issue 8, pp. 4631 - 4647

We consider a nonlinear PDEs system of two equations of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a biological...

Global existence | Stability | Parabolic–Elliptic systems | Chemotaxis | Asymptotic behavior | Parabolic-Elliptic systems | EQUATIONS | BOUNDEDNESS | LIMIT | MODEL | POINT DYNAMICS | MATHEMATICS | KELLER-SEGEL SYSTEM | ASYMPTOTIC STABILITY | BLOW-UP | REINFORCED RANDOM-WALKS | AGGREGATION

Global existence | Stability | Parabolic–Elliptic systems | Chemotaxis | Asymptotic behavior | Parabolic-Elliptic systems | EQUATIONS | BOUNDEDNESS | LIMIT | MODEL | POINT DYNAMICS | MATHEMATICS | KELLER-SEGEL SYSTEM | ASYMPTOTIC STABILITY | BLOW-UP | REINFORCED RANDOM-WALKS | AGGREGATION

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 04/2016, Volume 135, pp. 57 - 72

We study a Keller–Segel type of system, which includes growth and death of the chemotactic species and an elliptic equation for the chemo-attractant. The...

Chemotaxis–growth system | Global solutions | Hyperbolic–elliptic system | Blowup | Parabolic–elliptic system | Parabolic-elliptic system | Chemotaxis-growth system | Hyperbolic-elliptic system | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS | BOUNDEDNESS | DIFFUSION | Chemotherapy | Cancer | Death | Nonlinearity | Convexity | Mathematical analysis | Smooth boundaries | Constraining

Chemotaxis–growth system | Global solutions | Hyperbolic–elliptic system | Blowup | Parabolic–elliptic system | Parabolic-elliptic system | Chemotaxis-growth system | Hyperbolic-elliptic system | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS | BOUNDEDNESS | DIFFUSION | Chemotherapy | Cancer | Death | Nonlinearity | Convexity | Mathematical analysis | Smooth boundaries | Constraining

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 6/2018, Volume 289, Issue 1, pp. 71 - 108

This paper deals with an initial-boundary value problem for the chemotaxis-(Navier–)Stokes system $$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot...

Small-convection limit | Primary 35B40 | Mathematics, general | Navier–Stokes | Secondary 35K55 | Mathematics | 92C17 | 35Q30 | Exponential stabilization | Chemotaxis | 35Q92 | GLOBAL EXISTENCE | PARABOLIC EQUATIONS | DECAY | STABILIZATION | BOUNDEDNESS | NONLINEAR DIFFUSION | FLUID EQUATIONS | KELLER-SEGEL MODELS | MATHEMATICS | LARGE TIME BEHAVIOR | Navier-Stokes | WEAK SOLUTIONS

Small-convection limit | Primary 35B40 | Mathematics, general | Navier–Stokes | Secondary 35K55 | Mathematics | 92C17 | 35Q30 | Exponential stabilization | Chemotaxis | 35Q92 | GLOBAL EXISTENCE | PARABOLIC EQUATIONS | DECAY | STABILIZATION | BOUNDEDNESS | NONLINEAR DIFFUSION | FLUID EQUATIONS | KELLER-SEGEL MODELS | MATHEMATICS | LARGE TIME BEHAVIOR | Navier-Stokes | WEAK SOLUTIONS

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 8/2016, Volume 55, Issue 4, pp. 1 - 39

The coupled chemotaxis fluid system $$\begin{aligned} \left\{ \begin{array}{lll} n_t=\Delta n-\nabla \cdot (n S(x,n,c)\cdot \nabla c)-u\cdot \nabla n, &{}\quad...

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | 35B35 | Analysis | Theoretical, Mathematical and Computational Physics | 35Q35 | Mathematics | 92C17 | 35B40 | 35K55 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BACTERIA | PARABOLIC EQUATIONS | INITIAL-VALUE PROBLEM | BOUNDED DOMAINS | FLUID MODEL | NONLINEAR DIFFUSION | WEAK SOLUTIONS | KELLER-SEGEL MODELS | ASYMPTOTIC-BEHAVIOR

Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | 35B35 | Analysis | Theoretical, Mathematical and Computational Physics | 35Q35 | Mathematics | 92C17 | 35B40 | 35K55 | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | BACTERIA | PARABOLIC EQUATIONS | INITIAL-VALUE PROBLEM | BOUNDED DOMAINS | FLUID MODEL | NONLINEAR DIFFUSION | WEAK SOLUTIONS | KELLER-SEGEL MODELS | ASYMPTOTIC-BEHAVIOR

Journal Article

Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, 08/2011, Volume 21, Issue 8, pp. 1631 - 1650

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic-parabolic system derived from the...

nonlocal | global existence | long-time behavior | blowup criterion | local existence | hyperbolic-parabolic system | Chemotaxis | REACTION-DIFFUSION EQUATIONS | EXISTENCE | MATHEMATICS, APPLIED | TRAVELING-WAVES | NONLINEAR STABILITY | AGGREGATION

nonlocal | global existence | long-time behavior | blowup criterion | local existence | hyperbolic-parabolic system | Chemotaxis | REACTION-DIFFUSION EQUATIONS | EXISTENCE | MATHEMATICS, APPLIED | TRAVELING-WAVES | NONLINEAR STABILITY | AGGREGATION

Journal Article

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Full Text
Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2016, Volume 439, Issue 1, pp. 197 - 212

In this paper we study this chemotaxis-system{ut=Δu−χ∇⋅(u∇v)+g(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, defined in a convex smooth and bounded domain Ω of Rn, with n≥1,...

Logistic source | Global existence | Chemotaxis | Nonlinear parabolic systems | MATHEMATICS | MATHEMATICS, APPLIED | LOWER BOUNDS | EQUATIONS | BOUNDEDNESS | MODEL | ATTRACTOR | TIME BLOW-UP

Logistic source | Global existence | Chemotaxis | Nonlinear parabolic systems | MATHEMATICS | MATHEMATICS, APPLIED | LOWER BOUNDS | EQUATIONS | BOUNDEDNESS | MODEL | ATTRACTOR | TIME BLOW-UP

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 10/2016, Volume 261, Issue 8, pp. 4524 - 4572

This paper deals with the following quasilinear attraction–repulsion chemotaxis...

Chemotaxis | A priori estimates | Finite-time blow up | Boundedness | STOKES SYSTEM | GLOBAL EXISTENCE | LARGE-TIME BEHAVIOR | MODELING CHEMOTAXIS | NONRADIAL SOLUTIONS | PARABOLIC-PARABOLIC TYPE | MATHEMATICS | KELLER-SEGEL SYSTEM | DEGENERATE DIFFUSION | LOGISTIC SOURCE | SINGULARITY FORMATION

Chemotaxis | A priori estimates | Finite-time blow up | Boundedness | STOKES SYSTEM | GLOBAL EXISTENCE | LARGE-TIME BEHAVIOR | MODELING CHEMOTAXIS | NONRADIAL SOLUTIONS | PARABOLIC-PARABOLIC TYPE | MATHEMATICS | KELLER-SEGEL SYSTEM | DEGENERATE DIFFUSION | LOGISTIC SOURCE | SINGULARITY FORMATION

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 06/2017, Volume 35, pp. 102 - 131

An interesting problem arising in many contexts of mathematical biology is the study of the relevance of chemotaxis in reaction–diffusion processes. We...

Fluids | Reaction terms | Corals | Chemotaxis | Keller–Segel | Asymptotic behavior | CHEMICAL-ASPECTS | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | STABILIZATION | SPERM-ATTRACTANT | EGGS | ASYMPTOTIC-BEHAVIOR | SYSTEMS | Keller-Segel | Analysis

Fluids | Reaction terms | Corals | Chemotaxis | Keller–Segel | Asymptotic behavior | CHEMICAL-ASPECTS | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | STABILIZATION | SPERM-ATTRACTANT | EGGS | ASYMPTOTIC-BEHAVIOR | SYSTEMS | Keller-Segel | Analysis

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 05/2017, Volume 449, Issue 1, pp. 872 - 883

We study a hyperbolic–parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two. We consider diffusions given by the fractional...

Global classical solutions | Chemotaxis | Hyperbolic–parabolic system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR STABILITY | BEHAVIOR | SENSITIVITY | MODEL | Hyperbolic-parabolic system | TRAVELING-WAVES | Analysis of PDEs | Mathematics

Global classical solutions | Chemotaxis | Hyperbolic–parabolic system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR STABILITY | BEHAVIOR | SENSITIVITY | MODEL | Hyperbolic-parabolic system | TRAVELING-WAVES | Analysis of PDEs | Mathematics

Journal Article

Journal of Mathematical Biology, ISSN 0303-6812, 9/2019, Volume 79, Issue 4, pp. 1455 - 1490

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller–Segel chemoattraction system, 0.1...

Classical solution | Mathematical and Computational Biology | 35B35 | Parabolic–elliptic chemotaxis system | Traveling waves | Mathematics | 92C17 | 35K57 | 35Q92 | Logistic source | Applications of Mathematics | 35B40 | Spreading speeds | Wave equation | Usage | Models | Mathematical models | Chemotaxis | Spreading | Continuity (mathematics)

Classical solution | Mathematical and Computational Biology | 35B35 | Parabolic–elliptic chemotaxis system | Traveling waves | Mathematics | 92C17 | 35K57 | 35Q92 | Logistic source | Applications of Mathematics | 35B40 | Spreading speeds | Wave equation | Usage | Models | Mathematical models | Chemotaxis | Spreading | Continuity (mathematics)

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 08/2018, Volume 42, pp. 93 - 119

The current paper is devoted to the study of traveling wave solutions of the following parabolic–parabolicchemotaxis system,...

Logistic source | Parabolic–parabolic chemotaxis system | Spreading speed | Traveling wave solution | MATHEMATICS, APPLIED | VARIATIONAL PRINCIPLE | STABILITY | FRONTS | KELLER-SEGEL SYSTEM | R-N | MODELS | GROWTH | SPREADING SPEEDS | PROPAGATION | Parabolic-parabolic chemotaxis system

Logistic source | Parabolic–parabolic chemotaxis system | Spreading speed | Traveling wave solution | MATHEMATICS, APPLIED | VARIATIONAL PRINCIPLE | STABILITY | FRONTS | KELLER-SEGEL SYSTEM | R-N | MODELS | GROWTH | SPREADING SPEEDS | PROPAGATION | Parabolic-parabolic chemotaxis system

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2015, Volume 258, Issue 2, pp. 302 - 338

This paper is devoted to the analytical study of initial–boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis...

Initial–boundary value problem | Classical solution | Global well-posedness | Chemotaxis | Hyperbolic balance laws | Long-time behavior | Initial-boundary value problem | PARABOLIC SYSTEM | EQUATIONS | MODEL | TRAVELING-WAVES | MATHEMATICS | NONLINEAR STABILITY | GLOBAL DYNAMICS | CONSERVATION-LAWS | AGGREGATION | DIFFUSION LIMIT

Initial–boundary value problem | Classical solution | Global well-posedness | Chemotaxis | Hyperbolic balance laws | Long-time behavior | Initial-boundary value problem | PARABOLIC SYSTEM | EQUATIONS | MODEL | TRAVELING-WAVES | MATHEMATICS | NONLINEAR STABILITY | GLOBAL DYNAMICS | CONSERVATION-LAWS | AGGREGATION | DIFFUSION LIMIT

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 03/2019, Volume 42, Issue 4, pp. 1210 - 1226

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded...

periodic solutions | subsolutions and super‐solutions | asymptotic behavior | parabolic‐elliptic systems of PDEs | chemotaxis | rectangle method | parabolic-elliptic systems of PDEs | subsolutions and super-solutions | MATHEMATICS, APPLIED | MATHEMATICAL-MODEL | ANGIOGENESIS | INITIATION | STABILITY | DIFFUSION | Organic chemistry | Asymptotic properties | Time functions | Biological evolution

periodic solutions | subsolutions and super‐solutions | asymptotic behavior | parabolic‐elliptic systems of PDEs | chemotaxis | rectangle method | parabolic-elliptic systems of PDEs | subsolutions and super-solutions | MATHEMATICS, APPLIED | MATHEMATICAL-MODEL | ANGIOGENESIS | INITIATION | STABILITY | DIFFUSION | Organic chemistry | Asymptotic properties | Time functions | Biological evolution

Journal Article

Discrete and Continuous Dynamical Systems - Series B, ISSN 1531-3492, 05/2013, Volume 18, Issue 3, pp. 601 - 641

This article surveys the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of...

Conservation laws | Angiogenesis | Fisher equation | Transformation | Stability | Wave speed | Bacteria | Traveling waves | Chemical diffusion | Chemotaxis | MATHEMATICS, APPLIED | DIFFUSION EQUATIONS | bacteria | INITIATION | angiogenesis | PATTERNS | MODEL | transformation | FRONTS | conservation laws | wave speed | NONLINEAR STABILITY | traveling waves | HYPERBOLIC-PARABOLIC SYSTEM | stability | chemical diffusion | GLOBAL-SOLUTIONS | BANDS | PROPAGATION

Conservation laws | Angiogenesis | Fisher equation | Transformation | Stability | Wave speed | Bacteria | Traveling waves | Chemical diffusion | Chemotaxis | MATHEMATICS, APPLIED | DIFFUSION EQUATIONS | bacteria | INITIATION | angiogenesis | PATTERNS | MODEL | transformation | FRONTS | conservation laws | wave speed | NONLINEAR STABILITY | traveling waves | HYPERBOLIC-PARABOLIC SYSTEM | stability | chemical diffusion | GLOBAL-SOLUTIONS | BANDS | PROPAGATION

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 02/2017, Volume 262, Issue 4, pp. 3250 - 3283

We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic–parabolic model of chemotaxis and prove...

Hyperbolic–parabolic system | Weak solutions | Fisher information | SMOOTH SOLUTIONS | FUNCTIONAL INEQUALITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | TRAVELING-WAVES | MATHEMATICS | NONLOCAL VELOCITY | Hyperbolic parabolic system | CRITICAL MASS | NONLINEAR STABILITY | DIFFUSION | KELLER-SEGEL MODEL | Analysis of PDEs | Mathematics

Hyperbolic–parabolic system | Weak solutions | Fisher information | SMOOTH SOLUTIONS | FUNCTIONAL INEQUALITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | TRAVELING-WAVES | MATHEMATICS | NONLOCAL VELOCITY | Hyperbolic parabolic system | CRITICAL MASS | NONLINEAR STABILITY | DIFFUSION | KELLER-SEGEL MODEL | Analysis of PDEs | Mathematics

Journal Article

JOURNAL OF DIFFERENTIAL EQUATIONS, ISSN 0022-0396, 06/2017, Volume 262, Issue 11, pp. 5635 - 5690

In the current paper, we consider the following parabolic elliptic semilinear Keller Segel model on R-N, {u(t) = del center dot(del(u) - chi(u)del(u)) + a(u) -...

Parabolic elliptic chemotaxis system | Classical solution | VARIATIONAL PRINCIPLE | Asymptotic behavior | EQUATIONS | BOUNDEDNESS | MODEL | GROWTH SYSTEM | TRAVELING-WAVES | Local existence | MATHEMATICS | PATTERN-FORMATION | FRONTS | Logistic source | Global existence | SPREADING SPEEDS | TIME BLOW-UP

Parabolic elliptic chemotaxis system | Classical solution | VARIATIONAL PRINCIPLE | Asymptotic behavior | EQUATIONS | BOUNDEDNESS | MODEL | GROWTH SYSTEM | TRAVELING-WAVES | Local existence | MATHEMATICS | PATTERN-FORMATION | FRONTS | Logistic source | Global existence | SPREADING SPEEDS | TIME BLOW-UP

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 07/2016, Volume 39, Issue 11, pp. 2787 - 2798

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of RN, {N = 2;3}, under different boundary conditions, with...

blow‐up time | nonlinear parabolic systems | lower bounds | chemotaxis | nonlinear cross‐diffusion | nonlinear cross-diffusion | blow-up time | MATHEMATICS, APPLIED | KELLER-SEGEL SYSTEM | FINITE-TIME BLOWUP | BOUNDEDNESS | MODEL | Lower bounds | Time dependence | Boundary conditions | Mathematical analysis

blow‐up time | nonlinear parabolic systems | lower bounds | chemotaxis | nonlinear cross‐diffusion | nonlinear cross-diffusion | blow-up time | MATHEMATICS, APPLIED | KELLER-SEGEL SYSTEM | FINITE-TIME BLOWUP | BOUNDEDNESS | MODEL | Lower bounds | Time dependence | Boundary conditions | Mathematical analysis

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 11/2016, Volume 261, Issue 9, pp. 5035 - 5070

This paper is concerned with the boundary layer problem for a hyperbolic system transformed via a Cole–Hopf type transformation from a repulsive chemotaxis...

Weighted [formula omitted]-estimates | Chemotaxis | BL-thickness | Boundary layers | Weighted L | estimates | PARABOLIC SYSTEM | TRAVELING-WAVES | MATHEMATICS | KELLER-SEGEL SYSTEM | NAVIER-STOKES EQUATIONS | ZERO-VISCOSITY LIMIT | NONLINEAR STABILITY | GLOBAL DYNAMICS | CONSERVATION-LAWS | REINFORCED RANDOM-WALKS | Weighted L-2-estimates | DIFFUSION LIMIT | Analysis | Boundary layer

Weighted [formula omitted]-estimates | Chemotaxis | BL-thickness | Boundary layers | Weighted L | estimates | PARABOLIC SYSTEM | TRAVELING-WAVES | MATHEMATICS | KELLER-SEGEL SYSTEM | NAVIER-STOKES EQUATIONS | ZERO-VISCOSITY LIMIT | NONLINEAR STABILITY | GLOBAL DYNAMICS | CONSERVATION-LAWS | REINFORCED RANDOM-WALKS | Weighted L-2-estimates | DIFFUSION LIMIT | Analysis | Boundary layer

Journal Article