Global well-posedness and stability of constant equilibria in parabolic–elliptic chemotaxis systems without gradient sensing

Ahn, Jaewook; Yoon, Changwook

doi:10.1088/1361-6544/aaf513

Cited by 117 publications

(139 citation statements)

References 23 publications

Supporting

Mentioning

127

Contrasting

Order By: Relevance

“…The above result still holds true if one replaces assumption (A1) by the following Remark 2.3. Our result generalizes the corresponding boundedness result in [1] established for the simplified parabolic-elliptic system with special motility v −k with any k > 0 to more general functions satisfying (A0), (A1) and (A2), for example,…”

Section: Resultssupporting

confidence: 84%

“…First, local existence and uniqueness of classical solutions to system (1.4) can be established by the standard fixed point argument and regularity theory for parabolic equations. Similar proof can be found in [1,Lemma 3.1] or [18, Lemma 2.1] and hence here we omit the detail here.…”

Section: Preliminariesmentioning

confidence: 53%

“…Next, we recall the following lemma given in [1,4] about estimates for the solution of Helmholtz equations. Let a + = max{a, 0}.…”

Section: Preliminariesmentioning

confidence: 99%

“…Then the global classical solution is uniform-in-time bounded. [2] for γ(v) = v −k to more general motility functions satisfying (A2), for example,…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, global existence without any smallness assumption or logistic sources was only achieved in the simplified parabolic-elliptic case, i.e., ε = 0. With a specific motility γ(v) = v −k , global existence of classical solution with a uniform-in-time bound was established by delicate energy estimates in [1] when n ≤ 2 for any k > 0 or n ≥ 3 for k < 2 n−2 . In all work mentioned above, the upper bound of v was established via deriving the L p −integrability of u with p > n 2 by energy method.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Global existence for a kinetic model of pattern formation with density-suppressed motilities

Fujie

Jiang

2020

Journal of Differential Equations

View full text Add to dashboard Cite

This paper is concerned with global well-posedness to the following fully parabolic kinetic systemin a smooth bounded domain Ω ⊂ R n , n ≥ 1 with no-flux boundary conditions. This model was recently proposed in [8,20] to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function γ(·) which is decreasing in v and will vanish as v tends to infinity. The major difficulty in analysis comes from the possible degeneracy as v ր +∞. In this work we develop a new comparison method different from the conventional energy method in literature which reveals a striking fact that there is no finite-time degenercay in this system. More precisely, we use comparison principles for elliptic and parabolic equations to prove that degeneracy cannot take place in finite time in any spatial dimensions for all smooth motility functions satisfying γ(s) > 0, γ ′ (s) ≤ 0 when s ≥ 0 and lim s→+∞ γ(s) = 0.Then we investigate global existence of classical solutions to (0.1) when n ≤ 3 and discuss the uniform-in-time boundedness under certain growth conditions on 1/γ.In particular, we consider system (0.1) with γ(v) = e −v , which shares the same set of equilibria as well as the Lyapunov functional as the classical Keller-Segel model. In the two-dimensional setting, we observe a critical-mass phenomenon which is distinct from the well-known fact for the classical Keller-Segel model. We prove that classical solution always exists globally which is uniformly-in-time bounded with arbitrary initial data of sub-critical mass. On the contrary, with certain initial data of super-critical mass, the solution will become unbounded at time infinity which differs from the finite-time blowup behavior of the Keller-Segel model.

show abstract

Section: Resultssupporting

confidence: 84%

Section: Preliminariesmentioning

confidence: 53%

“…Next, we recall the following lemma given in [1,4] about estimates for the solution of Helmholtz equations. Let a + = max{a, 0}.…”

Section: Preliminariesmentioning

confidence: 99%

“…Then the global classical solution is uniform-in-time bounded. [2] for γ(v) = v −k to more general motility functions satisfying (A2), for example,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Global existence for a kinetic model of pattern formation with density-suppressed motilities

Fujie

Jiang

2020

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

A singular growth phenomenon in a Keller–Segel–type parabolic system involving density‐suppressed motilities

Wang,

Winkler

2024

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Instabilities of standing waves and positivity in traveling waves to a higher‐order Schrödinger equation

Díaz Palencia

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

The aim of this paper is to explore a Schrödinger equation that incorporates a higher‐order operator. Traditional models for electron dynamics have utilized a second‐order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher‐order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher‐order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as , above which the first minimum remains positive. For values of the traveling wave speed significantly greater than , the solutions can be entirely positive.

show abstract

Global well-posedness and stability of constant equilibria in parabolic–elliptic chemotaxis systems without gradient sensing

Cited by 117 publications

References 23 publications

Global existence for a kinetic model of pattern formation with density-suppressed motilities

Global existence for a kinetic model of pattern formation with density-suppressed motilities

A singular growth phenomenon in a Keller–Segel–type parabolic system involving density‐suppressed motilities

Instabilities of standing waves and positivity in traveling waves to a higher‐order Schrödinger equation

Contact Info

Product

Resources

About