Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system

Nishino, Teruto; Yokota, Tomomi

doi:10.1016/j.jmaa.2019.06.068

Cited by 5 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thereafter, even though in the literature there are several papers concerning estimates for lower bounds of blow-up time for solutions to general evolutive problems whose formulation relies on the hypothesis on the divergence of certain energy functions (see, for instance, [ given in Theorem 3.3. (An equivalent approach is employed in [18] for unbounded solutions to the same fully parabolic chemotaxis problem analyzed in [3]. )…”

Section: Starting Point and Presentation Of The Main Theoremsmentioning

confidence: 99%

A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

2020

Self Cite

View full text Add to dashboard Cite

This paper deals with unbounded solutions to the following zero-flux chemotaxis systemwhere α > 0, Ω is a smooth and bounded domain of R n , with n ≥ 1, t ∈ (0, Tmax), where Tmax the blow-up time, and m1, m2 real numbers. Given a sufficiently smooth initial data u0 := u(x, 0) ≥ 0 and set M := 1 |Ω| Ω u0(x) dx, from the literature it is known that under a proper interplay between the above parameters m1, m2 and the extra condition Ω v(x, t)dx = 0, system ( ) possesses for any χ > 0 a unique classical solution which becomes unbounded at t ր Tmax. In this investigation we first show that for p0 > n 2 (m2 − m1) any blowing up classical solution in L ∞ (Ω)-norm blows up also in L p 0 (Ω)-norm. Then we estimate the blow-up time Tmax providing a lower bound T .

show abstract

Section: Starting Point and Presentation Of The Main Theoremsmentioning

confidence: 99%

A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…After transforming the NPE into vector form, several variables in the equation are treated with linear algebraic discretization [5][6][7]. e RS and IE of NPE variables are constructed.…”

Section: Multivariable Linear Algebraic Discretementioning

confidence: 99%

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Zuo¹,

Mei²

2022

Computational Intelligence and Neuroscience

View full text Add to dashboard Cite

Since the nonlinear parabolic equation has many variables, its calculation process is mostly an algebraic operation, which makes it difficult to express the discrete process concisely, which makes it difficult to effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion. The extended mixed finite element method is a common method for solving reaction-diffusion equations. By introducing intermediate variables, the discretized algebraic equations have great nonlinearity. To address this issue, the paper proposes a multivariable linear algebraic discretization method for NPEs. First, the NPE is discretized, the algebraic form of the nonlinear equation is transformed into the vector form, and the rough set (RS) and information entropy (IE) are constructed to allocate the weights of different variable attributes. According to the given variable attribute weight, the multiple variables in the equation are discretized by linear algebra. It can effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion and has good adaptability in this field.

show abstract

“…The lower bound estimate appeared much later but also draw much attention recently (see e.g. [1,2,4,6,16,21,[24][25][26][27][28][29][30]33]). The main approach in these works was the energy method.…”

Section: Introduction 1historical Workmentioning

confidence: 96%

Prevention of blowup via Neumann heat kernel

Yang,

Zhou

2019

Preprint

View full text Add to dashboard Cite

Consider the heat equation ut − ∆u = 0 on a bounded C 2 domain Ω in R n (n ≥ 2) with any positive initial data. If a superlinear radiation law ∂u ∂n = u q with q > 1 is imposed on a partial boundary Γ1 ⊆ ∂Ω which has a positive surface area, then it has been known that the solution u blows up in finite time. However, if the partial boundary, on which the superlinear radiation law is prescribed, is shrinking and is denoted as Γ1,t at time t, then the solution may exist globally as long as the surface area |Γ1,t| of Γ1,t decays fast enough. This paper asks the question that how fast should |Γ1,t| decay in order to have a bounded global solution? This question is of significant importance in realistic situations, such as the temperature control within a certain safe range. By taking advantage of the Neumann heat kernel, we conclude that a polynomial decay |Γ1,t| ∼ |Γ1|(1 + Ct) −β with any β > n − 1 suffices to ensure a bounded global solution.

show abstract

Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system

Cited by 5 publications

References 29 publications

A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Prevention of blowup via Neumann heat kernel

Contact Info

Product

Resources

About