scholarly journals Global existence of classical solutions for a hyperbolic chemotaxis model and its parabolic limit

2006 ◽  
Vol 55 (1) ◽  
pp. 289-316 ◽  
Author(s):  
Hyung Ju Hwang ◽  
Kyungkeun Kang ◽  
Angela Stevens
Keyword(s):  
Global Existence ◽  
Classical Solutions ◽  
Chemotaxis Model ◽  
Parabolic Limit

Global boundedness and asymptotic behavior in a quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source

2020 ◽  
Vol 30 (13) ◽  
pp. 2619-2689
Author(s):  
Guoqiang Ren ◽  
Bin Liu
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Global Existence ◽  
Classical Solutions ◽  
Chemotaxis Model ◽  
Signal Production ◽  
Nonlinear Signal ◽  
Nonlinear Signal Production ◽  
Global Boundedness

In this work, we consider the quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.

Global existence and boundedness of classical solutions for a chemotaxis model with logistic source

2013 ◽  
Vol 351 (15-16) ◽  
pp. 585-591 ◽  
Author(s):  
Khadijeh Baghaei ◽  
Mahmoud Hesaaraki
Keyword(s):  
Global Existence ◽  
Classical Solutions ◽  
Logistic Source ◽  
Chemotaxis Model

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

scholarly journals Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pierre Roux ◽  
Delphine Salort
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Mean Field ◽  
Classical Solutions ◽  
Field Limit ◽  
Mean Field Limit ◽  
Diffusive Approximation ◽  
The Mean ◽  
Weak Connectivity ◽  
Time Existence

<p style='text-indent:20px;'>The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.</p>

scholarly journals Global Existence of Classical Solutions to a Cancer Invasion Model

2012 ◽  
Vol 03 (04) ◽  
pp. 382-388 ◽  
Author(s):  
Khadijeh Baghaei ◽  
Mohammad Bagher Ghaemi ◽  
Mahmoud Hesaaraki
Keyword(s):  
Global Existence ◽  
Cancer Invasion ◽  
Classical Solutions ◽  
Invasion Model

scholarly journals Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Jincheng Shi ◽  
Shengzhong Xiao
Keyword(s):  
Global Existence ◽  
Life Span ◽  
General Model ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Conditions ◽  
Short Wave ◽  
Classical Solutions ◽  
Global Classical Solutions

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.

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