Local and global solutions for a hyperbolic–elliptic model of chemotaxis on a network

2019 ◽  
Vol 29 (08) ◽  
pp. 1465-1509
Author(s):  
Francesca Romana Guarguaglini ◽  
Marco Papi ◽  
Flavia Smarrazzo
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elliptic System ◽  
Global Solutions ◽  
Transmission Conditions ◽  
The Other ◽  
Density Functions ◽  
Biological Models ◽  
Local And Global Solutions ◽  
Elliptic Model

In this paper, we study a hyperbolic–elliptic system on a network which arises in biological models involving chemotaxis. We also consider suitable transmission conditions at internal points of the graph which on one hand allow discontinuous density functions at nodes, and on the other guarantee the continuity of the fluxes at each node. Finally, we prove local and global existence of non-negative solutions — the latter in the case of small (in the [Formula: see text]-norm) initial data — as well as their uniqueness.

Global Existence and Blowup for the Cubic Nonlinear Klein-Gordon Equations in Three Space Dimensions

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Jian Zhang ◽  
Zaihui Gan ◽  
Boling Guo
Keyword(s):  
Global Existence ◽  
Invariant Manifolds ◽  
Gordon Equation ◽  
Global Solutions ◽  
The Other ◽  
Cubic Nonlinearity ◽  
Sharp Threshold ◽  
Klein Gordon ◽  
Three Space ◽  
Klein Gordon Equation

AbstractIn this paper, we apply a cross-constrained variational method to study the classic nonlinear Klein-Gordon equation with cubic nonlinearity in three space dimensions. By constructing a type of cross-constrained variational problem and establishing the so-called cross invariant manifolds, we obtain a sharp threshold for blowup and global existence of the solution to the equation under study which is different from that in [10] . On the other hand, we give an answer to the question that how small the initial data have to be for the global solutions to exist.

Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass

2018 ◽  
Vol 18 (2) ◽  
pp. 337-360
Author(s):  
Toshitaka Nagai ◽  
Tetsuya Yamada
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Elliptic System ◽  
Total Mass ◽  
Critical Mass ◽  
The Other ◽  
Boundedness Of Solutions ◽  
Nonnegative Solutions ◽  
The Cauchy Problem ◽  
Mass Case

AbstractWe consider the Cauchy problem for a parabolic-elliptic system in{\mathbb{R}^{2}}, called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is{8\pi}, the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [7] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [6] and López-Gómez, Nagai and Yamada [23]. In this paper, we investigate further the boundedness of nonnegative solutions.

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 Problem solving in MNCs: How local and global solutions are (and are not) created

2012 ◽  
Vol 43 (8) ◽  
pp. 746-771 ◽  
Author(s):  
Esther Tippmann ◽  
Pamela Sharkey Scott ◽  
Vincent Mangematin
Keyword(s):  
Problem Solving ◽  
Global Solutions ◽  
Local And Global Solutions

scholarly journals Size distributions in random triangles

2014 ◽  
Vol 51 (A) ◽  
pp. 283-295
Author(s):  
D. J. Daley ◽  
Sven Ebert ◽  
R. J. Swift
Keyword(s):  
Elliptic Integral ◽  
Unit Length ◽  
The Other ◽  
Density Functions ◽  
Size Distributions

The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

scholarly journals A remark on global solutions to random 3D vorticity equations for small initial data

2019 ◽  
Vol 24 (8) ◽  
pp. 4021-4030 ◽  
Author(s):  
Michael Röckner ◽  
◽  
Rongchan Zhu ◽  
Xiangchan Zhu ◽  
◽  
...  
Keyword(s):  
Initial Data ◽  
Global Solutions ◽  
Small Initial Data ◽  
Vorticity Equations
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