Existence of Solutions of the Cauchy Problem

2015 ◽  
pp. 283-312
Author(s):  
Helge Holden ◽  
Nils Henrik Risebro
Keyword(s):  
Cauchy Problem ◽  
Existence Of Solutions ◽  
The Cauchy Problem

scholarly journals Global existence of solution for multidimensional generalized double dispersion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaoqiang Dai
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Dispersion Equation ◽  
Existence Of Solutions ◽  
Existence Of Solution ◽  
New Methods ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Double Dispersion

Abstract In this paper, we study the Cauchy problem of multidimensional generalized double dispersion equation. To prove the global existence of solutions, we introduce some new methods and ideas, and fill some gaps in the established results.

Classical solutions to a dissipative hyperbolic geometry flow in two space variables

2019 ◽  
Vol 16 (02) ◽  
pp. 223-243
Author(s):  
De-Xing Kong ◽  
Qi Liu ◽  
Chang-Ming Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Scalar Curvature ◽  
Hyperbolic Geometry ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
The One ◽  
Uniformly Bounded

We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.

Global existence and non-existence for the degenerate and uniformly parabolic equations with gradient term

2013 ◽  
Vol 143 (3) ◽  
pp. 643-668 ◽  
Author(s):  
Haifeng Shang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Global Solvability ◽  
Gradient Term ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Limiting Case

We study the Cauchy problem for the degenerate and uniformly parabolic equations with gradient term. The local existence, global existence and non-existence of solutions are obtained. In the case of global solvability, we get the exact estimates of a solution. In particular, we obtain the global existence of solutions in the limiting case.

Existence results for the Kudryashov–Sinelshchikov–Olver equation

2020 ◽  
pp. 1-26 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Heat Transfer ◽  
Cauchy Problem ◽  
Existence Of Solutions ◽  
Gas Bubbles ◽  
Existence Results ◽  
Pressure Waves ◽  
The Cauchy Problem ◽  
Account Heat

Abstract The Kudryashov–Sinelshchikov–Olver equation describes pressure waves in liquids with gas bubbles taking into account heat transfer and viscosity. In this paper, we prove the existence of solutions of the Cauchy problem associated with this equation.

The Cauchy problem for the Aw–Rascle–Zhang traffic model with locally constrained flow

2017 ◽  
Vol 14 (03) ◽  
pp. 393-414 ◽  
Author(s):  
Mauro Garavello ◽  
Stefano Villa
Keyword(s):  
Cauchy Problem ◽  
Traffic Flow ◽  
Existence Of Solutions ◽  
Traffic Model ◽  
Construction Site ◽  
Tracking Method ◽  
The Cauchy Problem ◽  
Generalized Momentum ◽  
Zhang Model ◽  
Constrained Flow

We study the Cauchy problem for the Aw–Rascle–Zhang model for traffic flow with a flux constraint located at [Formula: see text]. The purpose of such a traffic model is to describe an obstruction in a road, such as a toll gate or a construction site. We consider here the concept of solutions introduced by Garavello and Goatin in 2011, which conserves the total number of cars flowing through the obstruction, but does not conserve the generalized momentum. We prove the existence of such solutions to the Cauchy problem by using the wavefront tracking method. We also compare our results with another concept of solutions introduced also by Garavello and Goatin in 2011 for the same model with a flux constraint. For this second formulation, Andreianov, Donadello, and Rosini had also proved the existence of solutions to the Cauchy problem.

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