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.

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.

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.

scholarly journals Global Existence of Solutions for a Nonstrictly Hyperbolic System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
De-yin Zheng ◽  
Yun-guang Lu ◽  
Guo-qiang Song ◽  
Xue-zhou Lu
Keyword(s):  
Global Existence ◽  
Existence Of Solutions ◽  
Bounded Function ◽  
Compensated Compactness ◽  
Resonant System ◽  
Existence Of Weak Solutions ◽  
Global Existence Of Solutions ◽  
Flux Approximation ◽  
The Cauchy Problem ◽  
Uniformly Bounded

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly boundedL∞estimatesz(ρδ,ε,uδ,ε)≤B(x)andw(ρδ,ε,uδ,ε)≤βwhena(x)is increasing (similarly,w(ρδ,ε,uδ,ε)≤B(x)andz(ρδ,ε,uδ,ε)≤βwhena(x)is decreasing) for theε-viscosity andδ-flux approximation solutions of nonhomogeneous, resonant system without the restrictionz0(x)≤0orw0(x)≤0as given in Klingenberg and Lu (1997), wherezandware Riemann invariants of nonhomogeneous, resonant system;B(x)>0is a uniformly bounded function ofxdepending only on the functiona(x)given in nonhomogeneous, resonant system, andβis the bound ofB(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

scholarly journals H4-Solutions for the Olver–Benney equation

2021 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Existence Of Solutions ◽  
Interaction Effects ◽  
Order Equation ◽  
Long Waves ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Benney Equation ◽  
Fifth Order

AbstractThe Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation.

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