On the Global and Local Solution of the Multidimensional Darboux Problem for Some Nonlinear Wave Equations

2007 ◽  
Vol 14 (1) ◽  
pp. 65-80
Author(s):  
Giorgi Bogveradze ◽  
Sergo Kharibegashvili
Keyword(s):  
Global Solution ◽  
Wave Equations ◽  
Nonlinear Term ◽  
Local Solvability ◽  
Spatial Dimension ◽  
Local Solution ◽  
Nonlinear Wave Equations ◽  
Darboux Problem ◽  
Power Nonlinearity ◽  
Global And Local

Abstract We consider a multidimensional analogue of the Darboux problem for wave equations with power nonlinearity. Depending on the spatial dimension of an equation, a power nonlinearity exponent and the sign in front of a nonlinear term, it is proved that the Darboux problem is globally solvable in some cases, but has no global solution in other cases though the local solvability of this problem remains in force.

scholarly journals On Glassey’s conjecture for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kimitoshi Tsutaya ◽  
Yuta Wakasugi
Keyword(s):  
Nonlinear Wave ◽  
Finite Time ◽  
Wave Equations ◽  
Nonlinear Term ◽  
Blow Up ◽  
Time Derivative ◽  
Upper Bounds ◽  
Nonlinear Wave Equations ◽  
Semilinear Wave Equations

AbstractConsider nonlinear wave equations in the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey’s conjecture for the time derivative nonlinearity. We also show blow-up results for the space derivative nonlinear term.

scholarly journals Existence, Asymptotic Behaviour, and Blow up of Solutions for a Class of Nonlinear Wave Equations with Dissipative and Dispersive Terms

2009 ◽  
Vol 64 (5-6) ◽  
pp. 315-326
Author(s):  
Necat Polat ◽  
Doğan Kaya
Keyword(s):  
Asymptotic Behaviour ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Term ◽  
Blow Up ◽  
Suitable Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Nonlinear Wave Equations

Abstract We consider the existence, both locally and globally in time, the asymptotic behaviour, and the blow up of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative and dispersive terms. Under rather mild conditions on the nonlinear term and the initial data we prove that the above-mentioned problem admits a unique local solution, which can be continued to a global solution, and the solution decays exponentially to zero as t →+∞. Finally, under a suitable condition on the nonlinear term, we prove that the local solutions with negative and nonnegative initial energy blow up in finite time.

On the existence and absence of global solutions of the first Darboux problem for nonlinear wave equations

2008 ◽  
Vol 44 (3) ◽  
pp. 374-389 ◽  
Author(s):  
G. K. Berikelashvili ◽  
O. M. Dzhokhadze ◽  
B. G. Midodashvili ◽  
S. S. Kharibegashvili
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Global Solutions ◽  
Nonlinear Wave Equations ◽  
Darboux Problem

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Tan ◽  
Zhao-Yang Yin
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equations ◽  
Rogue Waves ◽  
Homoclinic Solutions ◽  
Nonlinear Wave Equations ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

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