On the Cauchy Problem for the Third Order Benjamin-Ono Equation

1996 ◽  
Vol 53 (3) ◽  
pp. 512-528 ◽  
Author(s):  
Xueshang Feng ◽  
Xiaoyou Han
Keyword(s):  
Cauchy Problem ◽  
Third Order ◽  
The Third ◽  
The Cauchy Problem

scholarly journals Cauchy problem for hyperbolic equations of third order with variable exponent of nonlinearity

2020 ◽  
Vol 12 (2) ◽  
pp. 419-433
Author(s):  
O.M. Buhrii ◽  
O.T. Kholyavka ◽  
P.Ya. Pukach ◽  
M.I. Vovk
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Third Order ◽  
The Third ◽  
The Cauchy Problem

We investigate weak solutions of the Cauchy problem for the third order hyperbolic equations with variable exponent of the nonlinearity. The problem is considered in some classes of functions namely in Lebesgue spaces with variable exponents. The sufficient conditions of the existence and uniqueness of the weak solutions to given problem are found.

Global well-posedness of the Cauchy problem for the 3D Jordan–Moore–Gibson–Thompson equation

2020 ◽  
pp. 2050069
Author(s):  
Reinhard Racke ◽  
Belkacem Said-Houari
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Small Data ◽  
Third Order ◽  
Bootstrap Argument ◽  
The Cauchy Problem ◽  
Linear Decay

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained.

scholarly journals Time estimates for the Cauchy problem for a third-order hyperbolic equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 1073-1081 ◽  
Author(s):  
Vladimir Varlamov
Keyword(s):  
Cauchy Problem ◽  
Classical Solution ◽  
Acoustic Waves ◽  
Third Order ◽  
Time Estimates ◽  
Relaxing Medium ◽  
The Cauchy Problem ◽  
Small Times ◽  
Order Hyperbolic Equation ◽  
The Right

A classical solution is considered for the Cauchy problem:(utt−Δu)t+utt−αΔu=f(x,t),x∈ℝ3,t>0;u(x,0)=f0(x),ut(x,0)=f1(x), andutt(x)=f2(x),x∈ℝ3, whereα=const,0<α<1. The above equation governs the propagation of time-dependent acoustic waves in a relaxing medium. A classical solution of this problem is obtained in the form of convolutions of the right-hand side and the initial data with the fundamental solution of the equation. Sharp time estimates are deduced for the solution in question which show polynomial growth for small times and exponential decay for large time whenf=0. They also show the time evolution of the solution whenf≠0.

scholarly journals Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

2015 ◽  
Vol 12 (03) ◽  
pp. 535-579 ◽  
Author(s):  
Enrico Bernardi ◽  
Antonio Bove ◽  
Vesselin Petkov
Keyword(s):  
Cauchy Problem ◽  
Principal Symbol ◽  
Energy Estimates ◽  
Hyperbolic Operator ◽  
Third Order ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Loss Of Regularity ◽  
Well Posed ◽  
Lower Order Terms

We study a class of third-order hyperbolic operators P in G = {(t, x): 0 ≤ t ≤ T, x ∈ U ⋐ ℝn} with triple characteristics at ρ = (0, x0, ξ), ξ ∈ ℝn ∖{0}. We consider the case when the fundamental matrix of the principal symbol of P at ρ has a couple of non-vanishing real eigenvalues. Such operators are called effectively hyperbolic. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for P + Q is locally well posed for any lower order terms Q. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in G could have triple characteristics in G only for t = 0 or for t = T. We prove that the operators in our class are strongly hyperbolic if T is small enough. Our proof is based on energy estimates with a loss of regularity.

scholarly journals General decay and well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation with memory

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1745-1773
Author(s):  
Salah Boulaaras ◽  
Abdelbaki Choucha ◽  
Djamel Ouchenane
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Generalized Solution ◽  
General Decay ◽  
Small Data ◽  
Well Posedness ◽  
Third Order ◽  
The Cauchy Problem ◽  
With Memory

In this paper, we consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan-Moore-Gibson-Thompson (JMGT) equation with the presence of both memory. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result, and we show a local existence result in appropriate function spaces. Finally, we prove a global existence result for small data, and we prove the uniqueness of the generalized solution.

scholarly journals On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient non-linearity

2020 ◽  
Author(s):  
Maxim Korpusov ◽  
Alexandra Matveeva
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Critical Exponents ◽  
Weak Sense ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

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