scholarly journals On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order

2002 ◽  
Vol 186 (2) ◽  
pp. 394-419 ◽  
Author(s):  
Ferruccio Colombini ◽  
Tamotu Kinoshita
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equations ◽  
Higher Order ◽  
Well Posedness ◽  
The Cauchy Problem

scholarly journals Well-Posedness of the Cauchy Problem for Hyperbolic Equations with Non-Lipschitz Coefficients

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Akbar B. Aliev ◽  
Gulnara D. Shukurova
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equations ◽  
Well Posedness ◽  
Singular Coefficients ◽  
The Cauchy Problem ◽  
Finite Smoothness

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients. We prove well-posedness of the corresponding Cauchy problem in some functional spaces. These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothness with respect to variables corresponding to singular coefficients.

HOMOGENEOUS HYPERBOLIC EQUATIONS WITH COEFFICIENTS DEPENDING ON ONE SPACE VARIABLE

2007 ◽  
Vol 04 (03) ◽  
pp. 533-553 ◽  
Author(s):  
SERGIO SPAGNOLO ◽  
GIOVANNI TAGLIALATELA
Keyword(s):  
Cauchy Problem ◽  
Space Variable ◽  
Hyperbolic Equations ◽  
Diagonal Matrix ◽  
Well Posedness ◽  
Characteristic Roots ◽  
The Cauchy Problem

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition [Formula: see text] we succeed in constructing a smooth symmetrizer which behaves like a diagonal matrix: this allows us to prove the well-posedness in [Formula: see text].

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation

2020 ◽  
Vol 32 (1) ◽  
pp. 151-187
Author(s):  
Boling Guo ◽  
Zhaohui Huo
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Gauge Transformation ◽  
Energy Method ◽  
Higher Order ◽  
Well Posedness ◽  
Long Wave ◽  
The Cauchy Problem ◽  
Well Posed

AbstractIn this paper, the well-posedness of the higher-order Benjamin–Ono equationu_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u))is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in {H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equationu_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x},is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in {H^{3/4}}.

Sign in / Sign up

Export Citation Format

Share Document