scholarly journals Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients

2019 ◽  
Vol 18 (4) ◽  
pp. 1891-1919
Author(s):  
Wenming Hu ◽  
◽  
Huicheng Yin ◽  
Keyword(s):  
Hyperbolic Equations ◽  
Second Order ◽  
Well Posedness ◽  
Low Regularity ◽  
Low Regularity Solutions

scholarly journals LOW-REGULARITY SOLUTIONS FOR THE OSTROVSKY EQUATION

2006 ◽  
Vol 49 (1) ◽  
pp. 87-100 ◽  
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Well Posedness ◽  
Low Regularity ◽  
Ostrovsky Equation ◽  
Low Regularity Solutions

AbstractThe well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.

scholarly journals Scattering theory below energy space for two-dimensional nonlinear Schrödinger equation

2015 ◽  
Vol 17 (06) ◽  
pp. 1450052
Author(s):  
Changxing Miao ◽  
Jiqiang Zheng
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Scale Invariant ◽  
Low Regularity ◽  
Nonlinear Schrödinger ◽  
Low Regularity Solutions ◽  
Appl Math

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schrödinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see [J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Commun. Pure Appl. Math.62 (2009) 920–968; F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. École Norm. Sup.42 (2009) 261–290]), we establish global well-posedness and scattering for low-regularity solutions of the equation iut+ Δu = λ1|u|p1u + λ2|u|p2u under certain assumptions on parameters. This is the first result of this type for an equation which is not scale-invariant. In the first step, we establish global well-posedness and scattering for low regularity solutions of the equation iut+ Δu = |u|pu, for a suitable range of the exponent p extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math.62 (2009) 920–968].

scholarly journals WELL-POSEDNESS OF A NONLOCAL PROBLEM WITH INTEGRAL CONDITIONS FOR THIRD ORDER SYSTEM OF THE PARTIAL DIFFERENTIAL EQUATIONS

2018 ◽  
pp. 33-41
Author(s):  
Assanova Anar Turmaganbetkyzy ◽  
Alikhanova Botakoz Zhorakhanovna ◽  
Nazarova Kulzina Zharkimbekovna
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hyperbolic Equations ◽  
Nonlocal Problem ◽  
Second Order ◽  
Order System ◽  
Well Posedness ◽  
Integral Conditions ◽  
Third Order ◽  
Partial Differential

The nonlocal problem with integral conditions for the system of partial differential equations third-order is considered. The existence and uniqueness of classical solution to nonlocal problem with integral conditions for third-order system of partial differential equations are studied and the method for constructing their approximate solutions is proposed. Conditions of an unique solvability to nonlocal problem with integral conditions for third order system of partial differential equations are established. By introduction of new unknown functions, we have reduced the considered problem to an equivalent problem consisting of a nonlocal problem with integral conditions and parameters for a system of hyperbolic equations of second order and a integral relations. We have offered the algorithm for finding approximate solution to investigated problem and have proved its convergence. Sufficient conditions for the existence of unique solution to the equivalent problem with parameters are obtained. Well-posedness of the nonlocal problem with integral conditions for third order system of partial differential equations are established in the terms of well-posedness to nonlocal problem with integral conditions for system of hyperbolic equations second order. Key Words: third order partial differential equations, nonlocal problem, integral condition, system of hyperbolic equations second order, solvability, algorithm.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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