Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

2005 ◽  
Vol 20 (4) ◽  
pp. 441-447
Author(s):  
Zhou Fujun ◽  
Cui Shangbin
Keyword(s):  
Cauchy Problem ◽  
Local Solvability ◽  
Dispersive Equation ◽  
The Cauchy Problem ◽  
Fifth Order

scholarly journals The Cauchy Problem for a Fifth-Order Dispersive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjun Wang ◽  
Yongqi Liu ◽  
Yongqiang Chen
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Order Equation ◽  
Dispersive Equations ◽  
Dispersive Equation ◽  
The Cauchy Problem ◽  
Fifth Order ◽  
Well Posed

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev spaceHs(R)withs≥1/4. We also establish the ill-posedness for the initial data inHs(R)withs<1/4. Thus, the regularity requirement for the fifth-order dispersive equationss≥1/4is sharp.

scholarly journals Well-posedness of stochastic modified Kawahara equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
P. Agarwal ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Water Wave ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
Restriction Method ◽  
Fifth Order ◽  
Point Argument

AbstractIn this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R), $s\geq -1/4$s≥−1/4. Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

On the existence of a continuously differentiable solution to the Cauchy problem for implicit differential equations

2019 ◽  
pp. 376-383
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Point ◽  
Local Solvability ◽  
Time Interval ◽  
Phase Variable ◽  
Existence Of A Solution ◽  
Differentiable Functions ◽  
The Cauchy Problem

We study the question of the existence of a solution to the Cauchy problem for a differential equation unsolved with respect to the derivative of the unknown function. Differential equations generated by twice continuously differentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is sufficient for the local solvability of the Cauchy problem for any initial point in the class of continuously differentiable functions. It is shown that if the mapping defining the differential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping defining the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.

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