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.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

Sufficient conditions of a nonlocal solvability for a system of two quasilinear equations of the first order with constant terms

2020 ◽  
Vol 55 ◽  
pp. 60-78
Author(s):  
M.V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

We consider a Cauchy problem for a system of two quasilinear equations of the first order with constant terms. The study of the solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms in the original coordinates is based on the method of an additional argument. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. We prove the existence and uniqueness of the local solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms, which has the same smoothness with respect to $x$ as the initial functions of the Cauchy problem. Sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms are found; this solution is continued by a finite number of steps from the local solution. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms relies on global estimates.

scholarly journals Symmetries and solutions to dissipative hyperbolic geometric flow

2021 ◽  
Author(s):  
Fang Gao ◽  
Zenggui Wang
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Riemann Surfaces ◽  
Blow Up ◽  
Hyperbolic Equations ◽  
Global Solutions ◽  
Geometric Flows ◽  
Geometric Flow ◽  
The Cauchy Problem ◽  
Symmetric Method

Based on the Lie-symmetric method, we study the solutions of dissipative hyperbolic geometric flows on Riemann surfaces; In the process of simplification, the mixed equations are produced. And the hyperbolic equations are obtained under limited conditions. Considering the Cauchy problem of the hyperbolic equation, the existence and uniqueness conditions of the global solutions are obtained. Finally, the phenomenon of blow up is discussed.

scholarly journals Fractional Operators in p -adic Variable Exponent Lebesgue Spaces and Application to p -adic Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Leonardo Fabio Chacón-Cortés ◽  
Humberto Rafeiro
Keyword(s):  
Cauchy Problem ◽  
Integral Operator ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Operators ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Uniqueness Of The Solution

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p -adic variable exponent Lebesgue spaces.

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