On the blow-up of the solution and on the local and global solvability of the Cauchy problem for a nonlinear equation in Hölder spaces

2021 ◽  
pp. 125469
Author(s):  
M.O. Korpusov ◽  
A.A. Panin
Keyword(s):  
Cauchy Problem ◽  
Nonlinear Equation ◽  
Blow Up ◽  
Global Solvability ◽  
Hölder Spaces ◽  
The Cauchy Problem ◽  
Holder Spaces

scholarly journals Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1266-1271
Author(s):  
Mohamed Jleli
Keyword(s):  
Cauchy Problem ◽  
Nonlinear Equation ◽  
Fractional Derivatives ◽  
Blow Up ◽  
Second Order ◽  
Nonlinear Capacity ◽  
The Cauchy Problem ◽  
Mixed Fractional Derivatives ◽  
Order Nonlinear Equation ◽  
Capacity Method

Abstract In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

scholarly journals PROPERTIES OF INTEGRALS WHICH HAVE THE TYPE OF DERIVATIVES OF VOLUME POTENTIALS FOR DEGENERATED $\overrightarrow{2\lowercase{b}}$ - PARABOLIC EQUATION OF KOLMOGOROV TYPE

2021 ◽  
Vol 9 (2) ◽  
pp. 7-21
Author(s):  
V. Dron' ◽  
I. Medyns'kyi
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Homogeneous Equation ◽  
Hölder Spaces ◽  
Kolmogorov Type ◽  
Estimates Of Solutions ◽  
The Cauchy Problem ◽  
The Given ◽  
Holder Spaces ◽  
Derivatives Of

In weight Holder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solution of the Cauchy problem for degenerated $\overrightarrow{2b}$-parabolic equation of Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weight Holder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weight Holder spaces.

scholarly journals Properties of integrals which have the type of derivatives of volume potentials for one ultraparabolic arbitrary order equation

2019 ◽  
Vol 11 (2) ◽  
pp. 268-280
Author(s):  
V.S. Dron' ◽  
S.D. Ivasyshen ◽  
I.P. Medyns'kyi
Keyword(s):  
Cauchy Problem ◽  
Arbitrary Order ◽  
Homogeneous Equation ◽  
Fundamental Solutions ◽  
Order Equation ◽  
Hölder Spaces ◽  
The Cauchy Problem ◽  
Weighted Hölder Spaces ◽  
Holder Spaces ◽  
Derivatives Of

In weighted Hölder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solutions of the Cauchy problem for one ultraparabolic arbitrary order equation of the Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weighted Hölder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weighted Hölder spaces.

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN

2019 ◽  
Vol 16 (04) ◽  
pp. 639-661 ◽  
Author(s):  
Zhen Wang ◽  
Xinglong Wu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Initial Density ◽  
Chaplygin Gas ◽  
Bmo Space ◽  
Well Posedness ◽  
Dimension Formula ◽  
The Cauchy Problem ◽  
Bounded Below ◽  
Blow Up Criterion

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in [Formula: see text] for any dimension [Formula: see text]. First, given [Formula: see text], [Formula: see text], we prove the well-posedness property for solutions [Formula: see text] in the space [Formula: see text] for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density [Formula: see text] is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data [Formula: see text] in [Formula: see text] if [Formula: see text], and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

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