On the cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces

1992 ◽  
Vol 32 (2) ◽  
pp. 238-264 ◽  
Author(s):  
R. Mikulevičius ◽  
H. Pragarauskas
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Hölder Spaces ◽  
The Cauchy Problem ◽  
Holder Spaces

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.

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

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