The Cauchy problem with impulse action and degeneration for
parabolic equations

I. D. Pukal’skii; B. O. Yashan

doi:10.30970/ms.52.1.63-70

We study the Cauchy problem for the second-order linear parabolic equations with impulse conditions in the time variable and power singularity in the coefficients of any order with respect to the time and space variables. By using the maximum principle and apriori estimates we establish the existence and uniqueness of solution of the problem in Holder spaces with power weight.

1. Perestyuk N.A., Plotnikov N.A., Samoilenko A.M., Skripnik N.V., Differential Equations with Impulse Effects, Berlin: Multivalued Right-hand Sides with Discontinuities – Walter de Gruyter Co, 2011. – 408 p.

2. Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations. – Singapore: World Scientific, 1995. – 462 p.

3. Boinov D.D., Simeonov P.S. Systems with impulse effects: Stability theory and applications. – New York etc.: John Wiley&Sons, 1989. – 225 p.

4. Perestyk N.A., Tkach A.B. Periodic solutions of a weakly linear system of partial differential equations with impulse action// Ukr. Mat. Zh. – 1997. – №4.(49) – P. 601–605.

5. Asanova A.T. On a nonlocal boundary value problem for systems of hyperbolic equations with impulse actions// Ukr. Mat. Zh. – 2013. – №3.(65) – P. 315–328.

6. Pykal’skii I.D., Isaryuk I.M. The boundary-value problem with impulse conditions and degenerations for parabolic equations// Ukr. Mat. Zh. – 2015. – №10.(67) – P. 1399–1408.

7. Pykal’skii I.D. The boundary value problem for parabolic equations with impulse conditions and degenerations// Маthеmаtical mеthоds and physicoмеchanical fields – 2015. – №2.(58) – P. 55–63.

8. Isaryuk I.M., Pukalskyi I.D. The boundary value problems for parabolic equations with a nonlocal condition and degenerations// Journal of Mathematical Sciences. – 2015. – №1.(207) – P. 26–38.

9. Pukalskyi I.D. Isaryuk I.M. Nonlocal parabolic boundary value problems with singularities// Journal of Mathematical Sciences. – 2015. – №3.(208) – P. 327–343.

10. Friedman A. Partial differential equations of parabolic type. – M.: World, 1968. – 427p.

11. Ladyzhenskaya O. A., Solonnikov V. A., Ural’tseva N. N. Linear and quasilinear equations of parabolic type. – М.: Science, 1967. – 736p.

12. Pukalskyi I.D. The boundary value problems for unevenly parabolic and elliptic equations with degeneration and singularities. – Chernivtsi: Ruta, 2008. – 253p.

	The Cauchy problem with impulse action and degeneration for parabolic equations
Author	I. D. Pukal’skii¹, B. O. Yashan² bohdanjaschan94@gmail.com² Yuriy Fedkovych Chernivtsi National University Chernivtsi, Ukraine
Abstract	We study the Cauchy problem for the second-order linear parabolic equations with impulse conditions in the time variable and power singularity in the coefficients of any order with respect to the time and space variables. By using the maximum principle and apriori estimates we establish the existence and uniqueness of solution of the problem in Holder spaces with power weight.
Keywords	degeneration; impulse action; apriori estimation; boundary condition
DOI	doi:10.30970/ms.52.1.63-70
Reference	1. Perestyuk N.A., Plotnikov N.A., Samoilenko A.M., Skripnik N.V., Differential Equations with Impulse Effects, Berlin: Multivalued Right-hand Sides with Discontinuities – Walter de Gruyter Co, 2011. – 408 p. 2. Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations. – Singapore: World Scientific, 1995. – 462 p. 3. Boinov D.D., Simeonov P.S. Systems with impulse effects: Stability theory and applications. – New York etc.: John Wiley&Sons, 1989. – 225 p. 4. Perestyk N.A., Tkach A.B. Periodic solutions of a weakly linear system of partial differential equations with impulse action// Ukr. Mat. Zh. – 1997. – №4.(49) – P. 601–605. 5. Asanova A.T. On a nonlocal boundary value problem for systems of hyperbolic equations with impulse actions// Ukr. Mat. Zh. – 2013. – №3.(65) – P. 315–328. 6. Pykal’skii I.D., Isaryuk I.M. The boundary-value problem with impulse conditions and degenerations for parabolic equations// Ukr. Mat. Zh. – 2015. – №10.(67) – P. 1399–1408. 7. Pykal’skii I.D. The boundary value problem for parabolic equations with impulse conditions and degenerations// Маthеmаtical mеthоds and physicoмеchanical fields – 2015. – №2.(58) – P. 55–63. 8. Isaryuk I.M., Pukalskyi I.D. The boundary value problems for parabolic equations with a nonlocal condition and degenerations// Journal of Mathematical Sciences. – 2015. – №1.(207) – P. 26–38. 9. Pukalskyi I.D. Isaryuk I.M. Nonlocal parabolic boundary value problems with singularities// Journal of Mathematical Sciences. – 2015. – №3.(208) – P. 327–343. 10. Friedman A. Partial differential equations of parabolic type. – M.: World, 1968. – 427p. 11. Ladyzhenskaya O. A., Solonnikov V. A., Ural’tseva N. N. Linear and quasilinear equations of parabolic type. – М.: Science, 1967. – 736p. 12. Pukalskyi I.D. The boundary value problems for unevenly parabolic and elliptic equations with degeneration and singularities. – Chernivtsi: Ruta, 2008. – 253p.
Pages	63-70
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The Cauchy problem with impulse action and degeneration for parabolic equations