On the first mixed problem in Banach spaces for the degenerate parabolic equations with changing time direction

  • Petrushko Igor M., petrushkoim@mpei.ru Moscow Power Engineering Institute, 14 Krasnokazarmennaya Street, Moscow 111250 Russia
  • Petrushko Maksim I., petrushkomi@mpei.ru Moscow Power Engineering Institute, 14 Krasnokazarmennaya Street, Moscow 111250 Russia
Keywords: degenerate equations, changing time direction, functional spaces, integral identities, first mixed problem, solvability

Abstract

The paper is devoted to the study of a section of nonclassical differential equations, namely, the solvability problems for second-order parabolic equations with changing time direction. It is well known that in ordinary boundary value problems for strictly parabolic equations the smoothness of the initial and boundary conditions completely ensures that the solutions belong to Holder spaces, but in the case of equations with changing time direction the smoothness of the initial and boundary conditions does not ensure that the solutions belong to such spaces. S. A. Tersenov, for a model parabolic equation with changing time direction, and S. G. Pyatkov, for a more general second-order equation, both obtained necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces, while the initial and boundary conditions were always assumed to be zero. In this paper, we consider cases where the initial and boundary conditions belong to Banach spaces. We introduce functional spaces in which solutions must be sought and obtain appropriate a priori estimates that make it possible to find solvability conditions for these problems. We also study the properties of the solutions obtained. In particular, we establish the equivalence of the Riesz and Littlewood–Paley conditions analogous to those for solutions of strictly elliptic and strictly parabolic equations of the second order. The unique solvability of the first mixed problem with boundary and initial functions from a Banach space is proved.

References


[1]
Gevrey M., “Sur les equations aux derivees partielles du type parabolique,” J. Math. Appl., 9, No. 6, 305–478 (1913).

[2]
Baouendi M. S. and Grisvard P., “Sur une equation d’evolution changeante de type,” J. Funct. Anal., 2, No. 3, 352–367 (1968).

[3]
Tersenov S. A., “On the first boundary problem for a parabolic equation with changing time direction,” Dokl. Math., 53, No. 3, 341–343 (1996).

[4]
Tersenov S. A., Parabolic Equations with Changing Time Direction [in Russian], Nauka, Novosibirsk (1985).

[5]
Pyatkov S. G., Operator Theory. Nonclassical Problems, VSP, Utrecht (2002).

[6]
Pyatkov S. G., “On the solvability of a boundary value problem for a parabolic equation with changing time direction,” Sov. Math., Dokl., 285, No. 6, 1322–1327 (1985).

[7]
Egorov I. E., “On modified Galerkin method for parabolic equations with changing evolution direction,” Uzbek. Mat. Zh., 13 (2013).

[8]
Popov S. V., “On smoothness of solutions to parabolic equations with changing evolution direction,” Dokl. Math., 400, No. 1, 29–31 (2005).

[9]
Popov S. V., “The Gevrey boundary value problem for a third order equation,” Mat. Zamet. SVFU, 24, No. 1, 43–56 (2017).

[10]
Antipin V. I. and Popov S. V., “Boundary problems for third order equations with changing time direction,” Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program., 40, No. 14, 19–28 (2012).

[11]
Popov S. V. and Potapova S. V., “Holder classes of solutions to 2n-parabolic equations with changing direction of evolution,” Dokl. Math., 79, No. 1, 100–102 (2009).

[12]
Egorov I. E. and Efimova E. S., “Stationary Galerkin method for a parabolic equation with changing time direction [in Russian],” Mat. Zamet. YAGU, 18, No. 2, 41–47 (2011).

[13]
Egorov I. E. and Tikhonova I. M., “Stationary Galerkin method for a mixed-type second order equation [in Russian],” Mat. Zamet. YAGU, 17, No. 2, 41–47 (2010).

[14]
Petrushko I. M. and Chernykh E. V., “On initial-boundary value problem for equations with changing time direction [in Russian],” Vestn. MPEI, No. 6, 60–70 (2000).

[15]
Petrushko I. M., “On the first problem for the degenerate parabolic equations with changing time direction [in Russian],” Vestn. MPEI, No. 1, 53–58 (2016).

[16]
Mikhailov V. P., “On boundary values of solutions to elliptic second-order equations on domains with smooth borders [in Russian],” Mat. Sb., Nov. Ser., 101, 163–188 (1976).

[17]
Ladyzhenskaya O. A., Solonnikov V. A., and Ural’ceva N. N., Linear and Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1967).

[18]
Gushchin A. K. and Mikhailov V. P., “On boundary values of solutions to second-order elliptic equations in $L_p, p > 1$ [in Russian],” Mat. Sb., Nov. Ser., 108, 3–21 (1979).

[19]
Egorov I. E., Pyatkov S. G., and Popov S. V., Nonclasical Differential-Operator Equations [in Russian], Nauka, Novosibirsk (2000).
How to Cite
Petrushko, I. and Petrushko, M. ( ) “On the first mixed problem in Banach spaces for the degenerate parabolic equations with changing time direction”, Mathematical notes of NEFU, 25(4), pp. 45-59. doi: https://doi.org/10.25587/SVFU.2018.100.20553.
Section
Mathematics