najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/22208054201561146153najo-1019Research ArticleCONTRIBUTED TALKSCONTRIBUTED TALKSStrong solutions and the initial data space for some non-uniformly parabolic equationsSkryabinM. A.

Saint Petersburg

mskryabin@my.ifmo.ruITMO UniversityRussian Federation20151508202561146153Copyright © Skryabin M.A., 20252025Skryabin M.A.Skryabin M.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1019

This paper is devoted to strong solutions of the first and second initialboundary problems for nonuniformly parabolic equations. These equations are used in mechanics, glaciology, rheology, image processing as well as for nanosystem modeling. The initial data space for these problems was explicitly described as Orlicz—Sobolev spaces.

nonuniformly parabolic equationstrong solutioninitial data spaceOrlicz—Sobolev spaceThe author is grateful to Professor Shulin Zhou for his comments about this work. This work was supported by Beijing International Center for Mathematical Research.ReferencesAdams R. A. Sobolev Spaces. Boston: Academic Press (1975).Adams R. A. Sobolev Spaces. Boston: Academic Press (1975).Ashyralyev A., Sobolevskii P. E. WellPosedness of Parabolic Difference Equations. Basel, Boston, Berlin: Birkh¨auser (1994).Ashyralyev A., Sobolevskii P. E. WellPosedness of Parabolic Difference Equations. Basel, Boston, Berlin: Birkh¨auser (1994).Cai Y., Zhou Sh. Existence and uniqueness of weak solutions for a nonuniformly parabolic equation. J. Functional Analysis, 257, P. 30213042 (2009).Cai Y., Zhou Sh. Existence and uniqueness of weak solutions for a nonuniformly parabolic equation. J. Functional Analysis, 257, P. 30213042 (2009).DiBenedetto E. Degenerate Parabolic Equations. Berlin: SpringerVerlag (1993).DiBenedetto E. Degenerate Parabolic Equations. Berlin: SpringerVerlag (1993).Donaldson T. K., Trudinger N. S. Orlicz–Sobolev spaces and imbedding theorems. J. Functional Analysis, 8, P. 52–75 (1971).Donaldson T. K., Trudinger N. S. Orlicz–Sobolev spaces and imbedding theorems. J. Functional Analysis, 8, P. 52–75 (1971).Fuchs M., Mingione G. Full C1;regularity for free and contrained local minimizers of elliptic variational integrals with nearly linear growth. Manuscripta Math., 102, P. 227250 (2000).Fuchs M., Mingione G. Full C1;regularity for free and contrained local minimizers of elliptic variational integrals with nearly linear growth. Manuscripta Math., 102, P. 227250 (2000).Krasnosel’skii M. A., Rutickii Ya. B. Convex Functions and Orlicz Spaces. Noordhoff, Groningen, The Netherlands (1961).Krasnosel’skii M. A., Rutickii Ya. B. Convex Functions and Orlicz Spaces. Noordhoff, Groningen, The Netherlands (1961).Lions J.L. Some Methods of Solution of Nonlinear BoundaryValue Problems. Moscow: Mir (1972).Lions J.L. Some Methods of Solution of Nonlinear BoundaryValue Problems. Moscow: Mir (1972).Lions J.L., Magenes E. NonHomogeneous Boundary Value Problems and Applications. Berlin: Springer (1972).Lions J.L., Magenes E. NonHomogeneous Boundary Value Problems and Applications. Berlin: Springer (1972).Luxemburg W. Banach function spaces. PhD Thesis, Technische Hogeschool te Delft, The Netherlands (1955).Luxemburg W. Banach function spaces. PhD Thesis, Technische Hogeschool te Delft, The Netherlands (1955).Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: SpringerVerlag (1983).Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: SpringerVerlag (1983).Perona P., Malik J. Scalespace and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intelligence, 12 (7), P. 629639 (1990).Perona P., Malik J. Scalespace and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intelligence, 12 (7), P. 629639 (1990).Shamin R.V. Spaces of initial data for differential equations in a Hilbert space. Sb. Math., 194, P. 1411–1426 (2003).Shamin R.V. Spaces of initial data for differential equations in a Hilbert space. Sb. Math., 194, P. 1411–1426 (2003).Shin K., Kang S. Doubly nonlinear parabolic equations involving pLaplacian Operators via timediscretization method. Bull. Korean Math. Soc., 49 (6), P. 11791192 (2012).Shin K., Kang S. Doubly nonlinear parabolic equations involving pLaplacian Operators via timediscretization method. Bull. Korean Math. Soc., 49 (6), P. 11791192 (2012).Tribel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North Holland (1978).Tribel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North Holland (1978).Umantsev A. Thermal effects of phase transformations: A review. Physica D, 235, P. 114 (2007).Umantsev A. Thermal effects of phase transformations: A review. Physica D, 235, P. 114 (2007).Wang L., Zhou Sh. Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis. J. Partial Differential Equations, 19 (2), P. 97112 (2006).Wang L., Zhou Sh. Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis. J. Partial Differential Equations, 19 (2), P. 97112 (2006).Zhang Ch., Zhou Sh. Entropy solutions for a nonuniformly parabolic equation. Manuscripta Math., 131 (34), P. 335354 (2010).Zhang Ch., Zhou Sh. Entropy solutions for a nonuniformly parabolic equation. Manuscripta Math., 131 (34), P. 335354 (2010).

The authors declare that there are no conflicts of interest present.