# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021123
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## Parabolic problems in generalized Sobolev spaces

 1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37, Prosp. Peremohy, Kyiv, Ukraine, 03056 2 Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivs'ka st., Kyiv, Ukraine, 01024

* Corresponding author

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology)

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $\varphi:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $\varphi$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.

Citation: Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021123
##### References:
 [1] M. S. Agranovich and M. I. Vishik, Elliptic problems with parameter and parabolic problems of general form, (Russian) Uspehi Mat. Nauk, 19 (1964), 53–161 [English translation in Russian Math. Surveys, 19 (1964), 53–157].  Google Scholar [2] Y. Ameur, Interpolation between Hilbert spaces, in Analysis of Operators on Function Spaces, Trends Math. (eds. A. Aleman etc.), Birkhäuser/Springer, Cham, (2019), 63–115. doi: 10.1007/978-3-030-14640-5_4.  Google Scholar [3] A. Anop, R. Denk and A. Murach, Elliptic problems with rough boundary data in generalized Sobolev spaces, Commun. Pure Appl. Anal., 20 (2021), 697-735.  doi: 10.3934/cpaa.2020286.  Google Scholar [4] Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968.  Google Scholar [5] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss., band 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [6] O. V. Besov, V. P. Il'in and S. M. Nikol'skij, Integral Representations of Functions and Embedding Theorems, (Russian) 2$^{nd}$ edition, Nauka, Moscow, 1996.  Google Scholar [7] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.   Google Scholar [8] R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [9] W. F. Donoghue, The interpolation of quadratic norms, Acta Math., 118 (1967), 251-270.  doi: 10.1007/BF02392483.  Google Scholar [10] S. D. Eidel'man, Parabolic equations, in Encyclopaedia Math. Sci., vol. 63 (Partial Differential Equations, VI. Elliptic and Parabolic Operators) (eds. Yu.V. Egorov and M.A. Shubin), Springer, Berlin, (1994), 205–316. Google Scholar [11] S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.  Google Scholar [12] M. Fan, Qudratic interpolation and some operator inequalities, J. Math. Inequal., 5 (2011), 413-427.  doi: 10.7153/jmi-05-36.  Google Scholar [13] W. Farkas and H. G. Leopold, Characterisations of function spaces of generalized smoothness, Ann. Mat. Pura Appl., 185 (2006), 1-62.  doi: 10.1007/s10231-004-0110-z.  Google Scholar [14] C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282.   Google Scholar [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.  Google Scholar [16] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.  Google Scholar [17] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Differential Operators with Constant Coefficients, Springer, Berlin, 1983.  Google Scholar [18] A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, (Russian) Uspekhi Mat. Nauk, 17 (1962), 3–146; English translation in Russian Math. Surveys, 17: 3 (1962), 1–143. Google Scholar [19] V. A. Il'in, The solvability of mixed problems for hyperbolic and parabolic equations, (Russian) Uspekhi Mat. Nauk, 15 (1960), 97–154; English translation in Russian Math. Surveys, 15: 1 (1960), 85–142. Google Scholar [20] S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, American Mathematical Society, Providence, RI, 1982.  Google Scholar [21] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar [22] J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. II, Springer, Berlin, 1972.  Google Scholar [23] V. M. Los, Anisotropic Hörmander spaces on the lateral surface of a cylinder, J. Math. Sci. (N. Y.), 217 (2016), 456-467.  doi: 10.1007/s10958-016-2985-9.  Google Scholar [24] V. M. Los, Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case, Ukrainian Math. J., 68 (2016), 894-909.  doi: 10.1007/s11253-016-1264-8.  Google Scholar [25] V. M. Los, Classical Solutions of Parabolic Initial-Boundary-Value Problems and Hörmander Spaces, Ukrainian Math. J., 68 (2017), 1412-1423.  doi: 10.1007/s11253-017-1303-0.  Google Scholar [26] V. Los, V. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.  Google Scholar [27] V. Los and A. Murach, Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces, Open Math., 15 (2017), 57-76.  doi: 10.1515/math-2017-0008.  Google Scholar [28] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser Verlag, Basel, 1995.  Google Scholar [29] V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian. Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.  Google Scholar [30] V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar [31] V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.  Google Scholar [32] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean spaces, Birkhäser, Basel, 2010. Google Scholar [33] V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep., 1 (1984), 349-515.   Google Scholar [34] B. Paneah, The Oblique Derivative Problem. The Poincaré problem, Wiley–VCH, Berlin, 2000.  Google Scholar [35] E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.  Google Scholar [36] L. N. Slobodeckii, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, (Russian) Leningrad. Gos. Ped. Inst. Uchen. Zap., 197 (1958), 54–112; English translation in Amer. Math. Soc. Transl. (2), 57 (1966), 207–275.  Google Scholar [37] V. A. Solonnikov, Apriori estimates for solutions of second-order equations of parabolic type, (Russian) Tr. Mat. Inst. Steklova, 70 (1964), 133–212. Google Scholar [38] H. Triebel, Interpolation Theory, Function Spaces, Differential, Operators, 2$^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [39] H. Triebel, The Structure of Functions, Birkhäser, Basel, 2001.  Google Scholar [40] L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, (Russian) Uspekhi Mat. Nauk, 20 (1965), 3–74; English translation in Russian Math. Surveys, 20 (1965), 1–73.  Google Scholar [41] N. V. Zhitarashu, Theorems on complete collection of isomorphisms in the $L_2$-theory of generalized solutions for one equation parabolic in Petrovski$\breve{l}$ sense, Mat. Sb., 128 (1985), 451-473.   Google Scholar

show all references

##### References:
 [1] M. S. Agranovich and M. I. Vishik, Elliptic problems with parameter and parabolic problems of general form, (Russian) Uspehi Mat. Nauk, 19 (1964), 53–161 [English translation in Russian Math. Surveys, 19 (1964), 53–157].  Google Scholar [2] Y. Ameur, Interpolation between Hilbert spaces, in Analysis of Operators on Function Spaces, Trends Math. (eds. A. Aleman etc.), Birkhäuser/Springer, Cham, (2019), 63–115. doi: 10.1007/978-3-030-14640-5_4.  Google Scholar [3] A. Anop, R. Denk and A. Murach, Elliptic problems with rough boundary data in generalized Sobolev spaces, Commun. Pure Appl. Anal., 20 (2021), 697-735.  doi: 10.3934/cpaa.2020286.  Google Scholar [4] Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968.  Google Scholar [5] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss., band 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [6] O. V. Besov, V. P. Il'in and S. M. Nikol'skij, Integral Representations of Functions and Embedding Theorems, (Russian) 2$^{nd}$ edition, Nauka, Moscow, 1996.  Google Scholar [7] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.   Google Scholar [8] R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [9] W. F. Donoghue, The interpolation of quadratic norms, Acta Math., 118 (1967), 251-270.  doi: 10.1007/BF02392483.  Google Scholar [10] S. D. Eidel'man, Parabolic equations, in Encyclopaedia Math. Sci., vol. 63 (Partial Differential Equations, VI. Elliptic and Parabolic Operators) (eds. Yu.V. Egorov and M.A. Shubin), Springer, Berlin, (1994), 205–316. Google Scholar [11] S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.  Google Scholar [12] M. Fan, Qudratic interpolation and some operator inequalities, J. Math. Inequal., 5 (2011), 413-427.  doi: 10.7153/jmi-05-36.  Google Scholar [13] W. Farkas and H. G. Leopold, Characterisations of function spaces of generalized smoothness, Ann. Mat. Pura Appl., 185 (2006), 1-62.  doi: 10.1007/s10231-004-0110-z.  Google Scholar [14] C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282.   Google Scholar [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.  Google Scholar [16] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.  Google Scholar [17] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Differential Operators with Constant Coefficients, Springer, Berlin, 1983.  Google Scholar [18] A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, (Russian) Uspekhi Mat. Nauk, 17 (1962), 3–146; English translation in Russian Math. Surveys, 17: 3 (1962), 1–143. Google Scholar [19] V. A. Il'in, The solvability of mixed problems for hyperbolic and parabolic equations, (Russian) Uspekhi Mat. Nauk, 15 (1960), 97–154; English translation in Russian Math. Surveys, 15: 1 (1960), 85–142. Google Scholar [20] S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, American Mathematical Society, Providence, RI, 1982.  Google Scholar [21] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar [22] J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. II, Springer, Berlin, 1972.  Google Scholar [23] V. M. Los, Anisotropic Hörmander spaces on the lateral surface of a cylinder, J. Math. Sci. (N. Y.), 217 (2016), 456-467.  doi: 10.1007/s10958-016-2985-9.  Google Scholar [24] V. M. Los, Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case, Ukrainian Math. J., 68 (2016), 894-909.  doi: 10.1007/s11253-016-1264-8.  Google Scholar [25] V. M. Los, Classical Solutions of Parabolic Initial-Boundary-Value Problems and Hörmander Spaces, Ukrainian Math. J., 68 (2017), 1412-1423.  doi: 10.1007/s11253-017-1303-0.  Google Scholar [26] V. Los, V. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.  Google Scholar [27] V. Los and A. Murach, Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces, Open Math., 15 (2017), 57-76.  doi: 10.1515/math-2017-0008.  Google Scholar [28] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser Verlag, Basel, 1995.  Google Scholar [29] V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian. Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.  Google Scholar [30] V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar [31] V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.  Google Scholar [32] F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean spaces, Birkhäser, Basel, 2010. Google Scholar [33] V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep., 1 (1984), 349-515.   Google Scholar [34] B. Paneah, The Oblique Derivative Problem. The Poincaré problem, Wiley–VCH, Berlin, 2000.  Google Scholar [35] E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.  Google Scholar [36] L. N. Slobodeckii, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, (Russian) Leningrad. Gos. Ped. Inst. Uchen. Zap., 197 (1958), 54–112; English translation in Amer. Math. Soc. Transl. (2), 57 (1966), 207–275.  Google Scholar [37] V. A. Solonnikov, Apriori estimates for solutions of second-order equations of parabolic type, (Russian) Tr. Mat. Inst. Steklova, 70 (1964), 133–212. Google Scholar [38] H. Triebel, Interpolation Theory, Function Spaces, Differential, Operators, 2$^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [39] H. Triebel, The Structure of Functions, Birkhäser, Basel, 2001.  Google Scholar [40] L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, (Russian) Uspekhi Mat. Nauk, 20 (1965), 3–74; English translation in Russian Math. Surveys, 20 (1965), 1–73.  Google Scholar [41] N. V. Zhitarashu, Theorems on complete collection of isomorphisms in the $L_2$-theory of generalized solutions for one equation parabolic in Petrovski$\breve{l}$ sense, Mat. Sb., 128 (1985), 451-473.   Google Scholar
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