Abstract
We study the solvability of nonlocal boundary-value problems for singular parabolic equations of higher order in which the coefficient of the time derivative belongs to a space Lp of spatial variables and possesses a certain smoothness with respect to time. No constraints are imposed on the sign of this coefficient, i.e., the class of equations also contains parabolic equations with varying time direction. We obtain conditions guaranteeing the solvability of boundary-value problems in weighted Sobolev spaces and the uniqueness of the solutions.
Similar content being viewed by others
References
M. Gevrey, “Sur les equations aux derivees partielles du type parabolique,” C. R. 152, 428–431 (1911).
M. Gevrey, “Sur les equations aux derivees partielles du type parabolique (suite),” Journ. de Math. (6) 10, 105–148 (1914).
W. Bothe, “Die Streueabsorption der Electronenstronenstrahlen,” Z. Phys. 5, 101–178 (1929).
C. Cercignani, Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969).
R. Beals and V. Protopescu, “Half-range completeness for the Fokker-Planck equation,” J. Statist. Phys. 32 (3), 565–584 (1983).
C. Knessel and J. B. Keller, “Ray solutions of forward-backward parabolic problems for data-handling systems,” European J. Appl. Math. 11(1), 1–12(2000).
V. N. Monakhov, “Counter flows in boundary layer,” in Dynamica of Continuous Media (Inst. Gidrodin. SO RAN, Novosibirsk, 1998), Vol. 113, pp. 107–113 [in Russian].
R. J. Hangelbroek, “Linear analysis and solution neutron transport problem,” Transport Theory Statist. Phys. 5(1), 1–85 (1976).
K. Latrach, “Compactness properties for linear transport operator with abstract boundary conditions in slab geometry,” Transport Theory Statist. Phys. 22 (1), 39–64 (1993).
K. Latrach and M. Mokhtar-Kharroubi, “Spectral analysis and generation results for streaming operator with multiplying boundary conditions,” Positivity 3 (3), 273–296 (1999).
G. Webb, “On an unbounded linear operator arfroming in the theory of growing cell population with inherited cycle length,” J. Math. Biol. 23 (2), 269–282 (1986).
N. A. Lar'kin, V. A. Novikov, and N. N. Yanenko, Nonlinear Equations of Variable Type (Nauka, Novosibirsk, 1983) [in Russian].
S. V. Popov and V. G. Markov, “Boundary value problems for parabolic equations of high order with a changing time direction,” J. Phys. Conf. Ser. 894 (1), 012075 (2017).
S. V. Popov and S. V. Potapova, “Holder classes of solutions to 2n-parabolic equations with a varying direction of evolution,” Dokl. Akad. Nauk 424 (5), 594–596 (2009) [Dokl. Math. 79 (1), 100–102 (2009)].
S. A. Tersenov, Parabolic Equations with Changing Time Direction (Nauka, Moscow, 1985) [in Russian].
M. S. Baouendi and P. Grisvard, “Surune equation d'evolution changeante de type,” J. Funct. Anal. 2 (3), 352–367 (1968).
C. D. Pagani, “On the parabolic equation sgn(x)xp uy - uxx = 0,” Ann. Mat. Pura Appl. (4) 99 (1), 333–399 (1974).
L. Boulton, M. Marletta, and D. Rule, “On the stability of a forward-backward heat equation,” Integral Equations and Operator Theory 73 (2), 195–216(2012).
S. G. Pyatkov, “Solvability of a boundary-value problem for a parabolic equation with changing time direction,” Dokl. Akad. Nauk SSSR 285 (6), 1327–1329 (1985).
K. Hollig, “Existence of infinitely many solutions for a forward backward heat equation,” Trans. Amer. Math. Soc. 278 (1), 299–316(1987).
P. I. Plotnikov, “Equations with a variable direction of parabolicity and the hysteresis effect,” Dokl. Akad. Nauk SSSR 330 (6), 691–693 (1987) [Dokl. Math. 47 (3), 604–608 (1993)].
I. V. Kuznetsov, “Entropy solutions to a second order forward-backward parabolic differential equation,” Sibirsk. Mat. Zh. 46 (3), 594–619 (2005) [Sib. Math. J. 46 (3), 467–488 (2005)].
I. V. Kuznetsov and S. A. Sazhenkov, “Quasi-solutions of genuinely nonlinear forward-backward ultra-parabolic equations,” J. Phys. Conf. Ser. 894(1), 012046 (2017).
B. T. Thanh, F. Smarrazzo, and A. Tesei, “Sobolev regularfromation the class of forward-backward parabolic equations,” J. Differential Equations 257 (5), 1403–1456(2014).
A. Terracina, Two-Phase Entropy Solutions of Forward-Backward Parabolic Problems with Unstable Phase, arXiv: 1310.7728(2013).
M. Bertsch, F. Smarrazzo, and A. Tesei, “On the class of forward-backward parabolic equations: properties of solutions,” SIAM J. Math. Anal. 49 (3), 2037–2060 (2017).
M. Bertsch, F. Smarrazzo, and A. Tesei, “Pseudo-parabolic regularfromation forward-backward parabolic equations: power-typenonlinearities,” J. Reine Angew. Math. 712, 51–80(2016).
S. Kim and B. Yan, “On Lipschitz solutions for some forward-backward parabolic equations,” Ann. Inst. H. Poincare Anal. Non Lineaire 35 (1), 65–100 (2018).
L. C. Evans and M. Portilheiro, “Irreversibility and hysteresis for a forward-backward diffusion equation,” Math. Models Methods Appl. Sci. 14(11), 1599–1620(2004).
S. N. Antontsev and I. V. Kuznetsov, “Singular perturbations of forward-backward p-parabolic equations,” J. Elliptic Parabol. Equ. 2(1-2), 357–370 (2016).
E. Bonetti, P. Colli, and G. Tomassetti, “A non-smooth regularfromation a forward-backward parabolic equations,” Math. Models Methods Appl. Sci. 27 (4), 641–661 (2017).
M. Chugunova and S. Pyatkov, “Compactly supported solutions for a rimming flow model,” Nonlinearity 27 (4), 803–822 (2014).
W. Greenberg, C. V. M. Van der Mee, and P. F. Zweifel, “Generalized kinetic equations,” Integral Equations Operator Theory 7(1), 60–95 (1984).
W. Greenberg, C. V. M. Van der Mee, and V Protopopescu, Boundary Value Problems in Abstract Kinetic Theory, in Oper. Theory Adv. Appl. (Birkhauser Verlag, Basel, 1987), Vol. 23.
R. Beals, “An abstract treatment of some forward-backward problems of transport and scattering,” J. Funct. Anal. 34 (1), 1–20 (1984).
N. V Kislov, “Nonhomogeneous boundary-value problems for differential-operator equations of mixed type, and their application,” Mat. Sb. 125 (167), 19–37 (1984) [Math. USSR-Sb. 53 (1), 17–35 (1986)].
I. M. Karabash, “Abstract kinetic equations with positive collision operators,” in Spectral Theory in Inner Product Spaces and Applications, Oper. Theory Adv. Appl. (Birkhauser Verlag, Basel, 2008), Vol. 188, pp. 175–195.
S. Pyatkov, S. Popov, and V Antipin, “On solvability of boundary value problems for kinetic operator-differential equations,” Integral Equations Operator Theory 80 (4), 557–580 (2014).
I. E. Egorov, S. G. Pyatkov, and S. V Popov, Nonclassical Operator-Differential Equations (Nauka, Novosibirsk, 2000) [in Russian].
N. L. Abasheeva, “Solvability of a periodic boundary-value problem for an operator-differential equation of mixed type,” Vestnik NGU Ser. Mat. Mekh. Inf. 1 (2), 3–18 (2001).
S. G. Pyatkov, “Boundary-value problems for some classes of singular parabolic equations,” Mat. Tr. 6 (2), 144–208 (2003) [Siberian Adv. Math. 14 (3), 63–125 (2004)].
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland Publ., Amsterdam, 1978).
H. Amann, “Compact embeddings of vector-valued Sobolev and Besov spaces,” Glas. Mat. Ser. III 35 (55) (1), 161–177 (2000).
K. Yosida, Functional Analysis (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965; Mir, Moscow, 1967).
Yu. M. Berezanskii, Eigenfunction Expansion of Self-Adjoint Operators (Naukova Dumka, Kiev, 1965) [in Russian].
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications (Springer-Verlag, Berlin, 1972), Vol. 1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 4, pp. 578–594.
Rights and permissions
About this article
Cite this article
Pyatkov, S.G. On Some Classes of Nonlocal Boundary-Value Problems for Singular Parabolic Equations. Math Notes 106, 602–615 (2019). https://doi.org/10.1134/S0001434619090281
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619090281