Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities. (English) Zbl 1411.35177

Summary: In this study, we consider nonlinear parabolic equations with variable exponents of the nonlinearities. The nonlocal problem for these equations is investigated, and the existence and uniqueness theorems are proved for the problem.


35K59 Quasilinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: DOI


[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York, San Francisco, London · Zbl 0314.46030
[2] Aizicovici, S.; Lee, H., Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett., 18, 401-407 (2005) · Zbl 1084.34002
[3] Antontsev, S.; Shmarev, S., Evolution PDEs with Nonstandard Growth Conditions. Existence, Uniqueness, Localization, Blow-up, Atlantis Studies in Diff. Eq., vol. 4 (2015), Atlantis Press: Atlantis Press Paris · Zbl 1410.35001
[4] Aubin, J.-P., Un theoreme de compacite, C. R. Hebd. Séances Acad. Sci., 256, 24, 5042-5044 (1963) · Zbl 0195.13002
[5] Bernis, F., Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279, 373-394 (1988) · Zbl 0609.35048
[6] Bokalo, T., Mixed problem for parabolic equation with dislocated action, Visnyk (Herald) Lviv Univ. Ser. Mech.-Math., 68, 43-58 (2008) · Zbl 1199.35179
[7] Bokalo, M. M.; Buhrii, O. M.; Mashiyev, R. A., Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonlinear Evol. Equ. Appl., 2013, 6, 67-87 (2014) · Zbl 1330.35109
[8] Borok, V.; Kengne, E., Classification of integral boundary value problems in a wide band, Izv. Vuzov Math., 5, 384, 3-12 (1994)
[9] Boucherif, A.; Precup, R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 16, 507-516 (2007) · Zbl 1154.34027
[10] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Springer: Springer New York, Dordrecht, Heidelberg, London · Zbl 1220.46002
[11] Buhrii, O. M., Finiteness of time vanishing of the solution of a nonlinear parabolic variational inequality with variable exponent of nonlinearity, Mat. Stud., 24, 2, 167-172 (2005) · Zbl 1091.35513
[12] Buhrii, O. M., On integration by parts formulaes for special type of exponential functions, Mat. Stud., 45, 2, 118-131 (2016) · Zbl 1516.26007
[13] Buhrii, O. M., Nonlocal problem for nonlinear parabolic equations with variable exponents of the nonlinearities, Trans. Inst. Math. NAS Ukr., 14, 3, 47-75 (2017) · Zbl 1399.35234
[14] Buhrii, O.; Buhrii, N., Integro-differential systems with variable exponents of nonlinearity, Open Math., 15, 859-883 (2017) · Zbl 1383.35237
[15] Buhrii, O.; Domans’ka, G.; Protsakh, N., Initial boundary value problem for nonlinear differential equation of the third order in generalized Sobolev spaces, Visnyk (Herald) Lviv Univ. Ser. Mech.-Math., 64, 44-61 (2005)
[16] Byszewski, L., Abstract nonlinear nonlocal problems and their physical interpretations, (Akca, Haydar; etal., Biomathematics, Bioinformatics and Applications of Functional Differential Difference Equations. Biomathematics, Bioinformatics and Applications of Functional Differential Difference Equations, Alanya, July 14-19, 1999 (2000), Akdeniz Univ. Pres: Akdeniz Univ. Pres Antalya), 77-86
[17] Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179, 630-637 (1993) · Zbl 0798.35076
[18] Diening, L.; Harjulehto, P.; Hästö, P.; Ru̇žička, M., Lebesgue and Sobolev Spaces with Variable Exponents (2011), Springer: Springer Heidelberg · Zbl 1222.46002
[19] Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[20] Fan, X.-L.; Zhao, D., On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m, p(x)}(\Omega)\), J. Math. Anal. Appl., 263, 424-446 (2001) · Zbl 1028.46041
[21] Fardigola, L. V., Criterion of correctness in a layer of boundary value problem with integral condition, Ukrainian Math. J., 42, 11, 1546-1551 (1990)
[22] Gajewski, H.; Groger, K.; Zacharias, K., Nonlinear Operator Equations and Operator Differential Equations (1978), Mir: Mir Moscow, (translated from: Akademie-Verlag, Berlin, 1974)
[23] Gordeziani, D.; Avalishvili, M.; Avalishvili, G., On the investigation of one nonclassical problem for Navier-Stokes equations, Appl. Math. Inform., 7, 2, 66-77 (2002) · Zbl 1062.35062
[24] Ignjatovic, M., On solving parabolic equation with homogeneous boundary and integral initial conditions, Mat. Vesnik, 67, 4, 258-268 (2015) · Zbl 1462.65107
[25] Il’kiv, V. S.; Ptashnik, B. I., Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators, Ukraïn. Mat. Zh., 58, 12, 1624-1650 (2006) · Zbl 1114.35001
[26] Ivanchov, M., Inverse Problems for Equations of Parabolic Type, Mathematical Studies, Monograph Series, vol. 10 (2003), VNTL Publ.: VNTL Publ. Lviv · Zbl 1147.35110
[27] Ivanchov, M. I.; Pabyrivs’ka, N. V., Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions, Ukrainian Math. J., 53, 5, 674-684 (2001) · Zbl 0991.35102
[28] Jackson, D., Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 172, 256-265 (1993) · Zbl 0814.35060
[29] Kalenyuk, P. I.; Kogut, I. V.; Nytrebych, Z. M., Problem with integral condition for partial differential equation of the first order in time, Mat. Metodi Fiz.-Mekh. Polya, 53, 4, 7-16 (2010) · Zbl 1240.35083
[30] Kirichenko, S. V., On a boundary value problem for mixed type equation with nonlocal initial conditions in the rectangle, Vestnik Samara State Tech. Univ. Ser. Phys.-Math., 3, 32, 185-189 (2013) · Zbl 1413.35347
[31] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\), Czechoslovak Math. J., 41, 116, 592-618 (1991) · Zbl 0784.46029
[32] Kozhanov, A. I., A time-nonlocal boundary problem for linear parabolic equations, Sib. Zh. Ind. Mat., 7, 1, 51-60 (2004) · Zbl 1049.35003
[33] Kozhanov, A. I., Solvability of boundary value problems for linear parabolic equations with an integral condition in a time variable, Mat. Zametki SVFU, 21, 4, 20-30 (2014) · Zbl 1374.35202
[34] Lions, J.-L., Some Methods of Solving of Nonlinear Boundary Value Problems (1972), Mir: Mir Moscow, (translated from: Dunod, Gauthier-Villars, Paris, 1969)
[35] Loayza, M., Global existence and blow up results for a heat equation with nonlinear nonlocal term, Differential Integral Equations, 25, 7-8, 665-683 (2012) · Zbl 1265.35122
[36] Mashiyev, R. A.; Buhrii, O. M., Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity, J. Math. Anal. Appl., 377, 450-463 (2011) · Zbl 1208.49007
[37] Medvid, O. M.; Symotyuk, M. M., Integral problem for linear partial differential equations, Mat. Stud., 28, 2, 115-140 (2007) · Zbl 1164.11327
[38] Mikhailov, V. P., Partial Differential Equations (1976), Nauka: Nauka Moscow
[39] Ntouyas, S. K.; Tsamatos, P. C., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210, 679-687 (1997) · Zbl 0884.34069
[40] Olmstead, W. E.; Roberts, C. A., The one-dimensional heat equation with nonlocal initial condition, Appl. Math. Lett., 10, 3, 89-94 (1997) · Zbl 0888.35042
[41] Pao, C. V., Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl., 195, 702-718 (1995) · Zbl 0851.35063
[42] Popov, A. Y.; Tikhonov, I. V., Exponential classes of solvability in heat conduction problem with nonlocal mean in time condition, Mat. Sb., 196, 9, 72-102 (2005)
[43] Pukalskyi, I. D.; Isariuk, I. M., Nonlocal problem with skew derivative for degenerate parabolic equations, Nonlinear Bound. Value Probl., 21, 135-147 (2012) · Zbl 1324.35093
[44] Pulkina, L. S., On a class of nonlocal problems and their connection with inverse problems, (Proc. of Third All-Russian Sci. Conf. Part 3, Matem. Mod. Kraev. Zadachi (2006)), 190-192
[45] Rădulescu, V.; Repovš, D., Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (2015), CRC Press: CRC Press Boca Raton, London, New York · Zbl 1343.35003
[46] Shelukhin, V. V., The problem of particle movement in the model Burgers system of a viscous gas, Dinamika Sploshn. Sredy, 87, 136-154 (1988) · Zbl 0702.76082
[47] Shelukhin, V. V., Problem with mean in time data for nonlinear parabolic equations, Sib. Math. J., 32, 2, 154-165 (1991)
[48] Shelukhin, V. V., A variational principle for linear evolution problems nonlocal in time, Sib. Math. J., 34, 2, 369-384 (1993) · Zbl 0835.34078
[49] Shelukhin, V., A non-local in time model for radionuclides propagation in Stokes fluid, Dinamika Sploshn. Sredy (Novosibirsk), 107, 180-193 (1993) · Zbl 0831.76086
[50] Shelukhin, V., A problem nonlocal in time for the equations of the dynamics of a barotropic ocean, Sib. Math. J., 36, 3, 608-630 (1995) · Zbl 0855.76089
[51] Smagin, V. V.; Huyen, N. T., Convergence Galerkin’s method of approximate solution to parabolic equation with integral condition on solution, Vestnik Voronezh State Univ. Ser. Phys.-Math., 1, 144-149 (2010) · Zbl 1475.65128
[52] Souplet, P., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153, 374-406 (1999) · Zbl 0923.35077
[53] Suhubi, E., Functional Analysis (2003), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht, Boston, London
[54] Walker, C., A note on a nonlocal nonlinear reaction-diffusion model, Appl. Math. Lett., 25, 1772-1777 (2012) · Zbl 1252.35044
[55] Walker, C., Global continua of positive solutions for some quasilinear parabolic equation with a nonlocal initial condition, J. Dynam. Differential Equations, 25, 1, 159-172 (2013) · Zbl 1264.35029
[56] Webb, G. F., Population models structured by age, size, and spatial position, (Magal, P.; Ruan, S., Structured Population Models in Biology and Epidemiology. Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936 (2008), Springer-Verlag: Springer-Verlag Berlin, Heidelberg)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.