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Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity

Yıl 2024, Cilt: 53 Sayı: 2, 367 - 381, 23.04.2024

Rabil Ayazoğlu , Gülizar Alisoy , Sezgin Akbulut , Tuba Ağırman Aydın

https://doi.org/10.15672/hujms.1106985
Cited By: 1

Öz

In this paper, we consider an initial boundary value problem for a class of $p(\cdot )$-Laplacian parabolic equation with nonstandard nonlinearity in a bounded domain. By using new approach, we obtain the global and decay of existence of the solutions. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived.

Anahtar Kelimeler

parabolic $p(\cdot )$-Laplacian variable exponents global existence decay property extinction

Kaynakça

  • [1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Archive for Rational Mechanics and Analysis 164, 213-259, 2002.
  • [2] M.K. Alaoui, S.A. Messaoudi and H.B. Khenous, A blow-up result for nonlinear generalized heat equation, Computers & Mathematics with Applications 68 (12), 1723-1732, 2014.
  • [3] S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publicacions Matemàtiques 53, 355-399, 2009.
  • [4] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, Journal of Mathematical Analysis and Applications 361 (2), 371-391, 2010.
  • [5] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$- Laplacian type, Advances in Differential Equations 10 (9), 1053-1080, 2005.
  • [6] S. Antontsev, M. Chipot and S. Shmarev, Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions, Communications on Pure and Applied Analysis 12 (4), 1527-1546, 2013.
  • [7] R. Ayazoglu (Mashiyev), E. Akkoyunlu, Extinction properties of solutions for a parabolic equation with a parametric variable exponent nonlinearity, Transactions Issue Mathematics, Azerbaijan National Academy of Sciences 42 (1), 1-16, 2022.
  • [8] Y. Bai, N.S. Papageorgiou, S. Zeng, A singular eigenvalue problem for the Dirichlet $(p,q)$-Laplacian, Mathematische Zeitschrift, 300 (1), 325-345, 2022.
  • [9] M.M. Bokalo, O.M. Buhrii and R.A. Mashiyev, Unique solvablity of initial-boundaryvalue problems for anisotropic elliptic-parabolic equations with variable exponents of nanlinearity, Journal of Nonlinear Evolution Equations and Applications, 2013 (6), 67-87, 2014.
  • [10] O.M. Buhrii and R.A. Mashiyev, Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Nonlinear Analysis: Theory, Methods & Applications 70 (6), 2325-2331, 2009.
  • [11] J. Cen, A.A. Khan, D. Motreanu, S. Zeng, Inverse problems for generalized quasivariational inequalities with application to elliptic mixed boundary value systems, Inverse Problems 38 (6), 1-28, 2022.
  • [12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics 66 (4), 1383-1406, 2006.
  • [13] L. Diening, P. Harjulehto, P. Hästö, M. Ružicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics Vol. 2017, Springer-Verlag, Heidelberg, 2011.
  • [14] X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, Journal of Mathematical Analysis and Applications 262 (2), 749-760, 2001.
  • [15] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo I. 13, 109-124, 1966.
  • [16] Y.C. Gao, Y. Gao and W.J. Gao, Existence, uniqueness, and nonexistence of solution to nonlinear diffusion equations with $p(x,t)$-Laplacian operator, Boundary Value Problems 2016, 1-10, 2016.
  • [17] J. Giacomoni, V. Radulescu and G. Warnault, Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization, Communications in Contemporary Mathematics 20 (08), 1-38, 2018.
  • [18] B. Guo, Y.J. Li and W.J. Gao, Singular phenomena of solutions for nonlinear diffusion equations involving $p(x)$-Laplace operator and nonlinear source, Zeitschrift für angewandte Mathematik und Physik 66, 989-1005, 2015.
  • [19] Y. Han, Long-time behavior of solutions to a class of parabolic equations with nonstandard growth condition, Bulletin of the Malaysian Mathematical Sciences Society 39, 1183-1200, 2016.
  • [20] S. Kaplan, On the growth of the solutions of quasilinear parabolic equations, Communications on Pure and Applied Mathematics 16 (3), 305-330, 1963.
  • [21] O. Kholyavka, O. Buhrii, M. Bokalo and R. Ayazoglu (Mashiyev), Initial-boundaryvalue problem for third order equations of Kirchhoff type with variable exponents of nonlinearity, Advances in Math. Sciences and Appl. 23 (2), 509-528, 2013.
  • [22] V. Komornik, Exact Controllability and Stabilization, in: RAM: Research in Applied Mathematics, John Wiley, Ltd., Chichester, Masson, Paris, 1994.
  • [23] O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal 41 (4), 592-618, 1991.
  • [24] H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+F(u)$, Archive for Rational Mechanics and Analysis 51 (5), 371-386, 1973.
  • [25] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (in French), Dunod, Gauthier Villars, Paris, 1969.
  • [26] B. Liu and M. Dong, A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions, Nonlinear Analysis: Real World Applications 48, 383-409, 2019.
  • [27] Y. Liu, Exictence and blow-up of solutions to a parabolic equation with nonstadard growth conditions, Bulletin of the Australian Mathematical Society 99 (2), 242-249, 2019.
  • [28] Y. Liu, S. Mig´orski, V.T. Nguyen, S. Zeng, Existence and convergence results for an elastic frictional contact problem with nonmonotone subd ifferential boundary conditions, Acta Mathematica Scientia 41 (4), 1151-1168, 2021.
  • [29] A.T. Lourêdo, M.M. Miranda and M.R. Clark, Variable exponent perturbation of a parabolic equation with $p(x)$-Laplacian, Electronic Journal of Qualitative Theory of Differential Equations 2019 (60), 1-14, 2019.
  • [30] R.A. Mashiyev and O.M. Buhrii, Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Journal of Mathematical Analysis and Applications 377 (2), 450-463, 2011.
  • [31] L.C. Nhan, Q.V. Chuong, L.X. Truong, Potential well method for $p(x)$-Laplacian equations with variable exponent sources, Nonlinear Analysis: Real World Applications 56, 103155, 1-21, 2020.
  • [32] L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics 22 (3-4), 273-303, 1975.
  • [33] J.P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Analysis: Theory, Methods & Applications 71 (3-4) (2009), 1094-1099.
  • [34] M.A. Ragusa, The Cauchy-Dirichlet Problem for Parabolic Equations with VMO Coefficients, Mathematical and Computer Modelling 42 (11-12), 1245-1254, 2005.
  • [35] M. Ružicka, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.
  • [36] J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura Appl. 146 (4), 65-96, 1987.
  • [37] X. Wu, The blow-up of solutions for m-Laplacian equations with variable sources under positive initial energy, Computers & Mathematics with Applications 72 (9), 2516-2524, 2016.
  • [38] S. Zeng, Y. Bai, L. Gasi´nski, P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calculus of Variations and Partial Differential Equations 59, 1-18, .2020.
  • [39] S. Zeng, V.D. Radulescu, P. Winkert, Double phase implicit obstacle prob lems with convection and multivalued mixed boundary value conditions, SIAM Journal on Mathematical Analysis 54 (2), 1898-1926, 2022.

Yıl 2024, Cilt: 53 Sayı: 2, 367 - 381, 23.04.2024

Rabil Ayazoğlu , Gülizar Alisoy , Sezgin Akbulut , Tuba Ağırman Aydın

https://doi.org/10.15672/hujms.1106985
Cited By: 1

Öz

Kaynakça

  • [1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Archive for Rational Mechanics and Analysis 164, 213-259, 2002.
  • [2] M.K. Alaoui, S.A. Messaoudi and H.B. Khenous, A blow-up result for nonlinear generalized heat equation, Computers & Mathematics with Applications 68 (12), 1723-1732, 2014.
  • [3] S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publicacions Matemàtiques 53, 355-399, 2009.
  • [4] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, Journal of Mathematical Analysis and Applications 361 (2), 371-391, 2010.
  • [5] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$- Laplacian type, Advances in Differential Equations 10 (9), 1053-1080, 2005.
  • [6] S. Antontsev, M. Chipot and S. Shmarev, Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions, Communications on Pure and Applied Analysis 12 (4), 1527-1546, 2013.
  • [7] R. Ayazoglu (Mashiyev), E. Akkoyunlu, Extinction properties of solutions for a parabolic equation with a parametric variable exponent nonlinearity, Transactions Issue Mathematics, Azerbaijan National Academy of Sciences 42 (1), 1-16, 2022.
  • [8] Y. Bai, N.S. Papageorgiou, S. Zeng, A singular eigenvalue problem for the Dirichlet $(p,q)$-Laplacian, Mathematische Zeitschrift, 300 (1), 325-345, 2022.
  • [9] M.M. Bokalo, O.M. Buhrii and R.A. Mashiyev, Unique solvablity of initial-boundaryvalue problems for anisotropic elliptic-parabolic equations with variable exponents of nanlinearity, Journal of Nonlinear Evolution Equations and Applications, 2013 (6), 67-87, 2014.
  • [10] O.M. Buhrii and R.A. Mashiyev, Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Nonlinear Analysis: Theory, Methods & Applications 70 (6), 2325-2331, 2009.
  • [11] J. Cen, A.A. Khan, D. Motreanu, S. Zeng, Inverse problems for generalized quasivariational inequalities with application to elliptic mixed boundary value systems, Inverse Problems 38 (6), 1-28, 2022.
  • [12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics 66 (4), 1383-1406, 2006.
  • [13] L. Diening, P. Harjulehto, P. Hästö, M. Ružicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics Vol. 2017, Springer-Verlag, Heidelberg, 2011.
  • [14] X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, Journal of Mathematical Analysis and Applications 262 (2), 749-760, 2001.
  • [15] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo I. 13, 109-124, 1966.
  • [16] Y.C. Gao, Y. Gao and W.J. Gao, Existence, uniqueness, and nonexistence of solution to nonlinear diffusion equations with $p(x,t)$-Laplacian operator, Boundary Value Problems 2016, 1-10, 2016.
  • [17] J. Giacomoni, V. Radulescu and G. Warnault, Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization, Communications in Contemporary Mathematics 20 (08), 1-38, 2018.
  • [18] B. Guo, Y.J. Li and W.J. Gao, Singular phenomena of solutions for nonlinear diffusion equations involving $p(x)$-Laplace operator and nonlinear source, Zeitschrift für angewandte Mathematik und Physik 66, 989-1005, 2015.
  • [19] Y. Han, Long-time behavior of solutions to a class of parabolic equations with nonstandard growth condition, Bulletin of the Malaysian Mathematical Sciences Society 39, 1183-1200, 2016.
  • [20] S. Kaplan, On the growth of the solutions of quasilinear parabolic equations, Communications on Pure and Applied Mathematics 16 (3), 305-330, 1963.
  • [21] O. Kholyavka, O. Buhrii, M. Bokalo and R. Ayazoglu (Mashiyev), Initial-boundaryvalue problem for third order equations of Kirchhoff type with variable exponents of nonlinearity, Advances in Math. Sciences and Appl. 23 (2), 509-528, 2013.
  • [22] V. Komornik, Exact Controllability and Stabilization, in: RAM: Research in Applied Mathematics, John Wiley, Ltd., Chichester, Masson, Paris, 1994.
  • [23] O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal 41 (4), 592-618, 1991.
  • [24] H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+F(u)$, Archive for Rational Mechanics and Analysis 51 (5), 371-386, 1973.
  • [25] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (in French), Dunod, Gauthier Villars, Paris, 1969.
  • [26] B. Liu and M. Dong, A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions, Nonlinear Analysis: Real World Applications 48, 383-409, 2019.
  • [27] Y. Liu, Exictence and blow-up of solutions to a parabolic equation with nonstadard growth conditions, Bulletin of the Australian Mathematical Society 99 (2), 242-249, 2019.
  • [28] Y. Liu, S. Mig´orski, V.T. Nguyen, S. Zeng, Existence and convergence results for an elastic frictional contact problem with nonmonotone subd ifferential boundary conditions, Acta Mathematica Scientia 41 (4), 1151-1168, 2021.
  • [29] A.T. Lourêdo, M.M. Miranda and M.R. Clark, Variable exponent perturbation of a parabolic equation with $p(x)$-Laplacian, Electronic Journal of Qualitative Theory of Differential Equations 2019 (60), 1-14, 2019.
  • [30] R.A. Mashiyev and O.M. Buhrii, Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Journal of Mathematical Analysis and Applications 377 (2), 450-463, 2011.
  • [31] L.C. Nhan, Q.V. Chuong, L.X. Truong, Potential well method for $p(x)$-Laplacian equations with variable exponent sources, Nonlinear Analysis: Real World Applications 56, 103155, 1-21, 2020.
  • [32] L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics 22 (3-4), 273-303, 1975.
  • [33] J.P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Analysis: Theory, Methods & Applications 71 (3-4) (2009), 1094-1099.
  • [34] M.A. Ragusa, The Cauchy-Dirichlet Problem for Parabolic Equations with VMO Coefficients, Mathematical and Computer Modelling 42 (11-12), 1245-1254, 2005.
  • [35] M. Ružicka, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.
  • [36] J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura Appl. 146 (4), 65-96, 1987.
  • [37] X. Wu, The blow-up of solutions for m-Laplacian equations with variable sources under positive initial energy, Computers & Mathematics with Applications 72 (9), 2516-2524, 2016.
  • [38] S. Zeng, Y. Bai, L. Gasi´nski, P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calculus of Variations and Partial Differential Equations 59, 1-18, .2020.
  • [39] S. Zeng, V.D. Radulescu, P. Winkert, Double phase implicit obstacle prob lems with convection and multivalued mixed boundary value conditions, SIAM Journal on Mathematical Analysis 54 (2), 1898-1926, 2022.
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Rabil Ayazoğlu 0000-0003-4493-2937

Gülizar Alisoy

Sezgin Akbulut 0000-0001-5299-8587

Tuba Ağırman Aydın 0000-0001-8034-0723

Erken Görünüm Tarihi 15 Ağustos 2023
Yayımlanma Tarihi 23 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 53 Sayı: 2

Kaynak Göster

APA Ayazoğlu, R., Alisoy, G., Akbulut, S., Ağırman Aydın, T. (2024). Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity. Hacettepe Journal of Mathematics and Statistics, 53(2), 367-381. https://doi.org/10.15672/hujms.1106985
AMA Ayazoğlu R, Alisoy G, Akbulut S, Ağırman Aydın T. Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity. Hacettepe Journal of Mathematics and Statistics. Nisan 2024;53(2):367-381. doi:10.15672/hujms.1106985
Chicago Ayazoğlu, Rabil, Gülizar Alisoy, Sezgin Akbulut, ve Tuba Ağırman Aydın. “Existence and Extinction of Solutions for Parabolic Equations With Nonstandard Growth Nonlinearity”. Hacettepe Journal of Mathematics and Statistics 53, sy. 2 (Nisan 2024): 367-81. https://doi.org/10.15672/hujms.1106985.
EndNote Ayazoğlu R, Alisoy G, Akbulut S, Ağırman Aydın T (01 Nisan 2024) Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity. Hacettepe Journal of Mathematics and Statistics 53 2 367–381.
IEEE R. Ayazoğlu, G. Alisoy, S. Akbulut, ve T. Ağırman Aydın, “Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity”, Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, ss. 367–381, 2024, doi: 10.15672/hujms.1106985.
ISNAD Ayazoğlu, Rabil vd. “Existence and Extinction of Solutions for Parabolic Equations With Nonstandard Growth Nonlinearity”. Hacettepe Journal of Mathematics and Statistics 53/2 (Nisan 2024), 367-381. https://doi.org/10.15672/hujms.1106985.
JAMA Ayazoğlu R, Alisoy G, Akbulut S, Ağırman Aydın T. Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity. Hacettepe Journal of Mathematics and Statistics. 2024;53:367–381.
MLA Ayazoğlu, Rabil vd. “Existence and Extinction of Solutions for Parabolic Equations With Nonstandard Growth Nonlinearity”. Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, 2024, ss. 367-81, doi:10.15672/hujms.1106985.
Vancouver Ayazoğlu R, Alisoy G, Akbulut S, Ağırman Aydın T. Existence and extinction of solutions for parabolic equations with nonstandard growth nonlinearity. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):367-81.

Cited By

Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
Open Mathematics
https://doi.org/10.1515/math-2024-0064

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