# American Institute of Mathematical Sciences

July  2013, 12(4): 1527-1546. doi: 10.3934/cpaa.2013.12.1527

## Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions

 1 CMAF, University of Lisbon, Portugal 2 Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich 3 University of Oviedo, Spain

Received  March 2011 Revised  September 2011 Published  November 2012

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: \begin{eqnarray} u_{t}=div(a(x,t,u)|u|^{\alpha(x,t)}|\nabla u|^{p(x,t)-2} \nabla u) +f(x,t) \end{eqnarray} with given variable exponents $\alpha(x,t)$ and $p(x,t)$. We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.
Citation: Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527
##### References:
 [1] Y. Alkhutov, S. Antontsev and V. Zhikov, Parabolic equations with variable order of nonlinearity,, Zb. Pr. Inst. Mat. NAN Ukr., 6 (2009), 23. [2] S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results,, Differ. Integral Equ., 21 (2008), 401. [3] S. Antontsev, M. Chipot and Y. Xie, Uniqueness results for equations of the $p(x)$-aplacian type,, Adv. Math. Sci. Appl., 17 (2007), 287. [4] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type,, Adv. Differential Equations, 10 (2005), 1053. [5] S. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Gidrodinamiki, Novosibirsk, 1986., (in Russian;, (). [6] S. Antontsev, J. I. Díaz and S. Shmarev, "Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics,", Bikhäuser, (2002). doi: 10.1115/1.1483358. [7] S. Antontsev and S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Elsevier, 2006., Handbook of Differential Equations. Stationary Partial Differential Equations, (): 1. doi: 10.1016/S1874-5733(06)80005-7. [8] S. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity,, Fundam. Prikl. Mat., 12 (2006). doi: 10.1016/S1874-5733(06)80005-7. [9] S. Antontsev and S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity,, J. Math. Anal. Appl., 361 (2010), 371. doi: 10.1016/j.jmaa.2009.07.019. [10] S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, Publ. Mat., 53 (2009), 355. [11] S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities,, Math. Comput. Simulation, 81 (2011), 2018. doi: 10.1016/j.matcom.2010.12.015. [12] S. Antontsev and S. Shmarev, Elliptic equations with triple variable nonlinearity,, Complex Var. Elliptic Equ., 56 (2011), 573. doi: 10.1080/17476933.2010.504844. [13] M. Chipot, "Elliptic Equations: An Introductory Course,", A series of Advanced Textbooks in Mathematics, (2009). doi: 10.1007/978-3-7643-9982-5_7. [14] M. Chipot and J.-F. Rodrigues, Comparison and stability of solutions to a class of quasilinear parabolic problems,, Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988), 275. doi: 10.1017/S0308210500022265. [15] Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations,, Mat. Sb., 67 (1965), 609. [16] J. Díaz and J. Padial, Uniqueness and existence of a solution in $BV_t(q)$ space to a doubly nonlinear parabolic problem,, Publ. Mat., 40 (1996), 527. [17] J. Díaz and F. Thélin, On a nonlinear parabolic problem arising in some models related to turbulent flows,, SIAM J. Math. Anal., 25 (1994), 1085. doi: 10.1137/S0036141091217731. [18] L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$,, Math. Inequal. Appl., 7 (2004), 245. doi: 10.7153/mia-07-27. [19] D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent,, Studia Math., 143 (2000), 267. [20] P. Harjulento and P. Hästoö, An overview of variable exponent Lebesgue and Sobolev spaces,, in Future trends in geometric function theory, (). [21] A. I. Ivanov and J. F. Rodrigues, Existence and uniqueness of a weak solution to the initial mixed boundary-value problem for quasilinear elliptic-parabolic equations,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, 259 (1999), 67. doi: 10.1023/A:1014488123746. [22] A. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169. doi: 10.1070/RM1987v042n02ABEH001309. [23] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, , Czechoslovak Math. J., 116 (1991), 592. [24] G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration,, Sibirsk. Mat. Zh., 38 (1997), 1335. doi: 10.1007/BF02675942. [25] J. Musielak, "Orlicz Spaces and Modular Spaces,", vol. 1034 of Lecture Notes in Mathematics, (1034). doi: 10.1007/BFb0072212. [26] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,, Integral Transforms Spec. Funct., 16 (2005), 461. doi: 10.1080/10652460412331320322. [27] S. Samko, Density $C^\infty_0 (R^n)$ in the generalized Sobolev spaces $W^{m,p(x)}(R^n)$,, Dokl. Akad. Nauk, 369 (1999), 451. [28] K. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems,, Nonlinear Anal., 65 (2006), 2103. doi: 10.1016/j.na.2005.11.053. [29] M. Sango, Local boundedness for doubly degenerate quasi-linear parabolic systems,, Appl. Math. Lett., 16 (2003), 465. doi: 10.1016/S0893-9659(03)00021-1. [30] A. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations,, Appl. Anal., 86 (2007), 755. doi: 10.1080/00036810701435711. [31] S. Degtyarev and A. Tedeev, $L_1$-$L_\infty$ estimates for the solution of the Cauchy problem for an anisotropic degenerate parabolic equation with double nonlinearity and growing initial data,, Mat. Sb., 198 (2007), 45. doi: 10.1070/SM2007v198n05ABEH003853. [32] P. Cianci, A. Martynenko and A. Tedeev, The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source,, Nonlinear Anal., 73 (2010), 2310. doi: 10.1016/j.na.2010.06.026. [33] C. Vázquez, E. Schiavi, J. Durany, J. I. Díaz and N. Calvo, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics,, SIAM J. Appl. Math., 63 (2003), 683. doi: 10.1137/S0036139901385345. [34] V. Zhikov, On Lavrentiev's effect,, Dokl. Akad. Nauk, 345 (1995), 10. [35] V. Zhikov, On Lavrentiev's phenomenon,, Russian J. Math. Phys., 3 (1995), 249. [36] V. Zhikov, On the density of smooth functions in Sobolev-Orlich spaces,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 1. doi: 10.1007/s10958-005-0497-0.

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##### References:
 [1] Y. Alkhutov, S. Antontsev and V. Zhikov, Parabolic equations with variable order of nonlinearity,, Zb. Pr. Inst. Mat. NAN Ukr., 6 (2009), 23. [2] S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results,, Differ. Integral Equ., 21 (2008), 401. [3] S. Antontsev, M. Chipot and Y. Xie, Uniqueness results for equations of the $p(x)$-aplacian type,, Adv. Math. Sci. Appl., 17 (2007), 287. [4] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type,, Adv. Differential Equations, 10 (2005), 1053. [5] S. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Gidrodinamiki, Novosibirsk, 1986., (in Russian;, (). [6] S. Antontsev, J. I. Díaz and S. Shmarev, "Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics,", Bikhäuser, (2002). doi: 10.1115/1.1483358. [7] S. Antontsev and S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Elsevier, 2006., Handbook of Differential Equations. Stationary Partial Differential Equations, (): 1. doi: 10.1016/S1874-5733(06)80005-7. [8] S. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity,, Fundam. Prikl. Mat., 12 (2006). doi: 10.1016/S1874-5733(06)80005-7. [9] S. Antontsev and S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity,, J. Math. Anal. Appl., 361 (2010), 371. doi: 10.1016/j.jmaa.2009.07.019. [10] S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, Publ. Mat., 53 (2009), 355. [11] S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities,, Math. Comput. Simulation, 81 (2011), 2018. doi: 10.1016/j.matcom.2010.12.015. [12] S. Antontsev and S. Shmarev, Elliptic equations with triple variable nonlinearity,, Complex Var. Elliptic Equ., 56 (2011), 573. doi: 10.1080/17476933.2010.504844. [13] M. Chipot, "Elliptic Equations: An Introductory Course,", A series of Advanced Textbooks in Mathematics, (2009). doi: 10.1007/978-3-7643-9982-5_7. [14] M. Chipot and J.-F. Rodrigues, Comparison and stability of solutions to a class of quasilinear parabolic problems,, Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988), 275. doi: 10.1017/S0308210500022265. [15] Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations,, Mat. Sb., 67 (1965), 609. [16] J. Díaz and J. Padial, Uniqueness and existence of a solution in $BV_t(q)$ space to a doubly nonlinear parabolic problem,, Publ. Mat., 40 (1996), 527. [17] J. Díaz and F. Thélin, On a nonlinear parabolic problem arising in some models related to turbulent flows,, SIAM J. Math. Anal., 25 (1994), 1085. doi: 10.1137/S0036141091217731. [18] L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$,, Math. Inequal. Appl., 7 (2004), 245. doi: 10.7153/mia-07-27. [19] D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent,, Studia Math., 143 (2000), 267. [20] P. Harjulento and P. Hästoö, An overview of variable exponent Lebesgue and Sobolev spaces,, in Future trends in geometric function theory, (). [21] A. I. Ivanov and J. F. Rodrigues, Existence and uniqueness of a weak solution to the initial mixed boundary-value problem for quasilinear elliptic-parabolic equations,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, 259 (1999), 67. doi: 10.1023/A:1014488123746. [22] A. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169. doi: 10.1070/RM1987v042n02ABEH001309. [23] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, , Czechoslovak Math. J., 116 (1991), 592. [24] G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration,, Sibirsk. Mat. Zh., 38 (1997), 1335. doi: 10.1007/BF02675942. [25] J. Musielak, "Orlicz Spaces and Modular Spaces,", vol. 1034 of Lecture Notes in Mathematics, (1034). doi: 10.1007/BFb0072212. [26] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,, Integral Transforms Spec. Funct., 16 (2005), 461. doi: 10.1080/10652460412331320322. [27] S. Samko, Density $C^\infty_0 (R^n)$ in the generalized Sobolev spaces $W^{m,p(x)}(R^n)$,, Dokl. Akad. Nauk, 369 (1999), 451. [28] K. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems,, Nonlinear Anal., 65 (2006), 2103. doi: 10.1016/j.na.2005.11.053. [29] M. Sango, Local boundedness for doubly degenerate quasi-linear parabolic systems,, Appl. Math. Lett., 16 (2003), 465. doi: 10.1016/S0893-9659(03)00021-1. [30] A. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations,, Appl. Anal., 86 (2007), 755. doi: 10.1080/00036810701435711. [31] S. Degtyarev and A. Tedeev, $L_1$-$L_\infty$ estimates for the solution of the Cauchy problem for an anisotropic degenerate parabolic equation with double nonlinearity and growing initial data,, Mat. Sb., 198 (2007), 45. doi: 10.1070/SM2007v198n05ABEH003853. [32] P. Cianci, A. Martynenko and A. Tedeev, The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source,, Nonlinear Anal., 73 (2010), 2310. doi: 10.1016/j.na.2010.06.026. [33] C. Vázquez, E. Schiavi, J. Durany, J. I. Díaz and N. Calvo, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics,, SIAM J. Appl. Math., 63 (2003), 683. doi: 10.1137/S0036139901385345. [34] V. Zhikov, On Lavrentiev's effect,, Dokl. Akad. Nauk, 345 (1995), 10. [35] V. Zhikov, On Lavrentiev's phenomenon,, Russian J. Math. Phys., 3 (1995), 249. [36] V. Zhikov, On the density of smooth functions in Sobolev-Orlich spaces,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 1. doi: 10.1007/s10958-005-0497-0.
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