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

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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

Stanislav Antontsev ^1, , Michel Chipot ^2, and Sergey Shmarev ^3,

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

Abstract
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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.

Keywords: double nonlinearity, Nonlinear parabolic equations, nonstandard growth conditions..

Mathematics Subject Classification: Primary: 35K55, 35K65, 35K6.

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 and 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), 23-50.
[2]	S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results, Differ. Integral Equ., 21 (2008), 401-419.
[3]	S. Antontsev, M. Chipot and Y. Xie, Uniqueness results for equations of the $p(x)$-aplacian type, Adv. Math. Sci. Appl., 17 (2007), 287-304.
[4]	S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type, Adv. Differential Equations, 10 (2005), 1053-1080.
[5]	S. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Gidrodinamiki, Novosibirsk, 1986. (in Russian; "Lokalizatsiya resheniĭ vyrozhdayushchikhsya uravneniĭ mekhaniki sploshnoĭ sredy").
[6]	S. Antontsev, J. I. Díaz and S. Shmarev, "Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics," Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48. 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, Elsevier, Vol. 3, Chapter 1, 1-100. 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), 371-391. doi: 10.1016/j.jmaa.2009.07.019.
[10]	S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat., 53 (2009), 355-399.
[11]	S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities, Math. Comput. Simulation, 81 (2011), 2018-1032. 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), 573-597. doi: 10.1080/17476933.2010.504844.
[13]	M. Chipot, "Elliptic Equations: An Introductory Course," A series of Advanced Textbooks in Mathematics, Birkhäuser, 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), 275-285. doi: 10.1017/S0308210500022265.
[15]	Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations, Mat. Sb., 67 (1965), 609-642.
[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), 527-560.
[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), 1085-1111. doi: 10.1137/S0036141091217731.
[18]	L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$, Math. Inequal. Appl., 7 (2004), 245-253. doi: 10.7153/mia-07-27.
[19]	D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267-293.
[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), 67-98, 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), 169-222. 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), 592-618.
[24]	G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration, Sibirsk. Mat. Zh., 38 (1997), 1335-1355. doi: 10.1007/BF02675942.
[25]	J. Musielak, "Orlicz Spaces and Modular Spaces," vol. 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. 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), 461-482. 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), 451-454.*
[28]	K. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems, Nonlinear Anal., 65 (2006), 2103-2134. 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), 465-468. 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), 755-782. 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), 45-66. 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), 2310-2323. 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), 683-707. doi: 10.1137/S0036139901385345.
[34]	V. Zhikov, On Lavrentiev's effect, Dokl. Akad. Nauk, 345 (1995), 10-14.
[35]	V. Zhikov, On Lavrentiev's phenomenon, Russian J. Math. Phys., 3 (1995), 249-269.
[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), 1-14. doi: 10.1007/s10958-005-0497-0.

show all references

References:

[1]	Y. Alkhutov, S. Antontsev and V. Zhikov, Parabolic equations with variable order of nonlinearity, Zb. Pr. Inst. Mat. NAN Ukr., 6 (2009), 23-50.
[2]	S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results, Differ. Integral Equ., 21 (2008), 401-419.
[3]	S. Antontsev, M. Chipot and Y. Xie, Uniqueness results for equations of the $p(x)$-aplacian type, Adv. Math. Sci. Appl., 17 (2007), 287-304.
[4]	S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type, Adv. Differential Equations, 10 (2005), 1053-1080.
[5]	S. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Gidrodinamiki, Novosibirsk, 1986. (in Russian; "Lokalizatsiya resheniĭ vyrozhdayushchikhsya uravneniĭ mekhaniki sploshnoĭ sredy").
[6]	S. Antontsev, J. I. Díaz and S. Shmarev, "Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics," Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48. 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, Elsevier, Vol. 3, Chapter 1, 1-100. 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), 371-391. doi: 10.1016/j.jmaa.2009.07.019.
[10]	S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat., 53 (2009), 355-399.
[11]	S. Antontsev and S. Shmarev, Parabolic equations with double variable nonlinearities, Math. Comput. Simulation, 81 (2011), 2018-1032. 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), 573-597. doi: 10.1080/17476933.2010.504844.
[13]	M. Chipot, "Elliptic Equations: An Introductory Course," A series of Advanced Textbooks in Mathematics, Birkhäuser, 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), 275-285. doi: 10.1017/S0308210500022265.
[15]	Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations, Mat. Sb., 67 (1965), 609-642.
[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), 527-560.
[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), 1085-1111. doi: 10.1137/S0036141091217731.
[18]	L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$, Math. Inequal. Appl., 7 (2004), 245-253. doi: 10.7153/mia-07-27.
[19]	D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267-293.
[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), 67-98, 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), 169-222. 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), 592-618.
[24]	G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration, Sibirsk. Mat. Zh., 38 (1997), 1335-1355. doi: 10.1007/BF02675942.
[25]	J. Musielak, "Orlicz Spaces and Modular Spaces," vol. 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. 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), 461-482. 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), 451-454.*
[28]	K. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems, Nonlinear Anal., 65 (2006), 2103-2134. 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), 465-468. 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), 755-782. 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), 45-66. 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), 2310-2323. 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), 683-707. doi: 10.1137/S0036139901385345.
[34]	V. Zhikov, On Lavrentiev's effect, Dokl. Akad. Nauk, 345 (1995), 10-14.
[35]	V. Zhikov, On Lavrentiev's phenomenon, Russian J. Math. Phys., 3 (1995), 249-269.
[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), 1-14. doi: 10.1007/s10958-005-0497-0.

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