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
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show all references

References:
[1]

Zb. Pr. Inst. Mat. NAN Ukr., 6 (2009), 23-50. Google Scholar

[2]

Differ. Integral Equ., 21 (2008), 401-419.  Google Scholar

[3]

Adv. Math. Sci. Appl., 17 (2007), 287-304.  Google Scholar

[4]

Adv. Differential Equations, 10 (2005), 1053-1080.  Google Scholar

[5]

S. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Gidrodinamiki, Novosibirsk, 1986., (in Russian;, ().   Google Scholar

[6]

Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48. doi: 10.1115/1.1483358.  Google Scholar

[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.  Google Scholar

[8]

Fundam. Prikl. Mat., 12 (2006), doi: 10.1016/S1874-5733(06)80005-7.  Google Scholar

[9]

J. Math. Anal. Appl., 361 (2010), 371-391. doi: 10.1016/j.jmaa.2009.07.019.  Google Scholar

[10]

Publ. Mat., 53 (2009), 355-399.  Google Scholar

[11]

Math. Comput. Simulation, 81 (2011), 2018-1032. doi: 10.1016/j.matcom.2010.12.015.  Google Scholar

[12]

Complex Var. Elliptic Equ., 56 (2011), 573-597. doi: 10.1080/17476933.2010.504844.  Google Scholar

[13]

A series of Advanced Textbooks in Mathematics, Birkhäuser, 2009. doi: 10.1007/978-3-7643-9982-5_7.  Google Scholar

[14]

Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988), 275-285. doi: 10.1017/S0308210500022265.  Google Scholar

[15]

Mat. Sb., 67 (1965), 609-642.  Google Scholar

[16]

Publ. Mat., 40 (1996), 527-560.  Google Scholar

[17]

SIAM J. Math. Anal., 25 (1994), 1085-1111. doi: 10.1137/S0036141091217731.  Google Scholar

[18]

Math. Inequal. Appl., 7 (2004), 245-253. doi: 10.7153/mia-07-27.  Google Scholar

[19]

Studia Math., 143 (2000), 267-293.  Google Scholar

[20]

P. Harjulento and P. Hästoö, An overview of variable exponent Lebesgue and Sobolev spaces,, in Future trends in geometric function theory, ().   Google Scholar

[21]

Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, 259 (1999), 67-98, doi: 10.1023/A:1014488123746.  Google Scholar

[22]

Russian Math. Surveys, 42 (1987), 169-222. doi: 10.1070/RM1987v042n02ABEH001309.  Google Scholar

[23]

Czechoslovak Math. J., 116 (1991), 592-618.  Google Scholar

[24]

Sibirsk. Mat. Zh., 38 (1997), 1335-1355. doi: 10.1007/BF02675942.  Google Scholar

[25]

vol. 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072212.  Google Scholar

[26]

Integral Transforms Spec. Funct., 16 (2005), 461-482. doi: 10.1080/10652460412331320322.  Google Scholar

[27]

Dokl. Akad. Nauk, 369 (1999), 451-454.  Google Scholar

[28]

Nonlinear Anal., 65 (2006), 2103-2134. doi: 10.1016/j.na.2005.11.053.  Google Scholar

[29]

Appl. Math. Lett., 16 (2003), 465-468. doi: 10.1016/S0893-9659(03)00021-1.  Google Scholar

[30]

Appl. Anal., 86 (2007), 755-782. doi: 10.1080/00036810701435711.  Google Scholar

[31]

Mat. Sb., 198 (2007), 45-66. doi: 10.1070/SM2007v198n05ABEH003853.  Google Scholar

[32]

Nonlinear Anal., 73 (2010), 2310-2323. doi: 10.1016/j.na.2010.06.026.  Google Scholar

[33]

SIAM J. Appl. Math., 63 (2003), 683-707. doi: 10.1137/S0036139901385345.  Google Scholar

[34]

Dokl. Akad. Nauk, 345 (1995), 10-14.  Google Scholar

[35]

Russian J. Math. Phys., 3 (1995), 249-269.  Google Scholar

[36]

Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 1-14. doi: 10.1007/s10958-005-0497-0.  Google Scholar

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