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.   Google Scholar

[2]

S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results,, Differ. Integral Equ., 21 (2008), 401.   Google Scholar

[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.   Google Scholar

[4]

S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type,, Adv. Differential Equations, 10 (2005), 1053.   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]

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.  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]

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.  Google Scholar

[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.  Google Scholar

[10]

S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, Publ. Mat., 53 (2009), 355.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations,, Mat. Sb., 67 (1965), 609.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$,, Math. Inequal. Appl., 7 (2004), 245.  doi: 10.7153/mia-07-27.  Google Scholar

[19]

D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent,, Studia Math., 143 (2000), 267.   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]

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.  Google Scholar

[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.  Google Scholar

[23]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, , Czechoslovak Math. J., 116 (1991), 592.   Google Scholar

[24]

G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration,, Sibirsk. Mat. Zh., 38 (1997), 1335.  doi: 10.1007/BF02675942.  Google Scholar

[25]

J. Musielak, "Orlicz Spaces and Modular Spaces,", vol. 1034 of Lecture Notes in Mathematics, (1034).  doi: 10.1007/BFb0072212.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. Zhikov, On Lavrentiev's effect,, Dokl. Akad. Nauk, 345 (1995), 10.   Google Scholar

[35]

V. Zhikov, On Lavrentiev's phenomenon,, Russian J. Math. Phys., 3 (1995), 249.   Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results,, Differ. Integral Equ., 21 (2008), 401.   Google Scholar

[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.   Google Scholar

[4]

S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x,t)$-Laplacian type,, Adv. Differential Equations, 10 (2005), 1053.   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]

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.  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]

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.  Google Scholar

[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.  Google Scholar

[10]

S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, Publ. Mat., 53 (2009), 355.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Ju. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations,, Mat. Sb., 67 (1965), 609.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

L. Diening, Maximal function on generalized Lebesgue spaces $L^p(\cdot)$,, Math. Inequal. Appl., 7 (2004), 245.  doi: 10.7153/mia-07-27.  Google Scholar

[19]

D. Edmunds and J. Rákosnĭk, Sobolev embeddings with variable exponent,, Studia Math., 143 (2000), 267.   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]

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.  Google Scholar

[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.  Google Scholar

[23]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, , Czechoslovak Math. J., 116 (1991), 592.   Google Scholar

[24]

G. I. Laptev, Solvability of second-order quasilinear parabolic equations with double degeneration,, Sibirsk. Mat. Zh., 38 (1997), 1335.  doi: 10.1007/BF02675942.  Google Scholar

[25]

J. Musielak, "Orlicz Spaces and Modular Spaces,", vol. 1034 of Lecture Notes in Mathematics, (1034).  doi: 10.1007/BFb0072212.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. Zhikov, On Lavrentiev's effect,, Dokl. Akad. Nauk, 345 (1995), 10.   Google Scholar

[35]

V. Zhikov, On Lavrentiev's phenomenon,, Russian J. Math. Phys., 3 (1995), 249.   Google Scholar

[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.  Google Scholar

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