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To the memory of Professor Igor V. Skrypnik
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 |
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;, ().
|
[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, (): 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), 371-391.
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;, ().
|
[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, (): 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), 371-391.
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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