no. 3
Existence of solutions to an initial Dirichlet problem of evolutional p(x)-Laplace equations
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 377-399

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equation

u t = div |∇u| p(x)-2 ∇u+f(x,t,u),
with inf p(x)>2. The problems describe the motion of generalized Newtonian fluids which were studied by some other authors in which the exponent p was required to satisfy a logarithmic Hölder continuity condition. The authors in this paper use a difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p(x). The uniqueness is also proved.

DOI : 10.1016/j.anihpc.2012.01.001
Keywords: Electrorheological fluids, $ p(x)$-Laplace, Degenerate, Parabolic 
@article{AIHPC_2012__29_3_377_0,
     author = {Lian, Songzhe and Gao, Wenjie and Yuan, Hongjun and Cao, Chunling},
     title = {Existence of solutions to an initial {Dirichlet} problem of evolutional $ p(x)${-Laplace} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {377--399},
     year = {2012},
     publisher = {Elsevier},
     volume = {29},
     number = {3},
     doi = {10.1016/j.anihpc.2012.01.001},
     zbl = {1255.35153},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2012.01.001/}
}
TY  - JOUR
AU  - Lian, Songzhe
AU  - Gao, Wenjie
AU  - Yuan, Hongjun
AU  - Cao, Chunling
TI  - Existence of solutions to an initial Dirichlet problem of evolutional $ p(x)$-Laplace equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 377
EP  - 399
VL  - 29
IS  - 3
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2012.01.001/
DO  - 10.1016/j.anihpc.2012.01.001
LA  - en
ID  - AIHPC_2012__29_3_377_0
ER  - 
%0 Journal Article
%A Lian, Songzhe
%A Gao, Wenjie
%A Yuan, Hongjun
%A Cao, Chunling
%T Existence of solutions to an initial Dirichlet problem of evolutional $ p(x)$-Laplace equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 377-399
%V 29
%N 3
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2012.01.001/
%R 10.1016/j.anihpc.2012.01.001
%G en
%F AIHPC_2012__29_3_377_0
Lian, Songzhe; Gao, Wenjie; Yuan, Hongjun; Cao, Chunling. Existence of solutions to an initial Dirichlet problem of evolutional $ p(x)$-Laplace equations. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 377-399. doi: 10.1016/j.anihpc.2012.01.001

[1] E. Acerbi, G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal. 164 no. 3 (2002), 213-259 | Zbl

[2] E. Acerbi, G. Mingione, G.A. Seregin, Regularity results for parabolic systems related to a class of non Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 1 (2004), 25-60 | EuDML | Zbl | Numdam

[3] S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat. 53 no. 2 (2009), 355-399 | EuDML | Zbl

[4] M. Berger, Nonlinearity and functional analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Pure and Applied Mathematics, Academic Press, New York–London (1977)

[5] E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York (1993) | Zbl

[6] X.L. Fan, D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 no. 2 (2001), 424-446 | Zbl

[7] O. Kováčik, J. Rákosník, On spaces L p(x) and W k,p(x) , Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | EuDML | Zbl

[8] O.A. Ladyzhenskaja, V.A. Solonikov, N.N. UralʼCeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island (1968)

[9] L.Q. Peng, Non-uniqueness for the p-harmonic heat flow with potential into homogeneous spaces, Chinese Ann. Math. Ser. A 27 no. 4 (2006), 443-448, Chinese J. Contemp. Math. 27 no. 3 (2006), 231-238

[10] M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. vol. 1748, Springer, Berlin (2000) | Zbl

[11] J.N. Zhao, Existence and nonexistence of solutions for u t = div (|∇u| p-2 ∇u)+f(∇u,u,x,t), J. Math. Anal. Appl. 172 no. 1 (1993), 130-146

[12] J.N. Zhao, On the Cauchy problem and initial traces for the evolution p-laplacian equations with strongly nonlinear sources, J. Diff. Eqs. 121 no. 2 (1995), 329-383 | Zbl

[13] V.V. Zhikov, Passage to the limit in nonlinear variational problems, Mat. Sb. 183 no. 8 (1992), 47-84, Russian Acad. Sci. Sb. Math. 76 no. 2 (1993), 427-459 | Zbl

[14] V.V. Zhikov, On the density of smooth functions in Sobolev–Orlicz spaces, Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67-81, J. Math. Sci. (N.Y.) 132 no. 3 (2006), 285-294 | EuDML | Zbl

Cité par Sources :