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

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equation ${u}_{t}=\mathrm{div}\left({|\nabla u|}^{p\left(x\right)-2}\nabla u\right)+f\left(x,t,u\right),$ with $\mathrm{inf}\phantom{\rule{0.166667em}{0ex}}p\left(x\right)>2$. 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 $p\left(x\right)$. The uniqueness is also proved.

DOI : https://doi.org/10.1016/j.anihpc.2012.01.001
Keywords: Electrorheological fluids, $p\left(x\right)$-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},
publisher = {Elsevier},
volume = {29},
number = {3},
year = {2012},
pages = {377-399},
doi = {10.1016/j.anihpc.2012.01.001},
zbl = {1255.35153},
language = {en},
url = {http://www.numdam.org/item/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, Volume 29 (2012) no. 3, pp. 377-399. doi : 10.1016/j.anihpc.2012.01.001. http://www.numdam.org/item/AIHPC_2012__29_3_377_0/

[1] E. Acerbi, G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal. 164 no. 3 (2002), 213-259 | Zbl 1038.76058

[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 | Numdam | Zbl 1052.76004

[3] S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat. 53 no. 2 (2009), 355-399 | Zbl 1191.35152

[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 0794.35090

[6] X.L. Fan, D. Zhao, On the spaces ${L}^{p\left(x\right)}\left(\Omega \right)$ and ${W}^{m,p\left(x\right)}\left(\Omega \right)$, J. Math. Anal. Appl. 263 no. 2 (2001), 424-446 | Zbl 1028.46041

[7] O. Kováčik, J. Rákosník, On spaces ${L}^{p\left(x\right)}$ and ${W}^{k,p\left(x\right)}$, Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | Zbl 0784.46029

[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 0968.76531

[11] J.N. Zhao, Existence and nonexistence of solutions for ${u}_{t}=\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right)+f\left(\nabla u,u,x,t\right)$, 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 0836.35081

[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 0791.35036

[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 | Zbl 1086.46026