Existence of solutions to an initial Dirichlet problem of evolutional $ p(x) $-Laplace equations

Songzhe Lian; Wenjie Gao; Hongjun Yuan; Chunling Cao

doi:10.1016/j.anihpc.2012.01.001

JournalsaihpcVol. 29, No. 3pp. 377–399

Existence of solutions to an initial Dirichlet problem of evolutional $p (x)$ -Laplace equations

Songzhe Lian
Institute of Mathematics, Jilin University, Changchun, Jilin 130012, Peopleʼs Republic of China
Wenjie Gao
Institute of Mathematics, Jilin University, Changchun, Jilin 130012, Peopleʼs Republic of China
Hongjun Yuan
Institute of Mathematics, Jilin University, Changchun, Jilin 130012, Peopleʼs Republic of China
Chunling Cao
Institute of Mathematics, Jilin University, Changchun, Jilin 130012, Peopleʼs Republic of China

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Abstract

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

u_{t} = div (∣ \nabla u ∣^{p (x) - 2} \nabla u) + f (x, t, u),

with $in f p (x) > 2$ . 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.

Cite this article

Songzhe Lian, Wenjie Gao, Hongjun Yuan, Chunling Cao, Existence of solutions to an initial Dirichlet problem of evolutional $p (x)$ -Laplace equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 3, pp. 377–399

DOI 10.1016/J.ANIHPC.2012.01.001