On anisotropic parabolic equation with nonstandard growth order

ABSTRACT

In this paper, the existence and the uniqueness of an evolutionary anisotropic $p_i\left(x\right)$-Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $L^{\mathrm{\infty }}$-estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $L^{\mathrm{\infty }}$-estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $p\left(x\right)-$Laplace equation with convection term via $L^{\mathrm{\infty }}-$Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

KEYWORDS:

• Anisotropic parabolic equation
• nonstandard growth order
• the Di Giorgi iteration technique
• $L^{\mathrm{\infty }}$-estimate
• uniqueness

MR Subject Classifications:

• primary 35D05 49N60
• secondary 35K10
• 35K55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of Fujian Province[2022J011242].