### Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity

#### Abstract

In this paper, we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\; t > 0,\; x \in \mathbb R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^m$ as $u\to 0,$ $m\geq 2$, $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.

#### Article information

Source
Adv. Differential Equations, Volume 23, Number 7/8 (2018), 489-522.

Dates
First available in Project Euclid: 11 May 2018

Mathematical Reviews number (MathSciNet)
MR3801829

Zentralblatt MATH identifier
06889035

#### Citation

Majdoub, Mohamed; Otsmane, Sarah; Tayachi, Slim. Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity. Adv. Differential Equations 23 (2018), no. 7/8, 489--522. https://projecteuclid.org/euclid.ade/1526004064