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 $u\to 0,$ $m$ integer and $N(m-1)/4\ge 2$, we show that the solution is global. Moreover, we obtain a 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.

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