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

2016/06/23 by Mohamed Majdoub, Majdoub, Mohamed, Sarah Otsmane +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1606.07320

Abstract

In this paper we prove local well-posedness in Orlicz spaces for the\nbiharmonic heat equation \∂_t u+ \Δ² u=f(u), ;t>0, ;x\∈ R^N,\nwith f(u)\∼ \e^u2 for large u. Under smallness condition on the\ninitial data and for exponential nonlinearity f such that f(u)\∼ u^m as\nu\→ 0, m integer and N(m-1)/4\≥ 2, we show that the solution is\nglobal. Moreover, we obtain a decay estimates for large time for the nonlinear\nbiharmonic heat equation as well as for the nonlinear heat equation. Our\nresults extend to the nonlinear polyharmonic heat equation.\n

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