Advances in Differential Equations

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

Mohamed Majdoub, Sarah Otsmane, and Slim Tayachi

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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

Permanent link to this document
https://projecteuclid.org/euclid.ade/1526004064

Mathematical Reviews number (MathSciNet)
MR3801829

Zentralblatt MATH identifier
06889035

Subjects
Primary: 35K91: Semilinear parabolic equations with Laplacian, bi-Laplacian or poly- Laplacian 35A01: Existence problems: global existence, local existence, non-existence 35B40: Asymptotic behavior of solutions 35K25: Higher-order parabolic equations 35K30: Initial value problems for higher-order parabolic equations 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

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


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