Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation

Khadijeh Baghaei

doi:10.5802/crmeca.102

Short paper

Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation

Khadijeh Baghaei ¹

¹ School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Comptes Rendus. Mécanique, Volume 350 (2022), pp. 47-56.

Abstract

This paper is concerned with the following fourth-order parabolic problem:

\{\begin{matrix} u_{t} + u_{x x x x} = {| u |}^{p - 1} u - \frac{1}{| Ω |} \int_{Ω} {| u |}^{p - 1} u d x, & x \in Ω, t > 0, \\ u_{x} = u_{x x x} = 0, & x \in \partial Ω, t > 0, \\ u (x, 0) = u_{0}, & x \in Ω \end{matrix}

with $Ω = (0, a)$ and $p > 1$ . Here, $u_{0} \in H^{2} (Ω)$ is the initial function which satisfies $\int_{Ω} u_{0} (x) d x = 0$ with $u_{0} (x) \neg \equiv 0$ . We prove that the solutions for the preceding problem blow-up at finite time $t_{*}$ provided that:

\frac{λ (2 (p + 1) - m)}{(p + 1)} {| Ω |}^{- (p - 1) / 2} ∥ u_{0} ∥_{2}^{p + 1} + \frac{m - 4}{2 c^{*}} {∥ u_{0} ∥}_{2}^{2} - m J (u_{0}) \geq 0 .

Here, $4 < m < 2 (p + 1), 0 < λ < 1$ and $c^{*}$ is a positive constant related to the Poincaré inequality. Here, $J (u)$ is the energy functional. The above condition trivially holds for $J (u_{0}) \leq 0$ . Thus, the blow-up result is valuable for arbitrary positive initial energy and suitable initial data. We also obtain upper and lower bounds for the blow-up time. Hence, the exact blow-up time is obtained under some conditions. Besides, if $J (u_{0}) > 0$ and the above condition holds in the strict sense or $J (u_{0}) \leq 0$ , then for every $q \geq {2, ∥ u (t) ∥}_{q}$ grows exponentially for all $0 < t < t_{*}$ , also, under the same conditions, the solution for this problem does not extinct in finite time if $∥ u_{0} ∥_{2} > 0$ . The non-extinction of solutions also holds in the equal sense of the above condition. These results extend the recent results obtained for this problem.

Received: 2021-08-03
Revised: 2021-12-12
Accepted: 2022-01-14
Published online: 2022-02-09

DOI: 10.5802/crmeca.102

Keywords: Fourth-order equation, Blow-up, Exact blow-up time, Non-extinction, Exponential growth

Author's affiliations:

Khadijeh Baghaei ¹

¹ School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMECA_2022__350_G1_47_0,
     author = {Khadijeh Baghaei},
     title = {Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {47--56},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {350},
     year = {2022},
     doi = {10.5802/crmeca.102},
     language = {en},
}

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Khadijeh Baghaei. Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 47-56. doi : 10.5802/crmeca.102. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.102/

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