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

$$\left\{\begin{array}{cc}{\displaystyle {u}_{t}+{u}_{xxxx}={\left|u\right|}^{p-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\int}_{\Omega}{\left|u\right|}^{p-1}u\phantom{\rule{4pt}{0ex}}\mathrm{d}x,}\hfill & x\in \Omega ,\phantom{\rule{4pt}{0ex}}t>0,\hfill \\ {u}_{x}={u}_{xxx}=0,\hfill & x\in \partial \Omega ,\phantom{\rule{4pt}{0ex}}t>0,\hfill \\ u(x,0)={u}_{0},\hfill & x\in \Omega \hfill \end{array}\right.$$ |

with $\Omega =(0,a)$ and $p>1$. Here, ${u}_{0}\in {H}^{2}\left(\Omega \right)$ is the initial function which satisfies ${\int}_{\Omega}{u}_{0}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x=0$ with ${u}_{0}\left(x\right)\neg \equiv 0$. We prove that the solutions for the preceding problem blow-up at finite time ${t}_{*}$ provided that:

$$\frac{\lambda \left(2\right(p+1)-m)}{(p+1)}{\left|\Omega \right|}^{-\left(p-1\right)/2}\parallel {u}_{0}{\parallel}_{2}^{p+1}+\frac{m-4}{2{c}^{*}}{\parallel {u}_{0}\parallel}_{2}^{2}-mJ\left({u}_{0}\right)\ge 0.$$ |

Here, $4<m<2(p+1),0<\lambda <1$ and ${c}^{*}$ is a positive constant related to the Poincaré inequality. Here, $J\left(u\right)$ is the energy functional. The above condition trivially holds for $J\left({u}_{0}\right)\le 0$. 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 $J\left({u}_{0}\right)>0$ and the above condition holds in the strict sense or $J\left({u}_{0}\right)\le 0$, then for every $q\ge {2,\parallel u\left(t\right)\parallel}_{q}$ grows exponentially for all $0<t<{t}_{*}$, also, under the same conditions, the solution for this problem does not extinct in finite time if $\parallel {u}_{0}{\parallel}_{2}>0$. 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.

Revised:

Accepted:

Published online:

Khadijeh Baghaei ^{1}

@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}, }

TY - JOUR AU - Khadijeh Baghaei TI - Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation JO - Comptes Rendus. Mécanique PY - 2022 SP - 47 EP - 56 VL - 350 PB - Académie des sciences, Paris DO - 10.5802/crmeca.102 LA - en ID - CRMECA_2022__350_G1_47_0 ER -

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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