Short paper
Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation
Comptes Rendus. Mécanique, Volume 350 (2022), pp. 47-56.

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

 $\left\{\begin{array}{cc}{u}_{t}+{u}_{xxxx}={|u|}^{p-1}u-\frac{1}{|\Omega |}{\int }_{\Omega }{|u|}^{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\left(x,0\right)={u}_{0},\hfill & x\in \Omega \hfill \end{array}\right\$

with $\Omega =\left(0,a\right)$ 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 ${u}_{0}\left(x\right)¬\equiv 0$. We prove that the solutions for the preceding problem blow-up at finite time ${t}_{*}$ provided that:

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

Here, $4 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, 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:
DOI: 10.5802/crmeca.102
Keywords: Fourth-order equation, Blow-up, Exact blow-up time, Non-extinction, Exponential growth

1 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
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title = {Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation},
journal = {Comptes Rendus. M\'ecanique},
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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/

[1] T. P. Schulze; R. V. Kohn A geometric model for coarsening during spiral-mode growth of thin films, Physica D, Volume 132 (1999) no. 4, pp. 520-542 | DOI | MR | Zbl

[2] E. A. Ortiz; H. Si Repetto A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, Volume 47 (1999) no. 4, pp. 697-730 | DOI | MR | Zbl

[3] B. B. King; O. Stein; M. Winkler A fourth-order parabolic equation modeling epitaxial thin film growth, J. Math. Anal. Appl., Volume 286 (2003), pp. 459-490 | DOI | MR | Zbl

[4] G. Xu; J. Zhou Global existence and finite time blow-up of the solution for a thin-film equation with high initial energy, J. Math. Anal. Appl., Volume 458 (2018), pp. 521-535 | DOI | MR | Zbl

[5] F. Sun; L. Liu; Y. Wu Finite time blow-up for a thin-film equation with initial data at arbitrary energy level, J. Math. Anal. Appl., Volume 458 (2018), pp. 9-20 | DOI | MR | Zbl

[6] J. Zhou Global asymptotical behavior and some new blow-up conditions of solutions to a thin-film equation, J. Math. Anal. Appl., Volume 464 (2018), pp. 1290-1312 | DOI | MR | Zbl

[7] Z. Dong; J. Zhou Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys., Volume 68 (2017), pp. 68-89 | DOI | MR | Zbl

[8] Y. Cao; C. Liu Global existence and non-extinction of solutions to a fourth-order parabolic equation, Appl. Math. Lett., Volume 61 (2016), pp. 20-25 | DOI | MR

[9] C. Qu; W. Zhou Blow-up and extinction for a thin-film equation with initial-boundary value conditions, J. Math. Anal. Appl., Volume 436 (2016), pp. 796-809 | DOI | MR | Zbl

[10] J. Zhou Blow-up for a thin-film equation with positive initial energy, J. Math. Anal. Appl., Volume 446 (2017), pp. 1133-1138 | DOI | MR | Zbl

[11] V. A. Galaktionov On a spectrum of blow-up patterns for a higher-order semilinear parabolic equation, Proc. Math. Phys. Eng. Sci., Volume 457 (2011) no. 2001, pp. 1623-1643

[12] V. A. Galaktionov Hermitian spectral theory and blow-up patterns for a fourth-order semilinear Boussinesq equation, Stud. Appl. Math., Volume 121 (2008) no. 4, pp. 395-431 | DOI | MR | Zbl

[13] Y. Han A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal. Real World Appl., Volume 43 (2018), pp. 451-466 | DOI | MR

[14] A. Khelghati; K. Baghaei Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy, Comput. Math. Appl., Volume 70 (2015), pp. 896-902 | DOI | MR | Zbl

[15] K. Baghaei Lower bounds for the blow-up time in a superlinear hyperbolic equation with linear damping term, Comput. Math. Appl., Volume 73 no. 4, pp. 560-564 | DOI | MR | Zbl

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