On the blow-up of solutions to a fourth-order pseudoparabolic equation

In this note, we consider a fourth-order semilinear pseudoparabolic differential equation including a strong damping term together with a nonlocal source term. The problem is considered under the periodic boundary conditions and a finite time blow-up result is established. Also a lower bound estimate for the blow-up time is obtained.

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[1] Al’shin AB, Korpusov MO, Sveshnikov AG. Blow-up in nonlinear Sobolev type equations. De Gruyter Series in Nonlinear Analysis and Applications 2011; 15 : Walter de Gruyter and Co., Berlin.
[2] Budd C, Dold J, Stuart A. Blow-up in a system of partial differential equations with conserved first integrals. Part II: positive initial energy. Applied Mathematics Letters 2011; 24 (5) :784-788.
[3] Cao Y, Liu CH. Global existence and non-extinction of solutions to problems with convection. SIAM Journal of Applied Mathematics 1994; 54 (3) : 610-640.
[4] Ding J. Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions. Computer and Mathematics with Appl.ications 2013; 65 (11): 1808-1822.
[5] El Soufi A, Jazar M, Monneau R. A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Annales Henri Poincare Analyse Non Lineaire 2007; 24: 17-39.
[6] Enache C. Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition. Applied Mathematics Letters 2011; 24 (3): 288-292.
[7] Evans LC. Partial Differential Equations, AMS, 1998.
[8] Gao W, Han Y. Blow-up of non-local semilinear parabolic equation with a fourth-order parabolic equation. Applied Mathematics Letters 2016 ; 60 : 120-25.
[9] Hu B. Blow-up theories for semilinear parabolic equations Springer 2011.
[10] Jazar M, Kiwan R. Blow-up non-local semilinear parabolic equation with Neumann boundary conditions. Annales Henri Poincare Analyse Nonlineare 2008; 25 : 215-218.
[11] Khelghati A, Baghaei K. Blow-up phenomena for a non-local semilinear parabolic equation with positive initial energy. Computer and Math.ematics with Applications 2015; 70: 896-902.
[12] King BB, Stein O, Winkler M. A fourth-order parabolic equation modelling epitaxial thin film growth. Journal of Mathematical Analysis and Applications 2003; 286 : 459-490.
[13] Levine HA. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form P ut = Au + F(u), Arch. Rational Mechanical Analysis. 1973;51: 371-386.
[14] Liu W, Wang M. Blow-up of the solutions for a p-Laplacian equation with positive initial energy. Acta Applicandae Mathematicae 2008; 103 (2): 141-146.
[15] Payne LA, Schaefer PW. Lower boundedness for blow-up time in parabolic problems under the Dirichlet boundary conditions. Journal of Mathematical Analysis and Applications 2007; 328 :1196-1205.
[16] Payne LA, Schaefer, PW. Lower bounded of blow-up time in parabolic problems under the Neumann conditions, Applicable Analysis 2006; 85: 1301-1311.
[17] Payne LA , Philippin GA. Blow-up in a class of nonlinear parabolic problems with time-dependent coefficients under Robin Boundary Conditions, Analysis and Applications 2012; 91: 2245-2256.
[18] Payne LE, Schaefer PW. Blow-up in parabolic problems under Robin boundary conditions. Applicable Analysis 2008; 87(6): 699-707.
[19] Philippin GA. Blow-up phenomena for a calss of fourth-order parabolic problems, Proceedings of the American Mathematical Society 2015; 143 ( 6) : 2507-2513.
[20] Polat M. A blow-up result for the nonlocal thin-film equation with the positive energy. Turkish Journal of Mathematics 2019; 43: 1797-1807.
[21] Qu, C, Zhou W. Blow-up and extinction for a thin-film equation with the initial boundary value conditions. Journal of Mathematical Analysis and Applications 2016; 436 (2): 796-809.
[22] Ortiz M, Repetto EA, Si H. A continuum model of kinetic roughening and coarsing in thin films. Journal of Mechanics and Physics of Solids 1999; 47: 697-730.
[23] Samarski AA et al. Blow-up in quasilinear parabolic equations. Walter de Gruyter, Newyork, Berlin, 1995.
[24] Sell GR, You Y. Dynamics of Evolutionary Equations, Springer 2002.
[25] Showalter RE, Ting TW. Pseudoparabolic partial differential equations. SIAM Journal of Mathematical Analysis 1970; 1(1): 1-26.
[26] Zangwill A. Some couses and consequences of epitaxial roughening. Journal Crystal Growth 1996; 163: 8-21.
[27] Zhou J. Blow-up for a thin-film equation with positive energy. J.ournal of Mathematical Analysis and Applications 2017; 446: 1133-1138.