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EXISTENCE AND BLOW-UP OF SOLUTIONS TO A PARABOLIC EQUATION WITH NONSTANDARD GROWTH CONDITIONS

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  11 December 2018

YANG LIU Open the ORCID record for YANG LIU [Opens in a new window]
YANG LIU*
Affiliation:
College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730124, PR China College of Mathematics, Sichuan University, Chengdu 610065, PR China email liuyangnufn@163.com
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Abstract

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We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.

Keywords

fourth-order parabolic equationexistenceblow-up

MSC classification

Primary: 35K35: Initial-boundary value problems for higher-order parabolic equations
Secondary: 35A01: Existence problems 35B44: Blow-up
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 242 - 249
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the Science and Technology Plan Project of Gansu Province in China (Grant No. 17JR5RA279) and the Talent Introduction Research Project of Northwest Minzu University in China (Grant No. xbmzdxyjrc201701).

References

Acerbi, E. and Mingione, G., ‘Regularity results for stationary electro-rheological fluids’, Arch. Ration. Mech. Anal. 164(3) (2002), 213–259.Google Scholar
Antontsev, S. and Shmarev, S., ‘Anisotropic parabolic equations with variable nonlinearity’, Publ. Mat. 53(2) (2009), 355–399.Google Scholar
Ayoujil, A. and El Amrouss, A. R., ‘On the spectrum of a fourth order elliptic equation with variable exponent’, Nonlinear Anal. 71(10) (2009), 4916–4926.Google Scholar
Ayoujil, A. and El Amrouss, A. R., ‘Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent’, Electron. J. Differential Equations 2011(24) (2011), 12 pp.Google Scholar
Diening, L., Harjulehto, P., Hästö, P. and Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents (Springer, Heidelberg, 2011).Google Scholar
Diening, L., Nägele, P. and Růžička, M., ‘Monotone operator theory for unsteady problems in variable exponent spaces’, Complex Var. Elliptic Equ. 57(11) (2012), 1209–1231.Google Scholar
Drábek, P. and Ôtani, M., ‘Global bifurcation result for the p-biharmonic operator’, Electron. J. Differential Equations 2001(48) (2001), 19 pp.Google Scholar
Giacomoni, J., Tiwari, S. and Warnault, G., ‘Quasilinear parabolic problem with p (x)-Laplacian: existence, uniqueness of weak solutions and stabilization’, Nonlinear Differ. Equ. Appl. 23 (2016), Article ID 24.Google Scholar
Harjulehto, P., Hästö, P., Lê, Út V. and Nuortio, M., ‘Overview of differential equations with non-standard growth’, Nonlinear Anal. 72(12) (2010), 4551–4574.Google Scholar
Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory (Springer, Berlin, 2000).Google Scholar
Simon, J., ‘Régularité de la solution d’une équation non linéaire dans ℝ N ’, in: Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Mathematics, 665 (Springer, Berlin, 1978), 205–227.Google Scholar
You, Y. and Kaveh, M., ‘Fourth-order partial differential equations for noise removal’, IEEE Trans. Image Process. 9(10) (2000), 1723–1730.Google Scholar
Zang, A. and Fu, Y., ‘Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces’, Nonlinear Anal. 69(10) (2008), 3629–3636.Google Scholar
Zhikov, V. V., ‘Averaging of functionals of the calculus of variations and elasticity theory’, Izv. Akad. Nauk SSSR Ser. Mat. 50(4) (1986), 675–710.Google Scholar