Browse > Article
http://dx.doi.org/10.4134/CKMS.c200035

BLOW-UP TIME AND BLOW-UP RATE FOR PSEUDO-PARABOLIC EQUATIONS WITH WEIGHTED SOURCE  

Di, Huafei (School of Mathematics and Information Science Guangzhou University)
Shang, Yadong (School of Mathematics and Information Science Guangzhou University)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.4, 2020 , pp. 1143-1158 More about this Journal
Abstract
In this paper, we are concerned with the blow-up phenomena for a class of pseudo-parabolic equations with weighted source ut - △u - △ut = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions in any smooth bounded domain Ω ⊂ ℝn (n ≥ 1). Firstly, we obtain the upper and lower bounds for blow-up time of solutions to these problems. Moreover, we also give the estimates of blow-up rate of solutions under some suitable conditions. Finally, three models are presented to illustrate our main results. In some special cases, we can even get some exact values of blow-up time and blow-up rate.
Keywords
Pseudo-parabolic equation; upper and lower bounds; blow-up rate; weighted source;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. Escobedo and M. A. Herrero, A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl. 165 (1993), no. 1, 315-336. https://doi.org/10.1007/BF01765854   DOI
2 Z. Fang and Y. Wang, Blow-up analysis for a semilinear parabolic equation with time- dependent coefficients under nonlinear boundary flux, Z. Angew. Math. Phys. 66 (2015), no. 5, 2525-2541. https://doi.org/10.1007/s00033-015-0537-7   DOI
3 M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudo-parabolic type in problems of mathematicial physics, Comput. Math. Math. Phys. 43 (2003), no. 12, 1765-1797; translated from Zh. Vychisl. Mat. Mat. Fiz. 43 (2003), no. 12, 1835-1869.
4 J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, 1969.
5 Y. Liu, Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions, Math. Comput. Modelling 57 (2013), no. 3-4, 926-931. https://doi.org/10.1016/j.mcm.2012.10.002   DOI
6 Y. Liu, W. Jiang, and F. Huang, Asymptotic behaviour of solutions to some pseudoparabolic equations, Appl. Math. Lett. 25 (2012), no. 2, 111-114. https://doi.org/ 10.1016/j.aml.2011.07.012   DOI
7 P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Methods Appl. Sci. 38 (2015), no. 12, 2636-2641. https://doi.org/10.1002/mma.3253   DOI
8 L. E. Payne, G. A. Philippin, and S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Z. Angew. Math. Phys. 61 (2010), no. 6, 999-1007. https://doi.org/10.1007/s00033-010-0071-6   DOI
9 X. Lv and X. Song, Bounds of the blowup time in parabolic equations with weighted source under nonhomogeneous Neumann boundary condition, Math. Methods Appl. Sci. 37 (2014), no. 7, 1019-1028. https://doi.org/10.1002/mma.2859   DOI
10 L. Ma and Z. Fang, Blow-up analysis for a reaction-diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux, Nonlinear Anal. Real World Appl. 32 (2016), 338-354. https://doi.org/10.1016/j.nonrwa.2016.05.005   DOI
11 L. E. Payne, G. A. Philippin, and S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Anal. 73 (2010), no. 4, 971-978. https://doi.org/10.1016/j. na.2010.04.023   DOI
12 L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Neumann conditions, Appl. Anal. 85 (2006), no. 10, 1301-1311. https://doi. org/10.1080/00036810600915730
13 L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007), no. 2, 1196-1205. https://doi.org/10.1016/j.jmaa.2006.06.015   DOI
14 J. C. Song, Lower bounds for the blow-up time in a non-local reaction-diffusion problem, Appl. Math. Lett. 24 (2011), no. 5, 793-796. https://doi.org/10.1016/j.aml.2010.12.042   DOI
15 X. Peng, Y. Shang, and X. Zheng, Blow-up phenomena for some nonlinear pseudoparabolic equations, Appl. Math. Lett. 56 (2016), 17-22. https://doi.org/10.1016/j. aml.2015.12.005   DOI
16 M. Peszynska, R. Showalter, and S.-Y. Yi, Homogenization of a pseudoparabolic system, Appl. Anal. 88 (2009), no. 9, 1265-1282. https://doi.org/10.1080/00036810903277077   DOI
17 J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (1992), no. 3, 249-264. https://doi.org/10.1093/imamat/48.3.249   DOI
18 R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal. 3 (1972), 527-543. https://doi.org/10. 1137/0503051   DOI
19 S. L. Sobolev, On a new problem of mathematical physics, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 18 (1954), no. 1, 3-50.
20 X. Song and X. Lv, Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source, Appl. Math. Comput. 236 (2014), 78-92. https://doi.org/10.1016/j.amc.2014.03.023   DOI
21 G. Tang, Y. Li, and X. Yang, Lower bounds for the blow-up time of the nonlinear nonlocal reaction diffusion problems in $R^N$ ($N{\geq}3$), Bound. Value Probl. 2014 (2014), 265, 5 pp. https://doi.org/10.1186/s13661-014-0265-5   DOI
22 R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013), no. 12, 2732-2763. https://doi.org/10.1016/j.jfa.2013.03.010   DOI
23 H. Di, Y. Shang, and X. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett. 64 (2017), 67-73. https://doi.org/10.1016/j.aml.2016.08.013   DOI
24 A. B. Al'shin, M. O. Korpusov, and A. G. Sveshnikov, Blow-up in nonlinear Sobolev type equations, De Gruyter Series in Nonlinear Analysis and Applications, 15, Walter de Gruyter & Co., Berlin, 2011. https://doi.org/10.1515/9783110255294
25 J. Bebernes and A. Bressan, Thermal behavior for a confined reactive gas, J. Differential Equations 44 (1982), no. 1, 118-133. https://doi.org/10.1016/0022-0396(82)90028-6   DOI
26 H. Di and Y. Shang, Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration, J. Differential Equations 269 (2020), no. 5, 4566-4597. https://doi.org/10.1016/j.jde.2020.03.030   DOI
27 E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J. 30 (1981), no. 6, 821-854. https://doi.org/10.1512/iumj.1981.30.30062   DOI
28 E. S. Dzektser, Generalization of equations of motion of underground water with free surface, Dokl. Akad. Nauk SSSR. 202 (1972), no. 5, 1031-1033.