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BLOW-UP AND GLOBAL SOLUTIONS FOR SOME PARABOLIC SYSTEMS UNDER NONLINEAR BOUNDARY CONDITIONS

Guo, Limin;Liu, Lishan;Wu, Yonghong;Zou, Yumei

  • Received : 2018.08.07
  • Accepted : 2019.03.04
  • Published : 2019.07.01

Abstract

In this paper, blows-up and global solutions for a class of nonlinear divergence form parabolic equations with the abstract form of $({\varrho}(u))_t$ and time dependent coefficients are considered. The conditions are established for the existence of a solution globally and also the conditions are established for the blow up of the solution at some finite time. Moreover, the lower bound and upper bound of the blow-up time are derived if blow-up occurs.

Keywords

blows-up and global solutions;parabolic equations;nonlinear boundary conditions;time dependent coefficients;abstract form of $({\varrho}(u))_t$

References

  1. J. M. Arrieta, A. N. Carvalho, and A. Rodriguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), no. 2, 376-406. https://doi.org/10.1006/jdeq.1998.3612
  2. J. Ding and H. Hu, Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions, J. Math. Anal. Appl. 433 (2016), no. 2, 1718-1735. https://doi.org/10.1016/j.jmaa.2015.08.046
  3. C. Enache, Lower bounds for blow-up time in some non-linear parabolic problems under Neumann boundary conditions, Glasg. Math. J. 53 (2011), no. 3, 569-575. https://doi.org/10.1017/S0017089511000139
  4. Z. B. 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.
  5. F. Li and J. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl. 385 (2012), no. 2, 1005-1014. https://doi.org/10.1016/j.jmaa.2011.07.018
  6. F. Li, X. Zhu and Z. Liang, Multiple solutions to a class of generalized quasilinear Schrodinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl. 443 (2016), no. 1, 11-38. https://doi.org/10.1016/j.jmaa.2016.05.005
  7. D. Liu, C. Mu, and Q. Xin, Lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 3, 1206-1212.
  8. L. E. Payne and G. A. Philippin, Blow-up in a class of non-linear parabolic problems with time-dependent coefficients under Robin type boundary conditions, Appl. Anal. 91 (2012), no. 12, 2245-2256. https://doi.org/10.1080/00036811.2011.598865
  9. L. E. Payne, G. A. Philippin, and P. W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl. 338 (2008), no. 1, 438-447. https://doi.org/10.1016/j.jmaa.2007.05.022
  10. 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.
  11. L. E. Payne and P. W. Schaefer, Bounds for blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 6, 1289-1296.
  12. L. E. Payne and J. C. Song, Blow-up and decay criteria for a model of chemotaxis, J. Math. Anal. Appl. 367 (2010), no. 1, 1-6. https://doi.org/10.1016/j.jmaa.2009.11.025
  13. L. E. Payne and J. C. Song, Lower bounds for blow-up in a model of chemotaxis, J. Math. Anal. Appl. 385 (2012), no. 2, 672-676. https://doi.org/10.1016/j.jmaa.2011.06.086
  14. 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.
  15. P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), no. 6, 1301-1334. https://doi.org/10.1137/S0036141097318900
  16. F. Sun, L. Liu, and Y. Wu, Infinitely many sign-changing solutions for a class of biharmonic equation with p-Laplacian and Neumann boundary condition, Appl. Math. Lett. 73 (2017), 128-135.
  17. F. Sun, L. Liu, and Y. Wu, Blow-up of a nonlinear viscoelastic wave equation with initial data at arbitrary high energy level, Appl. Anal.; DOI: 10.1080/00036811.2018.1460812.
  18. F. Sun, L. Liu, and Y. Wu, Finite time blow-up for a thin-film equation with initial data at arbitrary energy level, J. Math. Anal. Appl. 458 (2018), no. 1, 9-20.
  19. F. Sun, L. Liu, and Y. Wu, Finite time blow-up for a class of parabolic or pseudo-parabolic equations, Comput. Math. Appl. 75 (2018), no. 10, 3685-3701.
  20. F. Sun, L. Liu, and Y. Wu, Global existence and finite time blow-up of solutions for the semilinear pseudoparabolic equation with a memory term, Appl. Anal. 98 (2019), no. 4, 735-755.
  21. X. Yang and Z. Zhou, Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition, J. Differential Equations 261 (2016), no. 5, 2738-2783. https://doi.org/10.1016/j.jde.2016.05.011

Acknowledgement

Supported by : National Natural Science Foundation of China, Natural Science Foundation of Shandong Province of China, Changzhou institute of technology