AIMS Mathematics, 2017, 2(3): 400-421. doi: 10.3934/Math.2017.3.400

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

The viscosity solutions of a nonlinear equation related to the p-Laplacian

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, P. R. China

The viscosity solutions of a nonlinear equation related to the p-Laplacian are considered. Besides there is a damping term in the equation, a nonlocal function is added. By considering the regularized problem and using Moser iteration technique, we get the uniformly local bounded properties of the solutions and the Lp-norm for the gradients. By the compactness theorem, we prove the existence of the viscosity solution of the equation.
  Figure/Table
  Supplementary
  Article Metrics

References

1. M. Aassila, The influence of nonlocal nonlinearities on the long time behavior of solutions of diffusion problems, J. of Diff. Equ., 192 (2003), 47-69.    

2. A. V. Babin and M. I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, 1992.

3. M. Bertsch, R. Dal Passo and M. Ughi, Discontinuous viscosity solutions of a degenerate parabolic equation, Trans. Amer. Math. Soc., 320 (1990), 779-798.

4. M. Bertsch, R. Dal Passo and M. Ughi, Non-uniqueness of solutions of a degenerate parabolic equation, Ann. Math. Pura Appl., 161 (1992), 57-81.    

5. C. Chen, On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity, J. Math. Anal. Appl., 337 (2008), 318-332.    

6. C. Chen and R. Wang, Global existence and L∞ estimates of solution for doubly degenerate parabolic equation (in Chinese), ACTA Math. Sinica, 44 (2001), 1089-1098.

7. J.W. Cholewa and T. Dlotko, Global attractors in abstract parabolic problems, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 278 (2000) .

8. A. Dall'Aglioa, D. Giachetti, I. Peral and S. León, Global existence for some slightly super-linear parabolic equations with measure data, J. Math. Anal. Appl., 345 (2008), 892-902.

9. R. Dal Passo and S. Luckhaus, A degenerate diffusion problem not in divergence form, J. Diff. Equa., 69 (1987), 1-14.    

10. E. DiBenedetto, Degenerate parabolic equations, Universitext, Springer Verlag, 1993.

11. J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in R with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-88.    

12. L. C. Evans, Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics Number 74, 1998.

13. L. Gu, Second order parabolic partial differential equations, The Publishing Company of Xiamen University, China, 2002.

14. A. V. Ivanov, Hölder estimates for quasilinear parabolic equations, J.Soviet Mat., 56 (1991), 2320-2347.    

15. A. S. Kalashnikkov, Some problems of nonlinear parabolic equations of second order, USSR. Math, Nauk, T., 42 (1987), 135-176.

16. S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behavior for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-354.

17. O. A. Ladyzenskaja, New equations for the description of incompressible fluids and solvability in the large boundary value problem for them, Proc. Steldov Inst. Math., 102 (1976), 95-118.

18. K. Lee, A. Petrosyan and J.L. Vázquez, Large time geometric properties of solutions of the evolution p-Laplacian equation, J. Diff. Equ., 229 (2006), 389-411.    

19. K. Lee and J. L. V´azque, Geometrical properties of solutions of the Porous Medium Equation for large times, Indiana Univ. Math. J., 52 (2003), 991-1016.

20. P. Lei, Y. Li and P. Lin, Null controllability for a semilinear parabolic equation with gradient quadratic growth, Nonlinear Anal. T.M.A., 68 (2008), 73-82.    

21. J. L. Lions, Quelques méthodes de resolution des problè mes aux limites non linear, Dound Gauthier-Villars, Paris, 1969.

22. J. Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations, Electronic J. Diff. Equ., 1994 (1994), 1-16.

23. M. Nakao, Global solutions for some nonlinear parabolic equations with non-monotonic perturbations, Non. Anal. TMA, 10 (1986), 299-314.    

24. M. Nakao, Lp estimates of solutions of some nonlinear degenerate diffusion equation, J. Math. Soc. Japan, 37 (1985), 41-63.    

25. M. Nakao and C. Chen, Global existence and gradient estimate for the quasilinear parabolic equation of m-Laplacian type with a nonlinear convection term, J. Diff. Equ., 162 (2000), 224-250.    

26. Y. Ohara, L∞ estimates of solutions of some nonlinear degenerate parabolic equations, Nonlinear Anal. TMA, 18 (1992), 413-426.    

27. M. Pierre, Uniqueness of solution of $u_{t}-\triangle\varphi(u)=0$ with initial datum a measure, Nonlinear Analysis TMA., 6 (1982), 175-187.    

28. R. Temam, Infinite-dimensional dynamical in mechanics and physics, Springer-Verlag, New York, 1997.

29. M. Ughi, A degenerate parabolic equation modelling the spread of an epidemic, Ann. Math. Pura Appl., 143 (1986), 385-400.    

30. M. Winkler, Large time behavior of solutions to degenerate parabolic equations with absorption, NoDEA. 8 (2001), 343-361.    

31. Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear diffusion equations, Word Scientific Publishing, 2001.

32. J. Yuan, Z. Lian, L. Cao, J. Gao and J. Xu, Extinction and positivity for a doubly nonlinear degenerate parabolic equation, Acta Math. Sinica, Eng. Ser., 23 (2007), 1751-1756.    

33. H. Zhan, The Asymptotic Behavior of Solutions for a Class of Doubly Nonlinear Parabolic Equations, J. Math. Anal. Appl., 370 (2010), 1-10.    

34. Q. Zhang and P. Shi, Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms, Nonlinear Anal. T.M.A., 72 (2010), 2744-2752.    

35. J. Zhao and H. Yuan, The Cauchy problem of some doubly nonlinear degenerate parabolic equations (in Chinese), Chinese Ann. Math., A, 16 (1995), 179-194.

36. W. Zhou and S. Cai, The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form, J. Jilin University (Natural Sci.), 42 (2004), 341-345.

Copyright Info: © 2017, Qitong Ou, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved