Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 7
185
Views
9
CrossRef citations to date
0
Altmetric
Articles

The uniqueness of the solution to the diffusion equation with a damping term

Pages 1333-1346 | Received 27 Apr 2017, Accepted 23 Dec 2017, Published online: 08 Jan 2018

References

  • Wang J , Gao W , Su M . Periodic solutions of non-Newtonian polytropic filtration equations with nonlinear sources. Appl Math Comput. 2010;216:1996–2009.
  • Ye H , Yin J . Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection J Math Anal Appl. 2015;421:1225–1237.
  • Kalashnikov AS . Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations. Russ Math Surv. 1987;42(2):169–222.
  • DiBenedetto E . Degenerate parabolic equations. New York (NY): Spring-Verlag; 1993.
  • Nakao M . Lp estimates of solutions of some nonlinear degenerate diffusion equation. J Math Soc Japan. 1985;37:41–63.
  • Wu Z , Zhao J , Yin J , et al . Nonlinear diffusion equations. Singapore: Word Scientific Publishing; 2001.
  • Dambrosio W . Multiple solutions of weakly-coupled system with p-Laplacian operators. Res Math. 1999;36:34–54.
  • Nabana E . Uniqueness for positive solutions of p-Laplacian problem in an annulus. Ann Fac Sci Toulouse Math. 1999;8:143–154.
  • Lee K , Petrosyan A , Vazquez JL . Large time geometric properties of solutions of the evolution p-Laplacian equation. J Differ Equ. 2006;229:389–411.
  • Zhan H . Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Appl Math. 2008;53:521–333.
  • Zhao J . Existence and nonexistence of solutions for ut = div(|Δu|p-2Δu) + f(Δu,ux,t) . J Math Anal Appl. 1993;172(1):130–146.
  • Yin J . Continuous solutions of the first boundary value problems for nonlinear diffusion equations. Acta Math Sin New Ser. 1994;10:64–71.
  • Wang J , Gao W . Blow-up and boundedness of solutions at finite time for evolution p-equations with nonlinear boundary conditions. J Jilin Univ. (Sci. Ed). 2004;42:395–401.
  • Wang J , Wang Z , Yin J . A class of degenerate diffusion equations with mixed boundary conditions. J Math Anal. Appl. 2004;298:589–603.
  • Zhan H . On a parabolic equation related to the p-Laplacian. Bound Value Probl. 2016;2016:78. DOI:10.1186/s13661-016-0587-6
  • Benedikt J , Girg P , Kotrla L , et al . Nonuniqueness and multi-bump solutions in parabolic problems with the p-Laplacian. J Differ Equ. 2016;260:991–1009.
  • Bertsch M , Dal Passo R , Ughi M . Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798.
  • Zhou W , Cai S . The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin Univ (Nat Sci). 2004;42:341–345.
  • Zhang Q , Shi P . Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal TMA. 2010;72:2744–2752.
  • Antontsev SN , Shmarev SI . Parabolic equations with double variable nonlinearities. Math Comput Simul. 2011;81:2018–2032.
  • Li Z , Yan B , Gao W . Existence of solutions to a parabolic p-Laplace equation with convection term via-estimates. Electron J Differ Equ. 2015;2015(46):1–21.
  • Chen M , Zhao J . On the Cauchy problem of evolution p-Laplacian equation with nonlinear gradient term. Chin Ann Math. 2009;30(B):1–18.
  • Cianci P , Cirmi GR , D’Asero S , et al . Morrey estimates for solutions of singular quadratic non linear equations. Ann Mat Pura Appl. 2017;196:1739–1758.
  • Cirmi GR , Leonardi S . Higher differentiability for the solutions of nonlinear elliptic systems with lower order terms and L1,θ -data. Ann Mat Pura Appl. 2014;193(1):115–131.
  • Cirmi GR , D’Asero S , Leonardi S . Fourth order nonlinear equations with lower order term and natural growth conditions. Nonlinear Anal: Theory Methods Appl. 2014;108:66–86.
  • Cirmi GR , D’Asero S , Leonardi S . Gradient estimate for solutions of a class of nonlinear elliptic equations below the duality exponent. Math Methods Appl Sci. 2018;41(1):261–269.
  • Bobkov VE , Takac P . A strong maximum principle for parabolic equations with the p-Laplacian. J Math Anal Appl. 2014;419:218–230.
  • Benedikt J , Girg P , Kotrla L , et al . The strong maximum principle in parabolic problems with the p-Laplacian in a domain. Appl Math Lett. 2017;63:95–101.
  • Simon J . Compact sets in the space lp(0,t;b) . Ann Mat pura Appl IV Ser. 1952;146:65–96.
  • Gu L . Second order parabolic partial differential equations. Xiamen: The Publishing Company of Xiamen University; 2002. Chinese.
  • Taylor ME . Partial differential equations III. New York (NY): Springer-Verlag; 1999.
  • Zhou S . A priori L∞ –estimate and existence of solutions for some nonlinear parabolic equations. Nonlinear Anal. 2000;42:887–904.
  • Li Z , Yan B , Gao W . Existence of solutions to a parabolic p-Laplace equation with convection term via-estimates. Electron J Differ Equ. 2015;2015(46):1–21.
  • Chen M , Zhao J . On the Cauchy problem of evolution p-Laplacian equation with nonlinear gradient term. Chin Ann Math. 2009;30(B):1–18.
  • Zhan H . The solution of convection--diffusion equation. Chin Ann Math. 2013;34(2):235–256. Chinese.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.