Publication Cover
Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 8
260
Views
1
CrossRef citations to date
0
Altmetric
Articles

Gradient blowup rate for a heat equation with general gradient nonlinearity

, , &
Pages 1635-1644 | Received 09 Nov 2014, Accepted 29 Jun 2015, Published online: 25 Jul 2015

References

  • Alikakos ND, Bates PW, Grant CP. Blow up for a diffusion-advection equation. Proc. Roy. Soc. Edinburgh Sect. A. 1989;113:181–190.
  • Arrieta JM, Bernal R, Souplet Ph. Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Ann. Scuola. Norm. Super. Pisa Cl. Sci. 2004;3:1–15.
  • Conner GR, Grant CP. Asymptotics of blowup for a convection-diffusion equation with conservation. Diff. Integral Equations. 1996;9:719–728.
  • Fila M, Lieberman GM. Derivative blow-up and beyond for quasilinear parabolic equations. Diff. Integral Equations. 1994;7:811–821.
  • Fila M, Taskinen J, Winkler M. Convergence to a singular steady state of a parabolic equation with gradient blow-up. Applied Mathematics Letters. 2007;20:578–582.
  • Guo J, Hu B. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete Contin. Dyn. Syst. 2008;20:927–937.
  • Li YX, Souplet Ph. Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. Commun. Math. Phys. 2010;293:499–517.
  • Lieberman GM. Second order parabolic differential Equations. Singapore: World Scientific; 2005.
  • Quittner P, Souplet Ph. Superlinear parabolic problems: blow-up. Birkhäuser: Global Existence and Steady States; 2007.
  • Souplet Ph. Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Diff. Integral Equations. 2002;15:237–256.
  • Souplet Ph. Recent results and open problems on parabolic equations with gradient nonlinearities. E. J. Differential Equations. 2001;2001(20):1–19.
  • Souplet Ph, Vázquez JL. Stability towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete Contin. Dyn. Syst. 2006;14:221–234.
  • Souplet Ph, Zhang QS. Global solutions of inhomogenous Hamilton-Jacobi equations. J. D’Analyse Math. 2006;99:335–396.
  • Zhang ZC, Hu B. Gradient blowup rate for a semilinear parabolic equation. Discrete Contin. Dyn. Syst. 2010;26:767–779.
  • Zhang ZC, Hu B. Rate estimates of gradient blowup for a heat equation with exponential nonlinearity. Nonlinear Analysis. 2010;72:4594–4601.
  • Zhang ZC, Li Y. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete Contin. Dyn. Syst. Ser. B. 2014;19:3019–3029.
  • Zhang ZC, Li YY. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Comm. Pure Appl. Anal. 2013;12:269–280.
  • Zhang ZC, Li YY. Boundedness of global solutions for a heat equation with exponential gradient source. Abstract and Applied Analysis. 2012;2012:1–10.
  • Zhang ZC, Li ZJ. A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations. Acta Mathematica Scientis. 2013;33B:678–686.
  • Zhu LP, Zhang ZC. Rate of approach to the steady state for a diffusion-convection equation on annular domains. Electron. J. Qual. Theory Differ. Equ. 2012;39:1–10.
  • Ladyzenskaja O, Solonnikov V, Uralceva N. Translations of mathematical monographs. Vol. 23. Providence (RI): American Mathematical Society; 1967. 7 Chapters, Linear and quasilinear equations of parabolic type; 648 p.
  • Lieberman GM. The first initial-boundary value problem for quasilinear second order parabolic equations. Ann. Scuola Norm. Sup. Pisa. 1986;13:347–387.

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.