Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion Authors Zhengce Zhang School of Mathematics and Statistics Xi’an Jiaotong University Article

First Online: 10 April 2013 Received: 20 November 2012 DOI :
10.1007/s00013-013-0505-4

Cite this article as: Zhang, Z. Arch. Math. (2013) 100: 361. doi:10.1007/s00013-013-0505-4
Abstract This paper is concerned with the gradient blowup rate for the one-dimensional p -Laplacian parabolic equation \({u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}\) with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.

Mathematics Subject Classification (2010) 35B35 35K58 35B20 This work was supported by the Fundamental Research Funds for the Central Universities of China and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References 1.

Attouchi A.: Well-posedness and gradient blow-up estimate near the boundary for a Hamilton–Jacobi equation with degenerate diffusion. Journal of Differential Equations

253 , 2474–2492 (2012)

MathSciNet MATH CrossRef 2.

A. Attouchi, Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term, 2012, arXiv:1209.5023 [math.AP].

3.

Barles G., Laurençot Ph., Stinner C.: Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton–Jacobi equation. Asymptotic Analysis

67 , 229–250 (2010)

MathSciNet MATH 4.

Guo J.-S., Hu B.: Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete Contin. Dyn. Sys.

20 , 927–937 (2008)

MathSciNet MATH CrossRef 5.

Kardar M., Parisi G., Zhang Y.C.: Dynamic scailing of growing interfaces. Phys. Rev. Lett.

56 , 889–892 (1986)

MATH CrossRef 6.

Krug J., Spohn H.: Universality classes for deterministic surface growth. Phys. Rev. A.

38 , 4271–4283 (1988)

MathSciNet CrossRef 7.

O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural’ceva, Linear and quasi-linear equations of parabolic type, Amer. Mathematical Society, 1968.

8.

Laurençot Ph.: Convergence to steady states for a one-dimensional viscous Hamilton–Jacobi equation with Dirichlet boundary conditions. Pacific J. Math.

230 , 347–364 (2007)

MathSciNet MATH CrossRef 9.

Laurençot Ph., Stinner C.: Convergence to separate variables solutions for a degenerate parabolic equation with gradient source. Journal of Dynamics and Differential Equations

24 , 29–49 (2012)

MathSciNet MATH CrossRef 10.

Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)

MATH CrossRef 11.

Stinner C.: Convergence to steady states in a viscous Hamilton–Jacobi equation with degenerate diffusion. Journal of Differential Equations

248 , 209–228 (2010)

MathSciNet MATH CrossRef 12.

Zhang Z.C., Hu B.: Gradient blowup rate for a semilinear parabolic equation. Discrete Contin. Dyn. Sys.

26 , 767–779 (2010)

MATH 13.

Zhang Z.C., Hu B.: Rate estimate of gradient blowup for a heat equation with exponential nonlinearity. Nonlinear Analysis

72 , 4594–4601 (2010)

MathSciNet MATH CrossRef 14.

Zhang Z.C., Li Y.Y.: Gradient blowup solutions of a semilinear parabolic equation with exponential source. Comm. Pure Appl. Anal.

12 , 269–280 (2013)

CrossRef 15.

Zhang Z.C., Li Z.J.: A note on gradient blowup rate of the inhomogeneous Hamilton–Jacobi equations. Acta Mathematica Scientia 33 , 678–686 (2013)

16.

Zhu L.P., Zhang Z.C.: Rate of approach to the steady state for a diffusion-convection equation on annular domains. E. J. Qualitative Theory Diff. Equations

39 , 1–10 (2012)

CrossRef