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Gradient blowup rate for a viscous Hamilton-Jacobi equation with degenerate diffusion. (English) Zbl 1264.35060
Summary: This paper is concerned with the gradient blowup rate for the one-dimensional $p$-Laplacian parabolic equation $u_t=(\vert u_x\vert^{p-2}u_x)_x+\vert u_x\vert^q$ 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.

35B44Blow-up (PDE)
35B20Perturbations (PDE)
35K51Second-order parabolic systems, initial bondary value problems
35B40Asymptotic behavior of solutions of PDE
35K92Quasilinear parabolic equations with $p$-Laplacian
Full Text: DOI
[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) · Zbl 1253.35075 · doi:10.1016/j.jde.2012.07.002
[2] A. Attouchi, Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term, 2012, arXiv:1209.5023 [math.AP]. · Zbl 1253.35075
[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) · Zbl 1211.35042
[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) · Zbl 1159.35009 · doi:10.3934/dcds.2008.20.927
[5] Kardar M., Parisi G., Zhang Y.C.: Dynamic scailing of growing interfaces. Phys. Rev. Lett. 56, 889--892 (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[6] Krug J., Spohn H.: Universality classes for deterministic surface growth. Phys. Rev. A. 38, 4271--4283 (1988) · doi:10.1103/PhysRevA.38.4271
[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) · Zbl 1221.35195 · doi:10.2140/pjm.2007.230.347
[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) · Zbl 1242.35160 · doi:10.1007/s10884-011-9238-x
[10] Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996) · Zbl 0884.35001
[11] Stinner C.: Convergence to steady states in a viscous Hamilton--Jacobi equation with degenerate diffusion. Journal of Differential Equations 248, 209--228 (2010) · Zbl 1181.35025 · doi:10.1016/j.jde.2009.09.019
[12] Zhang Z.C., Hu B.: Gradient blowup rate for a semilinear parabolic equation. Discrete Contin. Dyn. Sys. 26, 767--779 (2010) · Zbl 1191.35074
[13] Zhang Z.C., Hu B.: Rate estimate of gradient blowup for a heat equation with exponential nonlinearity. Nonlinear Analysis 72, 4594--4601 (2010) · Zbl 1189.35033 · doi:10.1016/j.na.2010.02.036
[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) · Zbl 1264.35061 · doi:10.3934/cpaa.2013.12.269
[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) · Zbl 1299.35050 · doi:10.1016/S0252-9602(13)60029-6
[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) · Zbl 06476189