Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion 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.

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CrossRef Authors and Affiliations 1. School of Mathematics and Statistics Xi’an Jiaotong University Xi’an People’s Republic of China