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Z. Naturforsch. 67a, 479 – 482 (2012)
doi:10.5560/ZNA.2012-0048
Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition
Junping Zhao
College of Science, Xi'an University of Architecture & Technology, Xi'an 710055, China
Received February 2, 2012 / revised May 4, 2012 / published online August 2, 2012
Reprint requests to: J. P. Z.; E-mail: junpingzhao@yeah.net
The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)∇u) + f(x, u,|∇ u|2, t) under nonlinear boundary condition ∂u/∂n + g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf's maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f.
Key words: Reaction-Diffusion Equation; Gradient Term; Nonlinear Boundary Conditions; Maximum Principles; Blow-Up of Solutions.
Mathematics Subject Classification 2000: 35K20; 35K55; 35K65
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