Author:
Liang Mengyang, ,Fang Zhong Bo,Yi Su-Cheol,
Abstract
<abstract><p>This paper deals with the blow-up phenomena of solution to a reaction-diffusion equation with gradient absorption terms under nonlinear boundary flux. Based on the technique of modified differential inequality and comparison principle, we establish some conditions on nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, some bounds for blow-up time are derived under appropriate measure in higher dimensional spaces $ \left({N \ge 2} \right). $</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference29 articles.
1. M. Ben-Artzi, P. Souplet, F. B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pure. Appl., 81 (2002), 343–378.
2. B. H. Gilding, M. Guedda, R. Kersner, The Cauchy problem for $ u_t = \Delta u + \left| {\nabla u} \right|^p$, J. Math. Anal. Appl., 284 (2003), 733–755.
3. H. A. Levine, L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differ. Equations, 16 (1974), 319–334.
4. J. Filo, Diffusivity versus absorption through the boundary, J. Differ. Equations, 99 (1992), 281–305.
5. P. Quittner, P. Souplet, Blow-up, global existence and steady states, In: Superlinear parabolic problems, Basel: Birkhauser, 2007.