Large time behavior of solutions for the porous medium equation with a nonlinear gradient source
- PDF / 458,838 Bytes
- 20 Pages / 595.276 x 793.701 pts Page_size
- 9 Downloads / 224 Views
RESEARCH
Open Access
Large time behavior of solutions for the porous medium equation with a nonlinear gradient source Nan Li1 , Pan Zheng2,3* , Chunlai Mu2 and Iftikhar Ahmed2 * Correspondence: [email protected] 2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China 3 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China Full list of author information is available at the end of the article
Abstract This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source ut = um + |∇ul |q , (x, t) ∈ × (0, ∞), where l ≥ m > 1 and 1 ≤ q < 2. When lq = m, we prove that the 1 global solution converges to the separate variable solution t– m–1 f (x). While m < lq ≤ m + 1, we note that the global solution converges to the separate variable 1 solution t– m–1 f0 (x). Moreover, when lq > m + 1, we show that the global solution also 1 converges to the separate variable solution t– m–1 f0 (x) for the small initial data u0 (x), and we find that the solution u(x, t) blows up in finite time for the large initial data u0 (x). MSC: 35K55; 35K65; 35B40 Keywords: large time behavior; separate variable solution; porous medium equation; gradient source; blow-up
1 Introduction In this paper, we investigate the large time behavior of non-negative solutions for the following initial-boundary value problem: ⎧ m l q ⎪ ⎨ut = u + |∇u | , (x, t) ∈ × (, ∞), u(x, t) = , (x, t) ∈ ∂ × (, ∞), ⎪ ⎩ u(x, ) = u (x), x ∈ ,
(.)
where l ≥ m > , ≤ q < , is a bounded domain of RN (N ≥ ) with smooth boundary ∂, and the initial function is u (x) ∈ C () = z ∈ C() : z = on ∂ ,
u (x) ≥ , u (x) ≡ .
(.)
Equation (.) arises in the study of the growth of surfaces and it has been considered as a mathematical model for a variety of physical problems (see [, ]). For instance, in ballistic deposition processes, the evolution of the profile of a growing interface is described by the diffusive Hamilton-Jacobi type equation (.) with m = l = (see []). One of the particular feature of problem (.) is that the equation is a slow diffusion equation with nonlinear source term depending on the gradient of a power of the solution. ©2014 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Li et al. Boundary Value Problems 2014, 2014:98 http://www.boundaryvalueproblems.com/content/2014/1/98
Page 2 of 20
In general, there is no classical solution. Therefore, it turns out that a suitable framework for the well-posedness of the initial-boundary value problem (.) is the theory of viscosity solutions (see [–]), so we first define the notion of solutions. Definition . A non-negative function u(x, t) ∈ C( × (, ∞))
Data Loading...
