Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source
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Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source Yuzhu Han · Wenjie Gao
Received: 9 May 2009 / Accepted: 21 January 2010 / Published online: 5 February 2010 © Springer Science+Business Media B.V. 2010
Abstract This paper deals with a class of localized and degenerate quasilinear parabolic systems ut = f (u)(u + av(x0 , t)),
vt = g(v)(v + bu(x0 , t))
with homogeneous Dirichlet boundary conditions. Local existence of positive classical solutions is proven by using the method of regularization. Global existence and blow-up criteria are also obtained. Moreover, the authors prove that under certain conditions, the solutions have global blow-up property. Keywords Global existence · Finite time blow-up · Localized source · Degenerate parabolic system · Global blow-up
1 Introduction and Main Results In this paper, we consider the positive solutions of the following degenerate quasilinear parabolic system with localized source ⎧ ut = f (u)(u + av(x0 , t)), ⎪ ⎪ ⎪ ⎨v = g(v)(v + bu(x , t)), t 0 ⎪ u(x, t) = v(x, t) = 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x),
x ∈ , t > 0, x ∈ , t > 0, x ∈ ∂, t > 0, x ∈ ,
The project is supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University. Y. Han · W. Gao () Institute of Mathematics, Jilin University, Changchun 130012, People’s Republic of China e-mail: [email protected] Y. Han e-mail: [email protected]
(1.1)
252
Y. Han, W. Gao
where is a bounded domain in R N (N ≥ 1) with smooth boundary ∂, x0 ∈ is a fixed point and a, b are positive constants. The initial data u0 (x) and v0 (x) are positive and smooth functions. Functions f (s) and g(s) satisfy assumptions (H3) and (H4) given below. This system can be used to describe the development of two groups in the dynamics of biological groups where u and v are their densities. It also arises in the study of the flow of a fluid through a homogeneous isotropic rigid porous medium with internal localized source, see [3, 6–8]. The studies on the blow-up and global existence of solutions for the equations or systems not in divergence form with local, nonlocal or localized terms can be found in [2, 4, 5, 9–12, 16–21]. Parabolic equations with nonlocal sources and homogeneous Dirichlet boundary condition of the following form ut = f (u) u + a(x) udx ,
have been studied in [4] and [9]. When a is a positive ∞constant, Deng et al. in [9] proved that the solution blows up in finite time if and only if δ sfds(s) < ∞ and a > 1/μ, where δ > 0 is a constant and μ is defined as μ = ϕ(x)dx,
and ϕ is the unique positive solution of the linear elliptic problem −ϕ = 1,
x ∈ ;
ϕ(x) = 0,
x ∈ ∂.
In [5], Chen and Wang investigated the corresponding system
ut = f (u)(u + a vdx), x ∈ , t > 0, vt = g(v)(v + b udx), x ∈ , t > 0.
(1.2)
They obtained positive solution of (1.2) blows up in finite time if and only if ∞ that the unique ∞ ds < ∞. Here δ is a positive constan
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