The Dirichlet problem for a class of nonlinear degenerate parabolic equations
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We study the first boundary value problem for a class of nonlinear degenerate parabolic ∇u)). We first consider its regularized problem and establish equations −∂u/∂t = div(A( some estimates. Based on these estimates, we prove the existence and uniqueness of the generalized solutions in BV space. 1. Introduction Let Ω ⊂ Rm (m ≥ 1) be a bounded set with smooth boundary ∂Ω. We are concerned with the Dirichlet problem −
∂u ∇u) (x,t) ∈ QT = Ω × (0,T), = div A( ∂t u(x,t) = 0 (x,t) ∈ ∂Ω × (0,T),
(1.1)
u(x,0) = u0 (x), = (A1 (p),... ,Am (p)) ∈ C 1 (Rm ,Rm ), u0 (x) is appropriately smooth on Ω and where A(p) certain compatibility conditions on the boundary of the lower base of QT are fulfilled. We suppose that 0≤
2 ∂Ai (p) ξi ξ j ≤ Λξ , ∂p j
· p, µ1 | p|q ≤ A(p)
∀ξ ∈ R m ,
A(p) ≤ µ2 pq−1 + 1 ,
(1.2) ∀ p ∈ Rm ,
(1.3)
where q ≥ 2, Λ, µ1 , µ2 are positive numbers. Under some conditions, Gregori [1] considered the elliptic problem ∇u) = 0 − div A(
x ∈ Ω,
u|∂Ω = 0 x ∈ ∂Ω,
(1.4)
and proved the existence and the uniqueness of BV solutions. In this note, we generalize the results of [1] to the parabolic case. The Dirichlet problem (1.1) arises from a variety Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 395–402 DOI: 10.1155/JIA.2005.395
396
Nonlinear degenerate parabolic equations
of diffusion phenomena which appear widely in nature. The non-Newtonian filtration equation p−2 ∂u = div ∇u ∇u , ∂t
p = 2
(1.5)
is a special case of problem (1.1). Problem (1.1) has been widely investigated, for example, see [2, 4, 5, 6] and references therein. For the one-dimensional case, Wu et al. [5] considered the Dirichlet problem −ut = (∂/∂x)(A((∂/∂x)B(u))) + ∂ f (u)/∂x, and proved the existence and uniqueness of the generalized solutions in BV space under some constrains. Our interest here is to treat the problem for a multi-dimensional case without absorption. Generally speaking, solutions of problem (1.1) are not continuous. The sense of satisfying the boundary value conditions for solutions is also special (see [3]). In present paper, we take some ideas from [6] and investigate the solvability in BV (QT ), where BV is the class of all integrable functions on QT , whose generalized derivatives are measures with bounded variation. The existence of solutions will be proved by means of the method of parabolic regularization. 2. Main results
Definition 2.1. A function u ∈ BV (QT ) L∞ (QT ) is said to be a generalized solution of problem (1.1), if the following conditions are fulfilled: (1) ut ∈ L∞ (0,T;L2 (Ω)), uxi ∈ Lq (QT ), i = 1,2,...,m. (2) For almost all x ∈ Ω, γu(x,0) = u0 (x), where γu is the trace of u. (3) For almost all t ∈ (0,T), γu(x,t) = 0 a.e. on ∂Ω. (4) u satisfies
∂ϕ1 sgn(u − k) (u − k) − A(∇u) · ∇ϕ1 dx dt ∂t QT
+
QT
sgn k u
∂ϕ2 − A(∇u) · ∇ϕ2 dx dt ≥ 0, ∂t
(2.1)
where ϕ1 ,ϕ2 ∈ C 1 (QT ), ϕ1 ,ϕ2 ≥ 0, ϕ1 = ϕ2 on ∂Ω × (0,T), suppϕ1 ,suppϕ2 ⊂ Ω × (0,T) and k ∈ R.
Remark 2.2
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