A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

Huashui Zhan

doi:10.1155/2018/3070738

Abstract

Consider the nonlinear parabolic equation ∂u/∂t-div(a(x)|∇u|p-2∇u)=f(x,t,u,∇u) with axx∈Ω>0 and a(x)x∈∂Ω=0. Though it is well known that the degeneracy of a(x) may cause the usual Dirichlet boundary value condition to be overdetermined, and only a partial boundary value condition is needed, since the nonlinearity, this partial boundary can not be depicted out by Fichera function as in the linear case. A new method is introduced in the paper; accordingly, the stability of the weak solutions can be proved independent of the boundary value condition.

Highlights

Introduction and the Main ResultsThe nonlinear parabolic equation ∂u ∂t − div (a (x) |∇u|p−2 ∇u) = f (x, t, u, (1)(x, t) ∈ QT = Ω × (0, T), comes from the theory of non-Newtonian fluid and had been studied widely; one can refer to [1–6] and the references therein
The main results of the paper are the following theorems
If p > 1, (14) is true, the stability of the weak solutions is true in the sense of (13). This is due to the fact that only we choose α ≥ (p−3)/(p− 2), and condition (15) is natural

Summary

Introduction and the Main Results

If p > 1, (14) is true, the stability of the weak solutions is true in the sense of (13) This is due to the fact that only we choose α ≥ (p−3)/(p− 2), and condition (15) is natural. ∫ dα (x) |u (x, t) − V (x, t)|2 dx (27) ≤ c ∫ dα (x) 󵄨󵄨󵄨󵄨u0 (x) − V0 (x)󵄨󵄨󵄨󵄨2 dx This is due to the fact that only if we choose α ≥ max{1, 1/(p − 1)}, conditions (16) and (18) in Theorem 5 are naturally true. As long as you like, you can choose the other general characteristic functions; for example, φ(x) = ea(x) − 1 and φ(x) = sin(a(x)/M), to obtain the corresponding theorems. There are many papers devoted to these equations; one can refer to [11–23] and the references therein

The Proof of Theorem 3

The Proof of Theorem 4

Findings

The Proof of Theorem 5