On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term

A kind of non-Newtonianfluid equation with a damping term and a source term is considered. After giving a result of the existence, if the diffusion coefficient is degenerate on the boundary, the local stability of theweak solutions is establishedwithout any boundary condition. If the diffusion coefficient is degenerate on a part of the boundary, by imposing the homogeneous value condition on the other part of the boundary, the local stability of the weak solutions is proved. Moreover, if the equation is with a damping term, other than the finite propagation property, the results of this paper reveal the essential differences between the non-Newtonian fluid equation and the heat conduction equation in a new way.


Introduction
related to the −Laplacian, with the initial value and the usual boundary value  (, ) = 0, (, ) ∈ Ω × (0, ) , where () ∈ (Ω) and () ≥ 0, (, ) ∈ (  ), (, , ) is a continuous function, Ω ⊂ R  is a bounded domain with a smooth boundary Ω,  > 1,  > 0. The equation comes from a host of applied fields such as the theory of non-Newtonian fluid, the water infiltration through porous media, and the oil combustion process; one can refer to [1][2][3][4] and the references therein.For the evolutionary −Laplacian equation   = div (|∇| −2 ∇) , (, ) ∈ Ω × (0, ) , and with the initial-boundary value conditions (2) and (3), the weak solution is unique and has finite propagation property [3].However, the damping term and the source term in (1) may change the situation.Bertsh et.al. [5] and Zhou et.al. [6] had discussed the existence and the properties of the viscosity solutions for the equation and shown that the uniqueness of the weak solution is not true, where  is a positive constant.Zhang et.al. [7] had discussed the existence and the properties of the viscosity solution for the equation and shown that the uniqueness of the weak solution is not true, where () ≥ 0 and at least there exists a point  0 ∈ Ω such that ( 0 ) > 0,  ≥ 1.
Definition .A function (, ) is said to be a weak solution of (1) with the initial value (2), if and for any function  ∈ and then ( ) with initial value ( ) has a nonnegative weak solution.Moreover, if then the initial-boundary value problem ( ), ( ), ( ) has a nonnegative solution in the sense of Definition .
Since () = 0 when  ∈ Ω, condition (16) implies that (, )| ∈Ω = 0; hereafter, the constants  may depend on .We think the existence of the weak solutions can be proved only if  > 2, and the condition  > 4 is just a makeshift.Also condition (16) may not be necessary, but we are not ready to pay so much attentions to the existence.We will focus on the uniqueness of the weak solution.
Compared with ( 6) and (7), Theorem 4 reveals that the degeneracy of () brings the new change about the property of the solutions.
In order to illustrate the problem more clearly, secondly, we assume that In this case, we consider the uniqueness of weak solution to (1) under a partial boundary value condition.This is the following theorem.
If we notice that () satisfies (22), according to [5][6][7], the uniqueness of the solution to the equation is not true when  ∈ Ω is near to Σ  , while Theorem 5 implies that the uniqueness of the solution to the equation is true provided that  > 2. This fact shows the differences between the heat conduction equation ( = 2) and the non-Newtonian fluid equation ( > 2) again.It is well-known that the heat conduction equation has the infinite propagation property, while the non-Newtonian fluid equation has the finite propagation property.
At last, we are able to prove (13) as in [17]; thus we have Theorem 3.

The Proof of Theorem 5
Lemma 7. If ∫ Ω () −1/(−1)  < ∞,  is a weak solution of ( ) with the initial condition ( ). en the trace of  on the boundary Ω can be defined in the traditional way.
The function (, ) is said to be the weak solution of (1) with the initial value (2) and the boundary value condition (3), if  satisfies Definition 1, and the boundary value condition (3) is satisfied in the sense of trace.