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

Consider the nonlinear parabolic equation ∂u/∂t−div(a(x)|∇u|p−2∇u) = f(x, t, u, ∇u)with a(x)|x∈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.

Certainly, the general characteristic method also can be used to study the stability of weak solutions to (1).We assume that (),   (), and (, ) are  1 functions, and  () > 0,  ∈ Ω,  () = 0,  ∈ Ω. ( Definition 1.A function (, ) is said to be a weak solution of (4) with the initial value (2), if The initial value is satisfied in the sense of that lim The existence of the solution can be proved in a similar way as that of the evolutionary -Laplacian equation [4]; we omit the details here.In order to study the stability of the weak solutions, let us introduce a new concept.Definition 2. A nonnegative continuous function  is said to be a general characteristic function of Ω, if and only if that One can see that Definition 2 is inspired by the usual characteristic function  of Ω, which is defined as Unlike , the general characteristic function  is not unique.For example, the distance function () = dist(, Ω) and the diffusion function () defined in (5) both are the general characteristic functions.Actually, we only borrow the concept of the characteristic function , but no more than that.The main results of the paper are the following theorems.Theorem 3. Let  and V be two weak solutions of (4) with the initial values  0 () and V 0 (), respectively,  > 1 and If there exists a general characteristic function  such that Theorem 4. Let  and V be two nonnegative solutions of (4) with the initial values  0 () and V 0 (), respectively.If  > 1, and there exists a general characteristic function  such that then the stability of the weak solutions is true in the sense of (13).Here, Ω  = { ∈ Ω : () > }.
A local stability of the weak solutions is given as follows.
Theorem 5. Let  and V be two solutions of ( 4) with the differential initial values  0 () and V 0 (), respectively.If  > 1, and there exists a general characteristic function  such that then From my own perspective, the geometric characteristic of the domain Ω and the degeneracy of the diffusion coefficient () can take place of the usual boundary value condition  (, ) = 0, (, ) ∈ Ω × (0, ) . ( The proofs of Theorems 3-5 are based on the general characteristic function, and we call this method as the general characteristic function method.Moreover, if we choose different general characteristic functions , we can obtain different results.For example, if we choose  =   (),  > 0, corresponding to Theorems 3-5, we have the following results.Theorem 6.Let  and V be two weak solutions of (4) with the initial values  0 () and V 0 (), respectively; suppose  > 1 and ( 11) is true, and then the stability ( 13) is true.
Theorem 7. Let  and V be two nonnegative solutions of (4) with the initial values  0 () and V 0 (), respectively.If  > 1, ( 14) is true, then the stability of the weak solutions is true in the sense of (13).
This is due to the fact that only we choose  ≥ (−3)/(− 2), and condition (15) is natural.Theorem 8. Let  and V be two solutions of (4) with the differential initial values  0 () and V 0 (), respectively.If  > 1, then there exists a constant  > 0 such that This is due to the fact that if  ≥ 2, condition ( 16) is natural.When  < 2, only we choose 0 <  ≤ 2/(2 − ), and condition ( 16) is also true.Once more, since  =   , condition ( 17) is natural.At the same time, if  ≥ 2, we can choose  ≥ 2 such that condition ( 18) is true.When  < 2, we can choose 2 ≤  ≤ 2/(2 − ) such that ( 18) is true.Then we have the conclusion of Theorem 8.
Corollary 9. Let  and V be two solutions of ( 4) with the differential initial values  0 () and V 0 (), respectively.Suppose  > 1 and ( 11) is true, and () satisfies then the stability ( 13) is true.

The Proof of Theorem 3
For small  > 0, let Obviously ℎ  () ∈ (R), and Proof of Theorem 3. Let  and V be two solutions of (4) with the different initial values  0 () and V 0 (), respectively.We choose   (( − V)) as the test function in Definition 1.Let  be a general characteristic function.Then Let us analyze every term in (30).
At the same time, by (11), Therefore, we have lim Now, let  → 0 in (30).Then By the Gronwall inequality, we have Theorem 3 is proved.

The Proof of Theorem 5
Proof of Theorem 5. Let  and V be two solutions of (4) with the initial values  0 () and V 0 (), respectively.From the definition of the weak solution, if () = , for any In particular, we choose where  [,] is the characteristic function on [, ] and  is a general characteristic function.Denoting   = Ω × [, ], then Clearly, In the first place, for the second term on the right-hand side of (66), we have In the second place, by ( 17)-( 18) The proof is complete.