The Dynamics of One Dimensional Singular Limiting Problem
In this chapter we study the dynamics of solutions to the singular parabolic system ( 1.12 ). We prove that every solution will converge to a stationary solution of ( 1.12 ) as time goes to infinity. We also give some interesting properties of these solut
- PDF / 237,647 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 3 Downloads / 165 Views
The Dynamics of One Dimensional Singular Limiting Problem
Abstract In this chapter we study the dynamics of solutions to the singular parabolic system (1.12). We prove that every solution will converge to a stationary solution of (1.12) as time goes to infinity. We also give some interesting properties of these solutions, see Sect. 4.2. The stationary problem is also studied in Sect. 4.1.
4.1 The Stationary Case In this section, we study the stationary case of differential inequalities (1.12): ⎧ 2 d ui ⎪ ⎪ − 2 ≤ ai ui − u2i in (0, 1); ⎪ ⎪ ⎪ dx ⎪ ⎪ ⎪ ⎪
⎨ d2 aj uj − u2j uj ≥ ai ui − u2i − − 2 ui − dx ⎪ j =i j =i ⎪ ⎪ ⎪ ⎪ ⎪ ui (0) = ui (1) = 0; ui ≥ 0, ⎪ ⎪ ⎪ ⎩ ui uj = 0 in [0, 1], for i = j (1 ≤ i, j ≤ M).
in (0, 1);
(4.1)
The main result of this section is Theorem 4.1.4. The Neumann boundary value problem can be treated similarly. We know that the above problem can have solutions with some identically zero components, thus it may have multiple solutions, but if the set where ui is positive has several components, we treat these as different species (because we are in the stationary case). Here, the number of species may become infinite. (Below we will exclude this possibility.) We have the following theorem. (Note that Lemma 4.1.2 below ensures that the set where ui is positive has only finitely many components.) Theorem 4.1.1 ∀M ∈ N, there exists at most one solution (u1 , u2 , . . . , uM ) of (4.1) (up to permutation) such that each ui is not identically zero and each of its supports is an interval. First, we need a well-known lemma. For later use and completeness, we give the proof here. K. Wang, Free Boundary Problems and Asymptotic Behavior of Singularly Perturbed Partial Differential Equations, Springer Theses, DOI 10.1007/978-3-642-33696-6_4, © Springer-Verlag Berlin Heidelberg 2013
45
46
4
The Dynamics of One Dimensional Singular Limiting Problem
Lemma 4.1.2 For each L >
π √ , 2 a
there exists a unique positive solution u of
⎧ 2 ⎪ ⎨− d u = au − u2 , dx 2 ⎪ ⎩ u(0) = u(L) = 0. There is no positive solution if L ≤
in (0, L)
(4.2)
π √ . 2 a
Proof The existence can be easily proven by, for example, the method of sub- and sup-solutions. In fact, we have the following conservation quantity:
du dx
2
2 + au2 − u3 ≡ c, 3
(4.3)
for some constant c. If we have two positive solutions u and v, then on the open set D := {u > v} (if not empty), we have ∂v ∂u 0≥ v− u= uv − vu ∂ν ∂D ∂ν D
− au − u2 v + av − v 2 u = D
uv(u − v)
= D
> 0,
which is a contradiction.
Corollary 4.1.3 The constant c defined in (4.3), seen as a function of L, is increasing in L. π Proof Assume L1 > L2 > 2√ with u1 and u2 the solution of (4.2) in [0, L1 ] and a [0, L2 ], respectively. Because
0 = u2 (L2 ) < u1 (L2 ), the same method as in the previous lemma gives that in [0, L2 ] u1 > u2 . This implies du1 du2 (0) > (0). dx dx
4.1 The Stationary Case
47
Note that here we have the strict inequality because otherwise we will have u1 ≡ u2 . Because 2 dui ci (Li ) = (0) , dx
our claim fol
Data Loading...
