# A free boundary problem arising in the ecological models with N -species

Let be given and let (2.5) hold for some . The aim of this section is to prove Theorem 2.1.

#### 3.1 An approximation problem

We will construct an approximation problem of (1.1a)-(1.3). We first construct some approximation functions.

For an arbitrary , choose such that in as . Then, for small enough *ε*,

(3.1)

Let be a smooth function with values between 0 and 1 such that for all , for and for , and let

Define

(3.2)

(3.3)

and

Then according to hypothesis (H)(ii) and (iii), it follows that the vector function

is mixed quasimonotone in with index vector , and

(3.4)

(3.5)

(3.6)

where . The definition of function implies that

(3.7)

(3.8)

In addition, it is obvious from (2.5) that

(3.9)

where , are the closure of and , respectively. Thus by (3.9) and the definition of functions , an argument similar to the one used in 16], Lemma 3.2] shows that

and

(3.10)

We next construct the approximation functions of . Let be a sufficiently smooth nonnegative function such that for and , and let be a sufficiently smooth nonnegative function taking values in such that for , for or , and for all . Define

Then hypothesis (H)(i) and 17], Chapter II] imply that

(3.11)

and

and (2.1) and (2.3) imply that

(3.12)

(3.13)

(3.14)

(3.15)

Employing the above approximation functions, we consider the following approximation

problem:

(3.16)

#### Lemma 3.1

*Problem* (3.16) *has a unique classical solution**in*, *and the following estimates hold*:

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

*where constants**and**C**are independent of**ε*.

#### Proof

In 15], by using the method of upper and lower solutions, together with the associated monotone

iterations and various estimates, we investigated the existence and uniqueness of

the global piecewise classical solutions of the quasilinear parabolic system with

discontinuous coefficients and continuous delays under various conditions including

mixed quasimonotone property of reaction functions. The same problem was also discussed

for the system with continuous coefficients without time-delay.

It is obvious that problem (3.16) is the special case of 15], problem (1.1)] without discontinuous coefficients and time delays. Hypothesis (H)(ii)

shows that , are the coupled weak upper and lower solutions of (3.16) in the sense of 15], Definition 2.2]. By (3.4)-(3.6) and (3.11)-(3.14), we conclude from 15], Theorem 4.1] that problem (3.16) has a unique classical solution in . Furthermore, using (3.1), (3.5), (3.6) and (3.12)-(3.14), the proof similar to that of 16], Lemma 3.3] shows that estimates (3.17) and (3.18) hold.

To prove (3.19), we first fix . In view of (3.7), (3.8), (3.10) and (3.15), we find that is the solution of the following problems for single equation:

(3.22)

and

(3.23)

By (3.22), (3.23), (3.12) and (3.14), the proof similar to that of 17], Chapter VI, Lemma 3.1] gives (3.19) for . The similar argument shows that (3.19) holds for .

We next prove (3.20). For any fixed , let

and let

By a direct computation we have

(3.24)

Then (3.16), (3.2), (3.3) and (3.10) imply that the function satisfies

(3.25)

A double integration by parts gives

(3.26)

Thus multiplying the equation in (3.25) by , integrating it on and using (3.5), (3.6), (3.24) and (3.26), we find that

Furthermore, by (3.12)-(3.14), (3.19) and Cauchy’s inequality, we deduce that for any ,

(3.27)

Consequently,

This, together with Gronwall’s inequality (see 17], Chapter II, Lemma 5.5]), implies that

Hence we deduce from (3.25), (3.27), (3.5) and (3.6) that

which, together with (3.24), yields (3.20) for .

For any fixed , we consider the equality

A similar argument gives (3.20) for . Therefore, (3.20) holds for all .

It remains to prove (3.21). For each , since is in , then (3.20) and 17], Chapter 2, formula (3.8)] show that

Thus (3.21) holds. □

#### Lemma 3.2

*Let*, *be the solutions in**for problem* (3.16) *corresponding to**and*, *respectively*. *Then*

(3.28)

#### Proof

Let , and let be fixed. We see from (3.16) that

(3.29)

In view of (3.15), we find . Multiplying the equation in (3.29) by and integrating by parts on , we deduce that, for any ,

(3.30)

Let us estimate , and . Since (3.2), (3.3) and (3.10) imply that

and

then it follows from Cauchy’s inequality that, for any ,

(3.31)

(3.32)

and

(3.33)

where

According to the definition of function , we see that if or . Thus by (3.21) we have

(3.34)

and

(3.35)

Summing equality (3.30) with respect to *l* from to , using (3.31)-(3.35) and Minkowski’s inequality, and choosing , we then conclude that

This, together with Gronwall’s inequality, yields

Combining the two inequalities above and (3.21) leads us to estimate (3.28). □

#### 3.2 The solutions of the diffraction problem

#### Proof of Theorem 2.1

We divide the proof into three steps.

Step 1. We prove the global existence of the solutions. Let us discuss the behavior

of the solution associated with by Theorem 2.1 as .

We first see from (3.16) that for any and any vector function ,

(3.36)

Furthermore, according to estimates (3.17), (3.20), (3.28) and the Arzela-Ascoli theorem, we conclude that there exists a subsequence (we retain

the same notation for it) such that

Thus **u** is in , **u** satisfies the parabolic condition (1.2), and estimates (2.6) and (2.7) hold.

We next show that for each , the sequences , , converge in to , and , respectively. Since

and

then it follows from (2.2), (3.3) and Lebesgue dominated convergence theorem that

and

and from (3.10) that

The similar argument shows that for each .

Based on the above arguments for sequences , , and , by letting , we conclude from (3.36) that (2.4) holds.

We also see from (3.20) that there exists a subsequence (denoted by still) such that for each , converge weakly in to . Recalling that in , we deduce . These, together with (3.20), imply that for each , for each , for each , and (2.8) holds. Thus (2.4) implies that **u** satisfies the equations in (1.1a) and (1.1b) for almost all and the inner boundary condition (1.3) for almost all (see 17], Chapter 3, Section 13]). As we have done in the derivation of (3.21), estimate (2.8) yields (2.9).

For fixed and , satisfies the linear equation

where

and

Then for any subdomains and satisfying and , we have . The parabolic regularity theory shows that . Hence **u** satisfies pointwise the equations in (1.1a) and (1.1b) for . Consequently, **u** is a solution in of problem (1.1a)-(1.3) and estimates (2.6)-(2.9) hold.

Step 2. In what follows, we will show that the solution in for problem (1.1a)-(1.3) is unique and estimates (2.10) and (2.11) hold.

Let , be the solutions in corresponding to and , respectively. Set . Then . We choose in (2.4) to find

and

For each , by a subtraction of the above equations for , we conclude that

and

Then

Setting and summing the above inequalities with respect to *l* from to , we have

Again by Gronwall’s inequality we deduce (2.10), which, together with 17], Chapter 2, formula (3.8)], gives (2.11). Therefore the solution in for problem (1.1a)-(1.3) associated with is unique.

Step 3. For , we discuss the regularity of **u**.

Then *v* satisfies

We will use the result of 18] to obtain the regularity of *v*. To do this, we need the estimate of for any fixed . Let be a smooth function with values between 0 and 1 such that for or , for and , and let

For any small enough Δ*t*, consider the equality . By employing the formula of integration by parts and 17], Chapter II, formula (4.7)], we get

where . Some tedious computation and Cauchy’s inequality yield

We choose and employ (2.8) to find

As we have done in the derivation of (3.20), by Gronwall’s inequality we get

Consequently,

By 17], Chapter II, Lemma 4.11], this inequality implies that

(3.37)

Hence . Using (3.37), hypothesis (H)(iii) and 18], Theorem 1.1], we deduce that is continuous with respect to *y* in and in for almost all , and is continuous in . Since and , then for almost all , and are continuous with respect to *x* in and in . □

The following corollary follows directly from Theorem 2.1.

#### Corollary 3.3

*Assume that*, , *satisfy* (2.5) *and the sequence**converges in**to*. *If*, **u***are the solutions in**of* (1.1a)-(1.3) *corresponding to**and*** φ** ,

*respectively*,

*then there exists a subsequence*(

*we retain the same notation for it*)

*such that*

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