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Volume 2009, Article ID 739097, 23 pagesdoi:10.1155/2009/739097 Research Article On Initial Boundary Value Problems with Equivalued Surface for Nonlinear Parabolic Equations Fengquan Li

Volume 2009, Article ID 739097, 23 pages

doi:10.1155/2009/739097

Research Article

On Initial Boundary Value Problems

with Equivalued Surface for Nonlinear

Parabolic Equations

Fengquan Li

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Fengquan Li,fqli@dlut.edu.cn

Received 6 January 2009; Revised 12 March 2009; Accepted 22 May 2009

Recommended by Sandro Salsa

We will use the concept of renormalized solution to initial boundary value problems withequivalued surface for nonlinear parabolic equations, discuss the existence and uniqueness ofrenormalized solution, and give the relation between renormalized solutions and weak solutions.Copyrightq 2009 Fengquan Li This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

There are many concrete physical sources for problem P, for example, in the

petroleum exploitation, u denotes the oil pressure, and At is the rate of total oil flux per unit length of the well at the time t; in the combustion theory, u denotes the temperature, for any fixed time t, the temperature distribution on the boundary is a constant to be determined, while, the total heat At through the boundary is given cf 1 7 For linear equations, theexistence, uniqueness of solution to the corresponding problem are well understoodcf 1

3, for the purpose, the Galerkin method was used For semilinear equations, the existence ofglobal smooth solution was obtained in7 in which a comparison principle was established

If a ij x, u is locally Lipschitz continuous with respect to the second variable, the existence

and uniqueness of bounded weak solution to problemP have been discussed in 8 under

the hypotheses of f ∈ L q Q and A ∈ L r 0, T with q > N/2 1, r > N 2 However, if

f ∈ L2Q and A ∈ L20, T, we cannot get a bounded weak solution In order to deal with

this situation, we will introduce the concept of renormalized solution to problem P anddiscuss the existence and uniqueness of renormalized solution

The paper is organized as follows In Section 2, we introduce the concept ofrenormalized solution and prove the existence of renormalized solution to problemP In

Section 3, uniqueness and a comparison principle of renormalized solution to problemP areestablished InSection 4, we discuss the relation between renormalized solutions and weaksolutions for problemP

In order to prove the existence of renormalized solution to problem P, we make thefollowing assumptions

Let a ij : Ω × R → R be Carath´eodory functions with 1 ≤ i, j ≤ N We assume that

a ij ·, 0 ∈ L∞Ω and for any given M > 0 there exist d M ∈ L∞Ω and a positive constant λ0

such that for every s, s1, s2∈ R, ξ  ξ1, , ξ N  ∈ R N , and a.e x ∈ Ω,

Under hypotheses2.1-2.2 and f ∈ L2Q, A ∈ L20, T, we cannot obtain an L∞

estimate on the determined function Ct; thus, we cannot prove the existence of bounded

weak solutions to problem P , hence a ij ·, uD j u may not belong to L2Q In order to

overcome this difficulty, we will use the concept of renormalized solution introduced byDiPerna and Lions in9 for Boltzmann equations see also 10–12

As usual, for k > 0, T kdenotes the truncation function defined by

0 L∞|u| and u ∈ L2Q The second term on the left

side of2.6 should be understood as

a ij x, T k uD j T k uD i hT k uξdxdt, 2.8

for k > 0 such that supp h ⊂ −k, k Since u ∈ L20, T; V , it is the same for huξ and

hut|Γξt|Γ The integral in2.7 should be understood as

a ij x, T m 1 uD j T m 1 uD i T m 1 udxdt. 2.9

Remark 2.3 Note that if u is a renormalized solution of problem  P , we get B h u 

u

0hrdr ∈ L20, T; V , B h u t ∈ L20, T; V L1Q; thus, B h u ∈ C0, T; L1Ω, hence

B h u·, 0  0 makes sense.

Remark 2.4 By approximation,2.6 holds for any h ∈ W 1,∞ R with compact support and all

ξ ∈ {ξ ∈ L20, T; V  | ξ t ∈ L2Q, ξ·, T  0}.

Now we can state the existence result for prolemP as follows.

Theorem 2.5 Under hypotheses 2.1-2.2 and f ∈ L2Q, A ∈ L20, T, problem  P  admits a

renormalized solution u ∈ L20, T; V  ∩ L∞0, T; L2Ω in the sense of Definition 2.1

In order to proveTheorem 2.5, we will consider the following problem:

Then problemP n admits a unique weak solution u n ∈ L20, T; V  ∩ C0, T; L2Ω

such that un ∈ L20, T; V and satisfies

In fact, here we can prove the existence of weak solution for problem P n via Galerkin

method Let us consider the operator

L2Ω By Riesz-Schauder’s theory, there is a completed orthogonal eigenvalues sequence

{w k } of the operator B Here we may take the special orthogonal system {w k}

above Galerkin equations such thatu m

n ∈ L20, T; V  Moreover, we can easily prove the

where C0is a positive constant independent of m.

The above estimates imply that there exists a subsequence of{u m

To deal with the time derivative of truncation function, we introduce a time

regularization of a function u ∈ L20, T; V  Let

u ν x, t 

t

−∞ν  u x, se ν s−t ds, ux, s  ux, sχ 0,T s, 2.17

where χ 0,T denotes the characteristic function of a set0, T and ν > 0 This convolution

function has been first used in14 see also 10, and it enjoys the following properties: u ν

belongs to C0, T; V , u ν x, 0  0, and u v converges strongly to u in L20, T; V  as ν tends

to the infinity Moreover, we have

Using the same method as10, we can obtain

u n −→ u a.e in Qup to some subsequence

Thus for any given k > 0,

T k u n   T k u weakly in L20, T; V , strongly in L2Q, a.e in Q. 2.25

By15, Lemma 2 and Lemma 3, we have

u n −→ u strongly in L q Q, ∀1 ≤ q < 2 4

u n −→ u strongly in L r Γ × 0, T, ∀2 ≤ r < 2 2

N . 2.27impling that

u n|Γ−→ u|Γ, a.e in 0, T. 2.28

For any given k > 0, it follows from 2.27-2.28 and Vitali’s theorem that

T k u n|Γ −→ T k u|Γ strongly in L20, T. 2.29Set

η ν u  T k u ν 2.30

Similar to 10, this function has the following properties: η ν u t  νT k u −

η ν u, η ν u0  0, |η ν u| ≤ k,

η ν u −→ T k u strongly in L20, T; V , as ν tends to the infinity. 2.31

For any fixed h and k with h > k > 0, let

w n  T 2k



u n − T h u n  T k u n  − η ν u. 2.32Then we have the following lemma

Lemma 2.6 Under the previous assumptions, we have

T

0

< u nt , w n > dt ≥ ω n, ν, h, 2.33

where lim h → ∞limν → ∞limn → ∞ ωn, ν, h  0.

Proof The proof ofLemma 2.6is the same as10, Lemma 2.1, and we omit the details

Lemma 2.7 Under the previous assumptions, for any given k > 0, we have

Now note that Dw n  0 if |u n | > h 4k; then if we set M  h 4k, splitting the integral

on the left side of2.35 on the sets {x, t ∈ Q : |u n x, t| > k} and {x, t ∈ Q : |u n x, t| ≤ k},

where limν → ∞limn → ∞ ωn, ν  0 Equations 2.38, 2.36, and 2.35 imply that

u − T h u T k u − η ν udxdt ω n, 2.41

where limn → ∞ ωn  0 2.31 and 2.41 imply that

Let n, ν, then and h tend to the infinity, respectively, we get

Using2.2, 2.25, and 2.46, we obtain 2.34

Proof of Theorem 2.5 For any given ξ ∈ W, h ∈ C1R, suppose that supp h ⊂ −k, k, taking

v  hu n tξt in 2.10 and integrating over 0, T, we have

Let n, m tend to the infinity in 2.58, respectively, then one can deduce that u satisfies 2.7.

Thus u is a renormalized solution to problem  P in the sense ofDefinition 2.1 This finishesthe proof ofTheorem 2.5

Remark 2.8 Using the same approach as before, we can deal with the nonzero initial value

u0/  0 In fact, we only replace η ν u  T k u ν by η ν u  T k u ν e −νt T k u0 in 2.30

In this section, we will present the uniqueness of renormalized solution to problemP Here

we will modify a method based on Kruzhkov’s technique of doubling variables in12 andprove uniqueness and a comparison principle of renormalized solution for problemP

Only simply modifying12, Lemma 3.1, we can obtain the following result.

Lemma 3.1 Let u be a renormalized solution to problem  P  for the data f, A Then

Lemma 3.2 For i  1, 2, let f i ∈ L2Q, A i ∈ L20, T, u i be a renormalized solution to problemP

for the data f i , A i  Then there exist K1∈ sign u1− u2 and K2 ∈ sign u1|Γ− u2|Γ such that

Note that for l sufficiently large,

x, s −→ ξ l x, t, s ∈ W, ∀t ∈ 0, T,

x, t −→ ξ l x, t, s ∈ W, ∀s ∈ 0, T. 3.5

Let h ∈ C1R, h ≥ 0, H ε ∈ W 1,∞ R be defined by H ε r  Hr/ε, where H ∈ W 1,∞ R,

Hr  0 for r ≤ 0, Hr  r for 0 < r < 1 and Hr  1 if r ≥ 1 As u1, u2 are renormalizedsolutions , according to2.6, for a.e t ∈ 0, T, we have

Note that ξ ∈ W, thus for l sufficiently large, φ l ∈ W Applying 3.1 with u  u1, ξ  φ l,

f  f1, At  A1s, and t  s, we have

It is easy to see that

Remark 3.3 In fact, by the density result, 3.3 is satisfied by any given ξ ∈ W1  {ξ ∈

W 1,∞ Q | ξT  0, ξt|Γ Ct an arbitary function of t} and ξ ≥ 0.

Now we state the uniqueness and comparison principle of renormalized solution toproblemP as follows

Theorem 3.4 Under hypotheses 2.1 and 2.2, for i  1, 2, let f i ∈ L2Q, A i ∈ L20, T, u i be a renormalized solution to problemP  for the data f i , A i  Then there exist K1∈ sign u1− u2 and

K2∈ sign u1|Γ− u2|Γ such that for a.e 0 < τ < T,

Proof For any given τ ∈ 0, T and any given ε > 0 sufficiently small, let α ε s be defined by

As for the second term in3.26, we have

By3.29 and 3.31, the second term in 3.26 tends to zero

Letting ε tend to zero in 3.27, 3.24 follows from 3.25 and 3.27

4 The Relation between Weak Solutions and

In this section, we will see that the concept of renormalized solution is an extension of theconcept of weak solution The main result in this section is the following theorem

Theorem 4.1 i Assume that u ∈ L20, T; V  ∩ L∞0, T; L2Ω and a ij ·, uD j u ∈ L2Q, i, j 

1, 2, N Then u is a weak solution to problem  P  if and only if u is a renormalized solution to

problemP .

ii If u ∈ L20, T; V  ∩ L∞Q, then u is a weak solution to problem  P  if and only if u is a

renormalized solution to problemP .

Proof i If u is a weak solution to problem  P, we have

u x, 0  0. 4.2

Noting that u ∈ L20, T; V , a ij x, uD j u ∈ L2Q, i, j  1, 2 N, we have u ∈ L20, T; V,

hence u ∈ C0, T; L2Ω, for any given h ∈ C1R and ξ ∈ W Taking v  ξhu in 4.1, wehave

For any given m, let S m r r

0T m 1 s−T m sds, it is easy to see that 0 ≤ S m r ≤ |r| Taking

v  T m 1 u − T m u in 4.1, we get

Conversely, assume that u is a renormalized solution Applying 2.6 with hu 

Hn 1 − |u|, where H ∈ C∞R, H ≥ 0, H  0 on −∞, 0, and H  1 on 1, ∞, as

Hence u is a weak solution to problem  P

ii Due to u ∈ L∞Q, assumption 3.1 and the definition of a renormalized solution

to problemP, ii is an immediate consequence of i

Remark 4.2 Theorems2.5and3.4improve those results of1,3,8

Acknowledgments

This work is supported by NCET of Chinano: 060275 and Foundation of Maths X of DLUT

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1 W X Shen, ? ?On mixed initial- boundary value problems of second order parabolic equations with

equivalued surface boundary. .. class="text_page_counter">Trang 17

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