Mô hình toán học và thuật toán giải số một lớp các bài toán biên trong thủy động lực học, trong truyền tải, khuếch tán và ô nhiễm môi trường

BÁO CÁO TÓM TẮTTên đề tài: Mồ hình toán học và thuật toán giải số một lớp các bài tóan biên trong thủy động lực học, trong truyền tải, khuếch tán và ô nhiễm môi trường.. M ục tiêu và nộ

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC T ự NHIÊN

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ĐỂ TÀI

M Ô H ÌN H TOÁN H Ọ C VÀ T H U Ậ T TOÁN G IẢI s ố M Ộ T L Ớ P CÁC

BÀI TOÁN BIÊN TRONG THỦY ĐỘNG L ự c HỌC, TRONG TRUYEN

TẢI, K H U ẾC H TÁN VÀ ổ N H IE M m ô i t r ư ờ n g

MẢ SỐ: QG 04-29

CHỦ T R Ì : PGS.TS T rầ n Huy Hổ

C Á C C Á N B Ộ T H A M G IA

1 PGS.TS Nguyẽn Thúy Thanh, ĐHKHTN, ĐHQG Hà Nội

2 PGS.TS Trần Gia Lịch, Viện Toán học

3 TS Phan Ngọc Vinh, Viện Cơ học

4 ThS Lê Huy Chuẩn, ĐH Osaka, Nhật bủn

5 PGS.TS Nguyễn Xuân Thao, Trường Đại học Thủy lợi Hà Nội

6 TS Lê Văn Thành, Viện cơ học

7 TS Pham Thanh Nam, Viện cơ học

HÀ NỘI - 2006

Ị rpỤNG THC-NG tin ĩ h u viện

BÁO CÁO TÓM TẮT

Tên đề tài: Mồ hình toán học và thuật toán giải số một lớp các bài tóan biên

trong thủy động lực học, trong truyền tải, khuếch tán và ô nhiễm môi trường

Mã sô: QG 04-29.

Chủ trì: PGS.TS Trần Huy Hổ.

Các cán bộ tham gia:

1 PGS.TS Nguyen Thủy Thanh, ĐHKHTN, ĐHQG Hà Nội

2 PGS.TS Trần Gia Lịch, Viện Toán học

3 TS Phan Ngọc Vinh, Viện Cơ học

4 ThS Lê Huy Chuẩn, ĐH Osaka, Nhật bản

5 PGS.TS Nguyễn Xuân Tháo, Trường Đại học Thủy lợi Hà Nội

6 TS Lê Văn Thành, Viện cơ học

7 TS Pham Thanh Nam, Viện Cơ học

M ục tiêu và nội dung:

Đồ tài nghicn cứu mô hình loán học trong một số vấn dồ liên quan đốn bao vệ môi lrường: Vấn đồ truyền tải vật chất Irong nước, trong không khí và sự xổi lở

bờ bicn gia lăng ở những nơi rừng ngập mặn đang dần bị phá hủy Đáy là những vân dề hôi sức quan trọng được dặt ra, khi mà ca nước la dang Irong lliời kỳ công nghiệp hóa và hiện đại hóa các nqành kinh tế Thực lc' phát Iricn kinh tố' ở Việt nam và các nước phút triển khác trong những năm qua cho thây, nhữim lợi ích mà phát triển kinh tế đcm lại không phải lúc nào cũng vượt trội so với những

gì những người dân sinh sống trong một vùng cụ thê phái gánh chịu VC mồi trường sống Ô nhiẽm môi trường được đặt ra như bài loán tất yếu cho sự phát triển kinh tế của mọi quốc gia Đề tài tập trung nghiên cứu vé mặt lý Ihuyéì một vài mô hình toán học, nhằm mô phỏng quá trình truyền lái vật chất trong không khí, trong môi trường nước Từ những nghicn cứu bài toán truyền tải nói trcn, chúng tôi nhận thày có sự liên quan chặt chẽ giữa nhữne hiện tượng xói lở bờ bicn với những khu rừng ngập mặn Đó là: ơ những nơi nào rừng ngập mặn phái triển, ở dó ít có hiện tượng xói lở bờ biển Bởi vậy, đề tài đề xuất mội mô hình toán học nghiên cứu sự phát triển của rừng ngập mặn thôim qua sự lan lỏa của sóng hiên, của đất, của nước và của hạt cây tro nu khu vực có rừng

Những kết quả chính:

1 Giải số bài tóan dòng chảy hai chiều với số liệu giả định, từ đó xác định mức

độ ô nhiễm môi trường khi có chất thải từ một nhà máy, xí nghiêp đang vận

2 Xây dựng mô hình toán học về sự lan tỏa cây giống trong mội khu rừng ngập mặn và giải số (với số liệu giả định) việc phát triển của rừng ngập mặn

Tình hình kinh phí

Tổng kinh phí: 60.000.000VNĐ, được chia làm hai năm: Năm 2004 được cấp

30.000.000VNĐ, năm 2005 được cấp 30.000.000VNĐ

hành

TRƯ Ờ N G ĐẠI H Ọ C K H O A H Ọ C T ự N H IÊN

Title o f the Project: Mathematical model and numerical solving algorithsmfor the boundary-value problems in hydro-dynamics, transporttation, diffusionand environmental polution

N am e of leader: Prof.Dr Tran Huy Ho

Index o f Code: QG 04-29.

M embers of Project:

1 Prof.Dr Nguyen Thuy Thanh, HUS, VNU, Vietnam

2 Prof.Dr Tran Gia Lich, Institute of Mathematics, Vietnam

3 Dr Phan Ngoe vinh, Institute of Mechanics, Vietnam

4 Ms Le Huy Chuan, Osaka University, Japan

5 Prof.Dr Nguyen Xuan Thao, Hanoi University of Water Resourscs,

Vietnam

6 Dr Le Van Thanh, Institute of Mathematics, Vietnam

7 Dr Pham Thanh Nam, Institute of Mathematics, Vietnam

The Arms and Results

1 Calculation of the hori/.oial iwo-dimcnsional unsteady flows by the mclhod

OÍ characteristics In lliis problem, wc sludy the cluuaclcrislics form of the Iwo-dimcnsional Sainl-vcnant equation syatcm, llic supplcmalcry equations

at ihc houndaics, Ihc method of characteristics for solving the equal ion system and some numerical experiments

crilcria We present Ihc algorithms for solving Ihc two-dimensional mailer propagation and its adjoint problems, the stability of the difference schemcs and the non-ncgalivc property of numerical solutions, ddetermination of Ihc plant locations so that some emvironmental criteria satisfied, and Ihc numerical cxperimenlsfor the test cases and for Halong Bav area

3 The model simulate and prcdictcs Icndcncics of accrction, erosion, changes

of the bottom topography in the coastal zone from Hoa Duan, Thuan An to Hai Duong, Thua Thien Hue Province

unsteady Jlows on the river and reservoirs, the discontinuous wave and unsteady flows after dam breaking, the numerical experiments for some test eases of natural Da river

cosidcr the dynamic system for forest kincmatic model

The results: One arcticlc published before this Project starting, three (3) articles

to be published in 2006, 2007 and for preprints

gì những người dân sinh sống Irong một vùng cụ thể phải gánh chịu vé mồi trường sống 0 nhiễm môi trường được đặt ra như là một bài toán tất yếu cho sự phát triển kinh tế của mọi quốc gia Đề tài tập trung nghiên cứu về mặt lý thuyết một vài mô hình toán học, nhằm mô phỏng quá trình truyền tải vật chất trong không khí, trong môi trường nước Từ những nghiên cứu bài loán truyền tải nói trên, chúng tôi nhận thấy có sự liên quan chặt chõ giữa những hiện tượng xói lở

bờ biển với những khu rừng ngập mặn Đó là: Ớ những nơi nào rừng ngập mặn phát triển, ở đó ít có hiện tượng xói lở bờ biển Bởi vậy, đề tài đề xuất mội mô hình loán học nghiên cứu sự phát triển của rừng ngập mặn thông qua sự lan tỏa của sóng biển, của đất, của nước và của hạt cây trong khu vực có rừng

NỘI DUNG

Đc tài đã hoàn thành nhưng mục liêu đề ra và đạt được nhũng kốl quá sau đây:

1 Tính toán số cho những dòng chảy hai chiều nằm nsang bằng phương pháp đặc trưng Trong bài toán này, chúng tôi nghiên cứu một dạng đặc trưng của hộ phương trình Saint-Venant, phương trình b ổ sung trốn bicn

và phương pháp đặc trưng đổ giai sô nhữim pluroìm trình ctậl ra

2 Xác định vị trí xáy dựng nhà máy, xí nghiệp để đảm bảo các điểu kiện về tiêu chuẩn môi trường Trong vấn đề này, chúng tôi giải bài toán truyền tải, khuyếch tán vật chất trong môi trườnc khí, xác định mức độ ô nhiễm không khí khi có nguồn như nhà máy, xí nghiệp Từ vấn đề đã nghiên cứu

ở trcn, có thể xác định được vị trí xây dựns xí nghiệp nhà máy trcn cơ sỏ' những điểu kiện về phát triển kinh tế từng vùng và đảm báo tiêu chuẩn môi trường cho phcp

3 Mô hình mô phỏng và dự báo xu thê' xói lở bờ biên và nhữnc biến đổi địa hình tại tầng đáy bờ biển khu vực tỉnh Thừa Thiên Huế Đề tài đưa ra các

mô hình mô phỏng và dự báo xu thế bồi tụ, xói lở bờ biển và biến đổi địa hình đáy: Lan truycn sóng, dòng cháy vcn bờ sông, dòng vận chuvển bùn cát, biến đổi địa hình đày và bồi xói ngang bờ Thuật loán giái các bài

to á n tr ê n b ằ n g phương p h á p sai p h â n theo sơ đồ ngược và biến đổi về phương tr ìn h ba đường chéo, và giải b ằ n g phương p h áp tr u y đuổi theo các trụ c tọa độ Phương ph áp này luôn cho nghiêm g ần đ ú n g và sai sô tích lũy nhỏ.

4 Đề tà i đề cập đến những dòng chảy không ổn định tr ê n sông, sóng không liên tục tro n g lũ lụ t do vỡ đập và n h ữ n g tín h tó an tr ê n sô" liệu giả đ ịnh cho đập sông Đà

5 Đề tà i ng h iên cứu rừ n g ngập m ặn Một mô h ìn h p h á t tr iể n của rừ n g được xây dựng T rên cơ sở mô h ìn h đưa ra, đề tà i xác đ ịn h được dáng điệu tiệm cận nghiệm của hệ động lực rừng ngập mặn Xác định được

m ột sô" điều kiện để rừ n g p h á t triể n hoặc bị diệt vong

6 Về m ặ t đào tạo: Đề tà i nằm trong khuôn khổ của Chương tr ìn h hdp tác giũa các trưòng đại học trọng điểm là ĐHQG H à Nội và Đ ại học Tổng hợp Osaka, N h ậ t bản Có một lu ậ n á n tiến sỹ to án học đã bảo vệ, vào

th á n g 12 n ă m 2006 tạ i N h ậ t bản

7 H ọat động chung: Đề tà i tổ chức và t à i trợ ba hội th ảo khoa học, tro n g

đó có m ột hội th ảo quôc tế:

+ T h á n g 10-2004: Hội th ảo quôc tê về ứng dụng tó a n tro n g các v ấn đê

về môi trường

+ T h á n g 5-2005: Hội th ảo tạ i Tam Đảo, Vĩnh Phúc

+ T h á n g 4-2006: Hội th ảo tạ i Ba Vì, Hà Tây

K Ế T L U Ậ N •

Dề tà i đề cập đến một số v ấn để liên q u a n đến môi trường, Đây là n h ữ n g vấn

tề thò sự tro n g điều kiện đ ấ t nước t a trong thờ kỳ gia tă n g tốc độ công nghiệp lóa và h iệ n đại hóa N hững v ấ n đề đưa ra và đã giải quyết tro n g p h ạ m vi của

tề tà i có ý n g h ĩa n h ấ t định về m ặ t lý th u y ế t và thực tiễn, v ề m ặ t lý thuyết, :ây dựng cơ sở to á n học và chứng m inh c h ặ t chẽ nhữ n g k ế t q uả n êu ra Một sô' tịnh lý to á n học đã được chứng minh, đặc biệt tro n g bài to á n về rừ n g ngập nặn Về m ặ t thực tế, do h ạ n c h ế về p h ạ m vi nghiên cứu, k in h phí cũng như

;hả n ă n g n h â n lực, đề tà i chỉ tín h to á n tr ê n cơ sở các sô' liệu giả định Vì vậy, ihững bài to á n về tr u y ề n tả i v ậ t ch ất ô nhiễm trong môi trường còn n h iều vấn

.ề bỏ ngỏ, chưa được giải quyết Tuy nhiên, những kết quả b a n đ ầ u về m ặ t

□án học, đã mở ra n h ữ n g hướng nghiên cứu tiếp theo, n h ằ m đáp ứng n h u cầu

em k ết q uả ng h iên cứu khoa học cơ b ả n vào thực tiễn

TÀI LIỆU THAM KHẢO

1 Đ ề cư ơ n g n g h iê n c ứ u đ ề t à i N C K H đ ặ c b iệ t c ấ p Đ ạ i học Q uốc G ia H à Nội n ă m 2004

2 A t s u s h i Y agi, M a t h e m a t i c a l A n a ly s is for N o n lin e a r D ifu s io n S y s te m ,

O s a k a U n iv e r s ity , J a p a n , 2005 (to b e p u b lis h e d )

CÁC ẤN PHẨM

A t s u s h i Y a g i

D e p a r t m e n t o f A p p lie d P h y sic s, O s a k a U niversity, S u ita , O s a k a 565-0871, J a p a n

(y a g i@ a p e n g o s a k a - u a c jp )

A b s t r a c t T l I is p a p e r s tu d ie s an a^<!-sl,ni<:Uiml, c o n tin u o u s space, forest, k i 1 1 (L 1 1 1 a I i (

m odel w h ic h was p re s e n te d by K u z n e ts o v c t ill [3] Not, only I.Ill’ global exist,(’]»•(• of

s o lu tio n is s h o w n in t h e tw o -d im e n sio n a l case b u t also a d y n a m ic a l syst em is const n u t ('(I

F u r t h e r m o r e , it is p ro v e d that, t h e d y n a m ic a l s y s te m possesses a b o u n d e d a b s o r b in g set

a n d t h a t e v e ry t r a j e c t o r y lias a n o n e m p t y w-liinit s e t in a s u ita b le w eak topology.

C o m m u n i c a t e d hy Y Y a in a d a ; R e c e iv e d J u n e 27, 2005; R e v is e d M a r c h 17 2006.

T h i s w o rk is s u p p o r t e d l>y G r a n t - i n - A i d for Scien tific R e s e a rc h (No 1G34004G) Ij} J a i ) a n S o c ie t y for t Ilf'

P r o m o t i o n o f S c ic n c r a n d is p a r t l y s u p p o r t e d by t h e C o ro U n iv e rs ity P r o g r a m b o t w w n J a p a n S o c ie ty for t h e P r o m o t i o n of S c ie n c e a n d V i e t n a m e s e A c a d e m y of S cience a n d T crlm o lo ^ y

A M S S u b j e c t C la s s if ic a tio n : 3ÕK57, 37LÕ0, 92D4U.

1 Introduction

In the study of forest kinetics, mathematical methods on the basis of kinematic models are becoming one of indispensable approaches Observations of forest require us extrem e y long time and experiments cost immensely There are already many kinematic models which have been presented in different scopes, such as individual-based model, individual- based continuous space model, age-structured model and age-structured continuous space

model (see Ị16] for a survey of these models).

We arc conccrncd, in this paper, with the age-structurcd continuous spacc model which was presented by Kuznetsov, Antonovsky, Biktashev and Aponina [3] The age-structured continuous space model intends to describe the forestry ccosystcm by regeneration proccss, furthermore, this process is divided into four stages, namely, seed production, seed tran s­port, seed deposition and seed establishment, and by age-dependent tree relationships

In [3], they consider a prototype ecosystem, that is, the forest consists of a mono-species and the generation is divided into only two age classcs, the young age class and the old age class

Their model is written by the initial-boundary value problem for a parabolic-ordinary system

in a two-dimensional bounded domain ÍX The unknown functions u(x , t) and v(x , t)

denote the tree densities of young and old age Glasses, respectively, a t a position X e Í2

and at time I € [0, 00) The third unknown function w ( x , l ) denotes the density of seeds

in the air at X € and t G [0, oc) The third equation dcscribcs the kinctics of seeds;

d > 0 is a diffusion constant of seeds, and Q > 0 and p > 0 arc seed production and seed

deposition rates rcspcctivcly While the first and sccond equations describe the growth

of young and old trees respectively; 0 < Ổ ^ 1 is a seed establishment rate, 7 (1;) > 0 is

a mortality of young trees which is allowed to depend on the old-trce density V, / > 0

is an aging rate, and h > 0 is a mortality of old trees On w , the Neumann boundary conditions are imposed on the boundary ỠÍ1 Nonnegative initial functions Uo(x) ^ 0 , fo(x) ^ 0 and u/o(£) ^ 0 are given in Q.

Several authors have already been interested in such a model but only in the one- dimensional ease Wu [8] studied the stability of travelling wave solutions Wu and Lin

|9] discusscd the stability of stationary solutions Lin and Liu [4j extended this result to

a ease when the model includes nonlocal effects

The main purpose of this paper is to construct, in the two-dimensional case, a global so­

lution to ( 1 1) for each triplet (uo, Vo, w0) of initial functions and to construct a dynamical

system determined from the problem Furthermore, we show that the dynamical system possesses a bounded absorbing set and consequently every trajectory has a nonempty o/-limit set in a suitable weak topology.

We regard and handle the system (1.1) as a degenerate nonlinear diffusion system with

respect to [ u , v , w ) The word “degenerate” here means that the diffusion constants for u and V both vanish B ut the general methods for constructing local and global solutions

are available if we take an underlying space carefully In fact, we shall verify that the

ab stract result obtained in [7, Theorem 3.1] is still applicable for the present problem if

X is taken as

Nonnegativity of local solutions and a priori estimates for local solutions will be es­tablished in ordinary manners

We have to pay much attention, however, th a t, owing to the degeneracy of dissipation,

we have no longer smoothing effect of solutions W h a t is even worse, we observe a t least

numerically (see [6]) th a t, even if the initial functions {uo,Vo,Wo) are very sm ooth, the solution {u(t), v(t), V)(t)) can tend to a discontinuous stationary solution (ũ, Ữ, w) as t —*

00, Ũ and V being discontinuous and w being continuous in Í2 This suggests furtherm ore

th a t some trajcctorics of the dynamical system no longer possess any nonem pty cj-iimit

sets in the usual sense (sec [10], 113] and f 14]) in the underlying space X given by (1.2)

In fact, if a sm ooth trajecto ry (it(£),t;(£),tu(<)), 0 < t < 00, has a cluster point ( u , v , w )

in X , then it is impossible th a t Ũ and V are discontinuous in Í1 T h e dynamical system is

neither ox peeled to possess the global a ttra c to r in general Ry this reason wc will cont,onl,

ourselves with constructing nonempty cư-limii sets in a suitable weak topology of X only

As will be shown in the last section, the system ( 1.1) enjoys some Lyapunov function If

we use this function, we can deduce stronger results for the weak cư-limit set, see [1]

As a m a tte r of fact, wc can rigorously know cxistcncc of discontinuous stationary solutions to th e present system (1.1) (see [2]) The interface of a discontinuous stationary solution is then considered as an internal forest boundary or an ccotone of forest which has a significant meaning from the viewpoint of ccology ( |3]) In this sense also it is quite

n a tu ra l to choose an underlying space in the form (1.2)

T hro u g h o u t th e paper, ÍÌ is a bounded, convex or c 2 domain in ]R2 According to

= 0 o n d ĩ ì enjoys th e o p tim al shift property th a t V 6 L 2 ( f i ) alw ays im plies th a t

w € / / 2(i2) We assume as in [3, p 220] th a t the mortality of young trees is given by a

square function of the form

where a, 6, c > 0 are positive constants This means th a t the m ortality takes its minimum when th e old-age tree density is a specific value 6 As mentioned, d, f , h , a , p > 0 are all positive constants and 0 < Ố ^ 1.

We will organize th e p ap er as follows Section 2 is a preliminary section In Section

3, local solutions to (1.1) are constructed Nonnegativity of local solutions and a priori

estimates are shown in Section 4 and in Section 5, respectively The d y n a m i c a l system

determined from (1.1) is constructed in Section 6 In Section 7, we mention the Lyapunov

function for our dynamical system

2 Prelim inary

We shall first rccall some known results for semilincar a b stract evolution equations studied

in Ị7Ị Consider the initial value problem for a semilinear equation

in a Banach space X Here, A is a closed linear op erato r in A", the spectral set of which

is contained in a sectorial domain £ = {\ (E C; I arg A| < u>} with some angle 0 < u <

and the resolvent satisfies the estimate

with some constant M > 1 This assumption implies th a t —A generates an analytic semigroup e~lA ori X The initial value ƯQ is taken from T>(A^) with an estim ate

here n is some exponent such Uml 0 < II < 1 and /{ > 0 is a constant T h e nonlinear operator F(U) is a mapping from X>(J'lr?) to X , where 1 ] is a sccond exponent such th at

n < TỊ < 1, and is assumed to satisfy a Lipschitz condition of the form

where v?( ) is some continuous increasing function Then, the following theorem is known:

T h e o r e m 2.1 ([7, Theorem 3.1]) Let 0 < ụ < 7? < 1 and let (2.2), (2.3) and (2.4) be

satisfied Then (2.1) possesses a unique local solution in the function space

(2.1)

(2.2)

{2A) \\F(U) - F(V)\\ < <p(\\A»U\\ + n i l ' l l )

I u € e([ 0,t r\-V(A»)) n e 1 (( 0,Tn]'t X ) n e ( ( 0 ,Tr\i D (A ))t

( ’-'■|MƠ(Í)II + \\A“U(l)\\ <c„, 0 < ( < 7/i

Wc next list some well-known results in the theories of function spaccs and of scctorial

[15]) For 0 < So < s < Si < 2, coincides w ith th e complex interpolation space

Ị//'° ( íĩ) , where s = (1 — 0)so + 0s\, and th e estim ate

on the space H 1(ũ) From this form we can define realization A of the Laplace operator

—d A I- p in L 2(£l) under the Neumann boundary conditions on the boundary d ữ (see

[11, Chap VIỊ) T he realization A > p is a positive definite self-adjoint, operator of L 2(Q)

and its domain is characterized by

holds w ith some constants 0 < C\ < Ơ 2

-Finally, consider th e initial value problem for an ordinary differential equation

u ( 0 ) = u 0

in the Banach space L°°(Q) Here, p, q e e([0,71; L°°(Q)) are g i v e n L°°-valued continuous

(2.12) For each 0 < t < T, we set the function

its derivative is given by the m ultiplication operator: g t—> e * g on it follow s th at (0 € (^ ((o , t}\ l ° ° ( i i ) ) and

(ef‘1p(r)dru (s))' = ef’ p(r)dr( - p ( s ) ) u ( s ) + e ^ p{r)drú ( s ) = ef‘tp(r)drg(s).

Integrating this equality in [0, ij, we obtain the formula

Then, the problem (1.1) is rewritten in the form (2.1).

/ ( t i l - tta)

a { v 1 - v 2)

th a t the estim ate

(3.4) ||F ( i / ) - F(V0H < CiWxm - w 2\\Loo + I h - V2\\L2

+ (llttlllỉ- + llvl|li~ + IIw 2 |Il» + 11 ^ 2 Ili« + l)(llul - W 2 ||l“ + \\vi — V 2 |U“ )}

< C ( | | / W ||2 + WA^VW2 + \)\\A»{U - V)\\

holds with some constant c > 0 This shows th a t (2.4) is fulfilled By virtue of Theorem

2 1, we th en conclude the following result:

T h e o r e m 3 1 For any initial value (uo,i>o,iuo) € K ) the problem (1.1) possesses a unique

4 N o n n e g a tiv ity o f so lu tion s

We next verify th a t nonnegativity of initial functions implies th a t of the local solutions obtained in Theorem 3.1

T h e o r e m 4 1 Let ( uoi V q ^ wq ) € K and let ( u , v , w ) be the local solution of (1.1) obtained

in Theorem 3.1 Then u(t) > 0, v ( t ) > 0 and w(t) > 0 in Í2 f o r every 0 < t < To (< T\), where To > 0 is determined by the norm ||zío||l« + II^oÌIl°° + ll^olli/^ alone.

Proof By T heorem 3.1, (1.1) possesses a unique local solution { u , v , w ) defined on [0, T\]

in function space (3.5)

= 0 dn

By repeating the same arguments as in Section 3, we can deduce th a t (4.1) possesses a

unique local solution {u,v, w) defined on [0 ,T 2] in the function space (3.5), where T 2 is

determined by the norm ||u0||l« + IK IIl« + ||«to||j/2** alone Our goal is then to show

the nonnegativity of w,V and w In this case, x(^) = V therefore ( u , v , w ) is also a

local solution of (1.1) on [0,T2] Then, by the uniqueness of solutions, we conclude th a t

unique nonticgalivc local solution in Lho fiuiclion spacc (3.5), where 7’o is determined by the norm ||iio|U~ I ||volU» I Ikoll/pM

local solution of (4.1), then its complex conjugate is also a local solution; so, they must coincide; hence, (u ,v ,w ) is real valued

Let us now verify the nonnegativity For this purpose, wc use another cutoff function Let / / ( r ) be the c 1,1 function defined by

V'KO — —d I H " ( w ) \ v w \ 2d x — Ị3 Ị H ' { w ) w d x + a [ H ' ( w ) x ( v ) d x < 0.

Since ^i(O) — 0, it follows that ĩị}\(I) 0 for every t e [0,7!;], th a t is, w(l.) > 0 in |0,T.]

By the same argument, putting

Since w > 0 and ^2(0) = 0, it follows that u(t) > 0 in [0, 7-2] It is the same for V. Hence,

5 G lobal so lu tion s

In this scction, we prove th e existence of global solution by establishing a priori estim ates

of local solutions In addition, we verify Lipschitz continuity of solution in initial data

P r o p o s i t i o n 5 1 Let (ito, Vo, Wo) É K and let (\ 1 ,V,U)) be any local solution of (1.1) on

an interval [0, T u v w ) such that

fo < u}v € euo,^.,,); £“(«)) n

\ 0 < w e e ( [ 0 , H 2“( Q) ) n e'((0 ,T ,i, i, ) ;L 2( n ) ) n € ( ( 0 , 7 ^ H ị m

Then, the estimate

(5.1) ||ti(t)iu- + l|u(i)IU- + IMOII//2"

< C [ e p<(||wo||x,oo + ỊI^oIIl00 + l l ^ o l l w 2^) + lj> 0 < Í < T UtViVJ

Proof T h ro u g h o u t the proof, wc shall use a universal notation c (rcsp p) to denote

positive constants (resp positive exponents) which axe determ ined by the constants

a, 6, c, d, / , hj a , Ị3 and Ố a n d by Q So, it may change from occurrence to occurrence If

we want to distinguish th e constants (resp exponents), we shall write Cl, Ơ2, (resp

Pi, P2, •■•)•

Step 1 Estimate f o r | | u | | l 2, IM Il2> I M L 2- Multiply the first equation of ( 1.1) by u

and integrate th e p ro d u ct in Q Then we have

(5.2) [ u 2d x + / [ u 2d x ~ Ị ỉ ỗ I w u d x — [ 7 (v )u 2dx

f w 2dx + 0 f w 2dx - d [ |Vu;|2íù: f a / v w d x < ^ f w2lix + C -2 I v 2dx.

Let Ơ3 > 0 be constant such that C 1C3 < 4 Multiply (5.2) by Ơ3 and add the product

to the above inequality Then,

Let C4 > 0 be constant such that CAh > 2 C2 Multiply the above equation by CA and

add the product to the inequality (5.3) to obtain that

^ J (C 3 Ú 2 + C4Ư2 + w2)dx + p J (C 3 U 2 + Ơ4V2 + u;2)cte < c

Solving this, we conclude that

c3\\u(t)\\l, + C4||V(0ị|ỉ3 + Monk < c-^(C3||iio||Ỉ 2 + G\\\v0\\l> t KIIỈ 2 ) + a

Hcncc,

(5-1) IMOII/.i I NOII,.* I IMOII/.i

< I I I K I I , , ; , ) I ] I, 0 < t < 7'w

< C'[e- P li(||uo||L«5 + 11^0IU°° + IIWoIIf/3^) + l]j

where 0 < Pi < m in{ P rp} Thus, denoting p] by p, we have obtained th a t

< C[e n t (\\uo\\L- + I M U « + ||mo||//2p) 4- l],

where 0 < P 2 < m in{/,/£>} Denoting p 2 by /?, we conclude th a t

(5.6) ||u(í)||l°° C [e_pi(||uolU«> + llt'oliz,00 + ||wo||.//2m) + 1], 0 < t < TUiV<w.

Similarly, by the second equation of (1.1) an d (5.6), we also verify th a t

( 5 7 ) M O I U - — C ị e - ^ i ị ị u o ị ị L o o + I K I I l o - + ||w>o||tf2„) + 1]5 0 < t < T U'V<W,

As all immediate conscqucnce of a priori estimates, wc prove tiu' following Llicon'jn:

T h eorem 5.2 For any initial value (uo, Vo, ĩUo) € K , the problem (1.1) possesses a unique

global solution such that

ị 0 < u, V <E e([0, 00); L°°(n)) n e 2((0, oo);

j o < w e C([0,oo); H ^ ( Q ) ) n e 1((0,oo); L 2(Q)) n e((0 , oo); H ị ( Q ) )

Proof By Theorem 4.1, there exists a unique local solution (u , V, w) on an interval [0, T0]

Moreover, by Proposition 5.1, ||u(T0)||l«> + ll^(To)IIl°° + l|w(7o)||//2i* is estim ated by

||t*o|U°° + ||vo||l~ + IIWoII//2/i alone This then shows th a t the solution (uyv , w ) can be extended as a local solution on an interval fo, To + r], where T > 0 is determined by

llu T O I L - -f- llvfToJIli00 + ||tx7(7o)I Ỉ , and hence depends on ly on II uo II l 00 +

Wc next verify the Lipschitz continuity of solution in initial data For any R > 0, let

K u be the bounded set of initial values:

Of course, by Theorem 5.2, there exists a unique global solution to (1.1) for each ƯQ € K r

P r o p o s i t i o n 5.3 Let ư (resp V ) be the solution to ( 1.1) with initial value ƯQ G Kfi

depending on R and T alone such that

(5.9) Ol W { U ( t ) - V/ (í)}|| I \\U(t) - V(t)\\ < C R J \\Uo - Vo||, ( ) < / , < T.

Jo

From (3.4) and Proposition 5.1, it follows that

t ^ ị A ^ U ự ) - l/(i)}|| < A J U o - Voll I CnApV1 [ (t - s ) I I t / ( s ) - K(s)}||cfc,

Therefore, taking c > 0 sufficiently small, \\v conclude l.hat.

s u p p { t ) < C R \\U0 - V o I I •

0 < t< £

For £ < t < T , by (5.10) and the above estimate, we get

p(t) < ( a ? + C r J 0 ^ \\Uo - Vbll + Cr £->1 j Pit - s)-fip{s)ds.

Hence, it is deduced th a t

p { t ) < C R ' T \ \ U o — Vo l l ) £ < t < T

ThuSj we have obtained the first estim ate of (5.9)

The second estim ate of (5.9) then follows immediately from

F ( í ) - V ( t )II < Ao\\Uo - VoII + CR f | | ^ { ơ ( s ) - V(s)}\\ds, 0 < t < T.

Jo

6 D yn am ical sy ste m

As shown in Scction 5, for each ơo ê K , there exists a unique global solution ư{t\ ƯQ) —

(u(i),t;(£),u/(i)) to (1.1) and this solution is continuous with rcspcct to t,hc initial value Therefore, wc can define a semigroup { }i>0 ad.ing on K by s ụ ) ỉ ỉ tí (/(/.;(/()), and the m apping (i, i/o) ► S{t)ƯQ is continuous from [0, oo) X K into K , where K is equipped with the distance induced from (.he universal space X Hence, we have constructed a dynamical system (5 (í), Ky X ) determined from (1.1).

We now verify th a t (S (í), K , X ) adm its a bounded absorbing set Indeed, let R > 0 be any radius and let Uo be in K }1 which is defined in (5.8) Then, from (5.1) there exists a

time ÍR such th a t ||i/(i)||D (/^) < Ơ + 1 for every t > tR, where c is the constant appearing

is a bounded absorbing set of {S(t), K , X )

Since 23 itself is absorbed by 3 , there exists a tim e t-B > 0 such th a t c ĩ> for every t>t<b Wc then consider th e set

x = u S { t ) (b = ( J S(ty.B.

0 < « o o 0 < t < t B

ỈI is c le a r flin t X is an a b s o rb in g ;m l in v a ria n t b o m u li'il i * 'l o f A' \\) T l i r o i v m 1.1 we

then verify that

with a sufficiently small time T ị > 0 and a constant C ị > 0 In view of such a sm oothing

cíỉcct, wc introduce the set

X = 5(7~)X c X.

It is easy to see th a t this set is also an absorbing and invariant set In addition, X c

with the estimate

\ \ AƯ \\ \ \ A S ( T ỵ ) U o \ \ < C ị rq - \ ư S ị T ^ ư o e X , ư o € X

Wc have thus arrived at the following result:

T h e o r e m 6.1 The dynamical system (S(t), K, X ) determined from the problem (1.1)

can be reduced to a dynamical system ( S ( t ), X, A') in which the phase spare is a bounded

set of 'D(A).

Since X is a bounded set of 'Đ{A)J it is meaningful to replace the universal space A' by

Xo = T)(A0) with any exponent 0 < 0 < 1 and consider a dynamical system ( S ( t) , X, A'fl),

where X is now a metric spare with the distance do(ư, V) — II.4Ỡ(r^ — K)||.

C o ro lla ry 6 2 For cadi 0 < 0 < 1, { S ( t ) , x , Xg) ừ a dynamical system.

Proof By the moment, inequality (cf [15]) and the boundedness of X in rD ( A ) ) it follows

absorbing set of ( 5 ( i ) , x , X ) , it follows that there exists a sequence of time tn —> oo sucli

such th a t v{tn") —* V weak* in Finally, by the boundedness of sequence {w(<7i")}

in H 2ạ(íì)y there exists a subsequence {w (t n>»)} such I,hat w ( t n">) —> w strongly in L2(f2)

7 L yapunov function

We finally rem ark th a t one can construct a Lyapunov function for the system (1.1)

Let (u , v , w ) be the global solution to (1.1) obtained in Theorem 5.2 with initial value

(ito,vo,wo) G K Set ip{t) — f u ( t ) — hv(t), 0 < t < oo From the first and second

equations of ( 1.1) it is easily observed th a t

i> a ĩ y a p u n m f u n c t i o n M o t v o w r , w«* :| N«| l - a v r IĨIC f o l l o w i n g r n T ^ r l i m u t c

If wo utilize this Lyapunov fund ion a n d such an energy estimate wr a r c able lo (Icdwtr

in err results on th r structure of tlir u.'-liniit and weak u-’-linut sets (s<H' (1])

si rongrr

References

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|2] L 11 Chuan and A Yagi, Stationary solutionÒ' to forest kincmatic model Preprint.

|3| Yu A Kuznetsov, M Ya Antonovsky, V N Btktashcv and A Apomim A CIDSS-

diffusion model nf forest boundary dynamics, J Math Riol 32 (1H91), 21D-2.Ỉ2

ị-lị Y Lin and Y Liu, Qualitative analysis fo r a model uỊ forest with diffusion and

nonlocal effects, Nonlinear Analysis 39 (2000), 217-229.

(5| 10 Nakaguchi and A Yagi, Fully discrete approximation by Galcrkin Rungc-Kutla

methods Ịor quasilincar parabolic systems, Hokkaido Math J 31 (2002), 385 429.

|G| H Nakata, Numerical simulations for forest boundary dynamics model, M aster’s the­

sis, Osulm I Inivrrsil.y (201)1)

|7| K Osaki and A Ya&i, (iloh al exist c u re f o r a th.em oltui.s-i/iotulh 'iijslc.m III IK2,

Adv Math Sri Appl 12 ('2002), r>S7-ii0fi

[8Ị Y VVu, Stability of travelling wanes for a cross-diffusion model, J Malli Anal Appli

215 (1997), 388-414

|9| Y Wu and Y Lin, The stability of steady states fo r a model with diffusion and spatial

average, J Math Anal Appli 232 (1999), 259-271.

[10j A V Dabin and M I Vishik, Attractors o f Evolution Equations, North-IIollawl,

Amsterdam, 1992

fllj R Dautray and J L Lions, Mathematical Analysis and Numerical Methods fo r Sci­

ence and Technology, Vol 2, Springer-Vcrlag, Berlin, 1988.

[12| P Grisvard, Elliptic Problems in Nonsmooth Domains, Pitm an, London, 1985.

(13] J c Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ Press,

Cambridge, 2001

[14| R Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics 2nd

cd., Springer-Verlag, Berlin, 1997

[ 1 5 ] I I T l i e b d , I n t c r j i o l a l i o i i T h e o r y , F u n d i o n F p n t t D i J J i II Ii l i nl Oj )f i'iilor.', N o r l l i -

Holland, Amsterdam, 1078.

[16] A Yagi, T H Ho and c Due, A mathematical model fo r m a n yIVVC forest dynamics,

Annual Report of PY 2003, j VI p'lijila and p I[ Viet, (nl.v), Fuji I,a Laboratory of Osaka University, 21)05, 2(J!J-;ìU;ỉ

[17] K Yosida, Functional Analysis, find eel., Springer-Vrrlag, Berlin, 1980.

F unk cialaj E kvacioj, 49 (2006) 4 2 7 -4 4 9

Asymptotic Behavior of Solutions for Forest Kinematic Model

By

Le Huy C h u a n 1, T ohru Tsujikayva2 and Atsushi Y a g i3

(O saka U n iv ersity 1-3 and M iyazaki U niversity2, Japan)

three kinds o f f/j-limit sets, namely, (o(Uị)) c L 2-(u(Uịi) c \v*-(fj( lor each point u II

o f the dynam ical syslcm which \v;is constructed ill o u r piL'ccdiim p aper | l | Usiim 1

L yapunov function, we will then investigate basic properties o f these (U-Iimil sets

K ey W ords a n d Phrases A sym ptotic behavior o f solutions D ynam ical system,

L yapunov function, a>-limit sel Forestry ccosystem.

This system has been introduced by Kuznetsov et ill [3] in Older to describe

a prototype ccosystem o f a mono-specics which consists of only two age classes,

This w ork was su p p o rted by G r;int-in-Aid for Scientific- Research (No 16340046) by J a p a n Society lor the P ro m o tio n o f Science and C o o p era tiv e Research P io n n im ill llic form ol C ore University

P ro g ra m between ISPS an d V A S T (Vietnamese A cadem y o f Science and T e d m o lo e y ) by Jap a n

4 2 8 Le Hu> C h i a n T o h ru T s l j i k a w a a n d A tsushi V a g i

denote the tree densities of voung and old age classes, respectively, at a position

v e f i and time / e [0 X ) The third unknown function ir(.v./) denotes the

density of seeds in the air at V 6 ÍÌ and t e [0 X ) The third equation desciibes

llie kinctics of seeds; (I > 0 is a dillusion constant of seeds; and y > 0 and fi > 0

arc seed production and seed deposition rates, respectively While tlic Inst mid sccond equations describe the erowth ol young and old trees, respectively;

0 < Ố < I is a seed establishment rate; }’(r) > 0 is a mortality OỈ young trees which is allowed to depend on the old-trce density r: f > 0 is an aging I ate: and, h > 0 is a mortality of old trees It is observed in general that y(r) has a

where </./>.( > () are positive constants, sec 13] Ill this p;ipcr we assume that

We trcal this system as a degenerate parabolic system (“degenerate’ means

that the diffusion constants for u and r arc considered to vanish identically)

As wc showed in the preceding paper |l | one can in fact construct the global solutions by regarding (1.1) as the Cauchy problem for some appropriate abstract parabolic evolution equation ill the product space

for ail initial values Co's from the space

^ i/o '

u „ = i'd : 0 < i/ị) V0 e L ' (Q) and 0 < If , ) e L 2{£1)

(cf Remark 2.1) and can also define a dynamical system { S { t ) , K , X ) , cf

ị 10 13, 15| etc Note that It is very natural to choose the space L ' (D.) which

IS a Banach algebra L v-(Ũ.) under the usual function product (i.e ||m;||ỵ, <

ll^ll/ ll( II/ ) ‘*s underlying spaccs lor ionnukitiim the first and sccond ctjilMtions (I I) in view ol ihc nonliiiLMi function ( 1.2)

Wc cncountcr however, a very unlavorablc lact that our syslcm docs

numerical example presented in Scction 6 (cl also [6|) shows, we find in some case a solution which starts from continuous initial functions and convcrccs

A sym ptotic Behavior o f Solutions fo r Forest Kinematic M odel 429

is no rigorous analytical proof for this phenomenon, but vvc can ccrtainly prove

in such a case cxistcncc o f the discontinuous stationary solutions, II, v e L ' (fì)

unfavorable phenom enon is o f course affected by the degeneracy o f a diffusion

for u and the nonlincarity o f the term y(v)u To clarify the situation more, it may be interesting to compare our system with another parabolic-ordinary

will be reviewed in Appendix, even if the initial data are discontinuous functions,

the solution always has a time sequence t„ tending to infinity for which u(t„)

that the ordinary equation of Fitzhueh-N agum o is linear in contrast with (1.1)

In view o f the forest ccosystem, such a solution with discontinuous den­

continuous gap o f density is eallcd the ccotonc boundary and is regarded as an

ecotone boundary is formed by the parameters in ( 1 1) and to know how the ecotone boundary moves are one of main interests in the theory of mathematical forestry ccology, although they seem to be very hard problems

In this paper, wc are concerned with studying asymptotic behavior of

functions may CO 11 verge k> discontinuous stationary solution; this then suggests

that our dynamical system (S(t), K, X ) never possesses a ulobiil ullraclor in the

topology o f /V, lor one Ciinnol show continuity o f the semigroup S(l) in the

the weak* topology In view o f these situations, we arc rather led to investigate

asymptotic behavior of each trajectory o f ( S ( t ) , K , X ) We will introduce

t h r e e k i n d s o f co-limit sets, n a m e l y , ( o ( ư q ) c= L 2 - co(U[)) c w *- tf j (ơ o) f o r U{) e K

Here, (ú(U{)) is the usual co-limit set in the topology o f X but may be empty for some ơo e K, L 2-(u(U{)) is an co-limit set with respect to the L 2 topology, and

others, it is proved that L 2-ío(U0) consists of stationary solutions alone But

for the moment, it is an open problem to prove that w*-co( L'(|) consists of

stationary solutions alone

The next interesting problem may be to investigate the structure of

value o f // When 0 < h < f a ỗ / ( a b 2 + c + / ) , it can be proved that stationary

4 3 0 Lc Hu> C m AN T oliru T s l j i k a w a a n d A lsushi V a g i

solutions are only homoíĩcneous solutions in n As will be shown in Theorem

4.3 when /'*>■/(<• + / ) < h < X the zero solution (().().()) is the only stationary

solution of (1.1) To th e contrary, when / 7.Ổ/(ah' + c + J ) < h < J xờ/ (<■ + / )>

the structure of stationary solutions, including possibly iin infinite numbci ol discontinuous ones, seems to be very complicated This will be studied 111 theforthcoming paper |2]

For other results for (I I) we refer to [4, 8 9Ị in which a spatial averageclTect is incorporated

Throughout the paper, c stands for some constant which is determined by the initial constants a, h c í/, / //, 7 and fi and by the domain f l in a specific

Similarly, /)(■) stands for some continuous increasing function which is deter­

mined in a specific way in each occurrence

In this section, we shall list sonic known results for (1.1) which have already been obtained in I lie previous paper I ! |, and shall also describe some con- sequenccs cicduccd from these which will be needed in the present paper.The problem (1.1) is formulated as the Cauchy problem for an abstract evolution equation

where A is a realisation of the operator - i l A + f i in L 2{Q) under the hom o­

known that A is a positive definite scll-adjoint operator o f L 2(Q) with '/ { A ) —

ĩ l ị ( í ì ) (see [11 12]) where ỉ ỉ ị { Q ) is a closed subspacc o f / / 2(Q) consisting of

functions M S satisfying t h e homogeneous Neumann boundary conditions o n m

It is also known that C/{A") = J ỉ yi(íl) for 0 < 0 < 3 /4 and V{A ) = IỉỊỈ’(Q) for 3/4 < 0 < \, where /-/*"( Í2) is a closctl subspace o f H 2"(Q) consisting of luiKlions salislying the homogeneous Neumann boundary conditions on r Q (see

[5 Ift|).

Meanwhile /• is a nonlinear operator from fy { A 11) ink) X dvcn by

Asym ptotic Behavior o f Solutions fo r Forest Kinematic Model 431

in ơo’s were assumed to be in H 2f)(Q.) with some 1/2 < // < 3/4 But, actually,

this restriction is not neccssary, because, as [1, (3.3)] shows, it is possible to

apply [7, Theorem 3.1] (namely, [1, Theorem 2.1Ị) to (2.1) with /11 = 0 and

l / 2 < ; / < 1 (due lo the fact that ( / ( A '1) cz L v (Cl)) This means that it is

The solution satisfies the following integral equations (cf., [1, (2.12)]):

Here, C~'A denotes the linear semigroup generated by A Since A > /i, it f o l l o w s

We verify the following uniform estimates o f solutions which were essen­tially established in [1, Proposition 5.1]

Proposition 2.1 Let U{t) = {u{t),v{t),\v{t)) be the global solution to (2.1)

Proof We already know that

| | t ' ( / ) | | (.: < / > ( l | W | | t : ) 0 < l < X ,

From (2.6) it follows that

II»!’(/)II/ / i f < cjll/iV-'-'u-olL: + f ||/lve (,- ĩK,ltv)||A:í/.v|

In addition, we verify the uniform estimates for the derivative of solutions

P r o p o s i t i o n 2 2 / o r i h c lie I i v u I i r e u ' { l ) — ( l i 1 ( I) , I'1 { I ),

(2 11) ||i/'(/)ll/ < ( 1 + / 'O/^.dl^VIỈA), 0 < / < x ,

(2.13) II" (Oil/.-' + l|ll-(i)ll/,: < (I + f ')/>l(||t/(,llA0 0 < / < X ,

where p |{-) is an appropriate continuous iìiirctìsitiỊỊ function.

observe (2.12) We know that r e ^([0 x ) : L 2(Cl)) DY, 1 ((0 x ) ; Z.2(H)) with

the estimate (2.12) Then, (2.13) is dcduccd by the standard arguments for the

Wc next obtain uniform estimates for the second order derivative of solutions

Proposition 2.3 For the second order derivative L"'{t) = {u"[t), v"(l).

«r"(0 )

( 2 >4 ) ll""ư )||A < (1 + 1 1 _v)/^2( ! I ° < ' <

A sym ptotic Behavior o f Solutions fo r Forest Kinematic M odel 4 3 3

(2.15) ||o"(Olli- á ( i + r " ) f t ( l | í / o L - ) , 0 < / < CO,

(2.16) IK M IIz.: + IK M II ,,; á (1 + r 2)K ( ||ơ „ ||A.) 0 < / < X ,

where Pi{-) is an appropriate continuous increasing function.

v " ( t ) — f u ' ( t ) — h v ' { t ), 0 < / < X)

Then, v E fổ 2((0, CO);L®(Í2)) and the estimate (2.15) is seen by (2.11) and (2.12).With any Ĩ > 0, we consider the Cauchy problem for a linear evolution equation

Therefore, (2.16) is obtained in view of (2.13)

As a consequence o f (2.13) and (2.16), we have

I k ' M l l f s C | | !■'(/)II,A < (1 + r | - " ) / > ( | | ơ „ | | v ) 0 < / < X

Then, (2.14) is observed directly from

Wc conclude this section with describing the dynamical system determined

by the Cauchy problem (2.1) For any U[) G K, let ơ ( / ; ơ o ) be the global

solution of (2.1) We set S(t)ƯQ — U(t\ Uo) for every 0 < / < 00 Then S(t)

this semigroup is continuous on K in the sense that the mapping (t Co) 6

[0 , 00) X K — K is continuous Therefore, the set of all trajectories S(t)Uí)'s

4 3 4 Le Huy C h u a n , Tohru T sujikaw a and A tsushi V agi

defines a dynamical system in X with phase space K which is denoted by

According to [1 Theorem 6.1], there exists an invariant and absorbing set

X c l \ V 1 : 0 < u V e L Tj{Q) and 0 < I f e H ị ( Q )

with + \\i\\L, + \\U'\\H: < c ,

with some constant 0 < Cf < X Therefore, (S(t), ì \ X ) is also a dynamical

that of

3 Lyapunov function

In this section we shall construct a Lyapunov function V'(U) for the dynamical system ( S ( t ) , K , X ) and shall establish some results concerning the asymptotic behavior of trajectories S(t)U()'s.

Let Uti e K and let sụ) Ui, = LJ(t) = (!#(/),!?(/) \\'({)) for 0 < / < GO Set

is easily observed lliul

- Ệ = f /ỈỐU- - { ; ' ( ( ’) -f- / I h}(p - h { y ( v ) v + / ( ’}, 0 < / < CO.

Multiply this by <p{t) = cv/i't and integrate the product in Q Then

(3.1) 2 d ' \ ỵ dx + h dt L r{ v)dx ~ m L I "'dx

where / » = Jq {}’(r)r + f v } d v

While, multiplying the third equation of (1.1) by v w / c t and integrating the

product in Í2, we obtain that

Asym ptotic Behavior o f Solutions fo r Forest Kinematic M ode! 4 3 5

= - j n « w » ) + / + * } ( ! ) + //* > '( J ) 2 i/.Y < 0 , 0 < r < 30.

Note that

^ (/w — /?y)2 + IFti'j2 + / i a / » + U’2 — ( f a f ì ỗ ) v w > c

dfps

Jfl I ( / i/ - /i i> ) 2 + ^ | 1 7 i r | 2 + / ! « / »

+ " 1-^ U’2 — ( / a/ỉổ)ưw dx, U e ữ { A ]/2)

is a Lyapunov function for the present dynamical system (S(t), K, X )

From these arguments we obtain the following energy estimates

436 Le Huy C h ia n Tohru T s u iik a w a and A lsu sh i ^ ACỈ1

Theorem 3.2 For any trajectory S ( l ) L \ f = U(l) as t — J the (fcriralive

( đ i ' / ( h ) ( 1 ) might not c o n v e r g e to 0 ill L 2( i i ) as / — X Then there would

exist a number /: > 0 anti a time sequence {/„} tending to fj such that

This is a contradiction to the fact that \ \ ụ u /dt){l)\\]: is intcmablc in ( 1 x ),

In this section, we shall introduce three types of iu-limit sets, namely,

As well known, the (usual) w-limit set o f S(f)Ui), U[) e K, is defined by

w(ơo) = n {5'(r)ơo; t < T < go} (elosure in the topology o f X ) ,

t> 0

namely, u 6 oj(ưị)) if and only if there exists a time sequence {/„} tending; to X such that $(t„)ơ() —* Ư in the topology of X As will be presented in Scction 6, some numerical simulation suggests that there exists a trajectory which starts

from a continuous initial function Ư 0 — (t/o(.v), i’o(.v), m’o(.y)) e K but, as t —► CO,

converges lo a discontinuous stationary solution V — (z7(.v), f(.v), M’(.v)) If this

phenomenon is true, then any sequence S ( t n) U[) cannot converge lo V in the topology of X , namely, it is possible that oj(Uu) = 0

is said to be L 2 convergent to (//0, tf(), H'o) e X as / 1 —> CO, if

í’„ —> r<) strongly in L 2(i2)

Then, using this topology wc define the L 2-cu-limit set o f S(t)Uị), Ơ(| 6 AT, by

I U'„ —> U'o strongly in L 2(Q).

Usinq this lopolouy, we define the w*-a>-limit set of S{ t)U 0 1/(1 e A', by

w * -c ư (ơ o ) = f ) { S ( t )U()\ t < T < x } ( c l o s u r e in t he w e a k * topoloszy o f X )

I > 0

According to [I, Theorem 6.3], it is already known that \v*-w(ơ[j) # 0 for anv

initial data (Jo e K.

In izencral wc observe the following relations

A sym ptotic Behavior o f Solutions fo r Forest Kinematic M odel 437

4 3 8 Lc H uy CiiL'AN T ohru T s u jjk a w a an d A tsushi V agi

Theorem 4.1 For each L\\ e Ả (u(Uiì) c: <= \v -<u(C\t).

Let Ũ — (ũ ĩ, Ũ’) e L : -(rj( ư ị i ) Then, there exists a scqucncc {/„} tending

to X such that S{t„)U0 = («(/„), »»■(',.)) - i7 in the L 2 topology o f A'.

Let <y?eZJ(f2) For any / e L: (Q)

Hcncc, //(/„) —* /? in the weak* topology of L x (f2) Due to (2.8) it is the same

for the weak* convergence of / (/„) to V Thus we have u e w*-<ư(Uu) □

Wc do not know whether the converse relation w *-(o(ưo) c= L : -ífj(ơo) is

Theorem 4.2 For ư[) e K, let there exist a sequence {(„} tending to 00

such that S(i„)Uu — {u(t„).i'(t„), n(/„)) converges to a triplet o f functions u =

vergence implies I.2 convergence I or c;ich sequence of //(/„) r(/„) and ir(/,J.

The rest of this section is devoted to proviim some slriictural results for the sets under specific conditions assumed to hold lor the cocflicicnts of equations ill ( 1 1)

Theorem 4.3 Assume that h > f x i i / ( c + f ) Then, tìi(ưit) = L 2-to(Uịị)

— \ v *- oj ( U {)) = { ( 0 , 0 , 0 ) } f o r c c c r y uI, € K.

global solution Multiply the first equation of ( 1 1) by 2{c + f ) u and integrate

Asym ptotic Behavior o f Solutions fo r Forest Kinematic M odel 4 3 9

where £ = 2(c + f ) aốỤi — (f<xổ/(c+ f)))ỊỌ>f) > 0 We here notice that

2{((c 4- / ’)//)2 + (ocf'ir)2 + (//r>ir)2 — (<• 4- f)uoLỏv — (xồvỊithv - Ịiồ\\'(c + /')//} -I- 3/:r2

and r(f) to 0 in the L f topolocy In this way, we ultimately conclude that,

Theorem 4.4 Assume that ah2 < 3(c + / ) Then L 2-io( £/«) = w*-oj( Uo)

f o r e v e r y b\) 6 K

{/„} which tends to X as / / —* X By (2.9), |||»'(/,|)||//: is a bounded sequence;

so, wc can choose a subset]Licncc {/„'} for which {u (/„ )} is convergent to Ũ’ in

/ / l+/(i2) and licncc in L ‘ (Q) From the first and second equations o f (2.1) it

is easily observed tlu'.t

(y(r(/„')) + />(/„■) = lị/ỉ<hvụ,ệ’) - j t M - y (/„') j

-Here, wc introduce the cubic function

I \ v ) = ()-(/•) + / ) / • = a r - 2 a h i 2 + { a i r + c + f ) i \ - X < r < 00.

Ii is e isy l o s ee the f o l l o w i n g p r o p e r l y

1.0111111:1 4 5 Will'll a h - < 3 ( r I / ), ir — /*(/’) is Í/ m o n o t o n e incrciiM iiii

smooth function for If with uniformly bounded derivative in the whole leal

a x i s H' e ( — f -fj).

/"(I-) = - -4 + , + / , = 3 „ ( , - I ’V - ‘± : - * ‘- ± 1 1 > 0

A sym ptotic Behavior o f Solutions fo r Forest Kilicimilit M odel 4 41

which converges to some vector o f X in the L 2 topology Hence, the relation

5 Constituents of L 2-( o -limit sets

In this the section, we shall show that every L 2-6U-Iimit set consists

proposition

Proposition 5.1 For each ưo e K, L 2- uj (U o ) is an invariant set o f S(t) i.e.,

continuous from K into itself in the L 2 topoloey.

To see this, consider two initial values Ơ0I = (//(»1, I’m "'01) and U [)2 =

(»02, 1’02, "'02) in K , and let («1 (/), U|(/), H'l(f)) and (i/2(0 i 1’2Ơ)’ l|,2(0 ) be lllc solutions to (2.1) with the initial value Ơ()| and U() 2 , respectively Let T > 0 be

Then, from (2.4),

/ = 1 , 2 I)

In view of (2.7), (2.8) and (2.10), we obtain that

11*0(0 - i'l (OIL- — flz/0- - “01IIL- + QAW Ư 01II A" + II ^0211 A')

— l | | t : +

For any R > 0, there exists a constant C r > 0 such that |t'- - 1| < Ctfj^l holds

for all Id < R Using this estimate, we verify that