Asymptotic Behavior of Solutions of a Periodic Diffusion Equation

Volume 2010, Article ID 597569,12pages doi:10.1155/2010/597569

Research Article

Asymptotic Behavior of Solutions of a Periodic

Diffusion Equation

Jiebao Sun, Boying Wu, and Dazhi Zhang

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Jiebao Sun,[email protected]

Received 25 September 2009; Accepted 24 December 2009

Academic Editor: Shusen Ding

Copyrightq2010 Jiebao Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a degenerate parabolic equation with logistic periodic sources. First, we establish the existence of nontrivial nonnegative periodic solutions by monotonicity method. Then by using Moser iterative technique and the method of contradiction, we establish the boundedness estimate of nonnegative periodic solutions, by which we show that the attraction of nontrivial nonnegative periodic solutions, that is, all non-trivial nonnegative solutions of the initial boundary value problem, will lie between a minimal and a maximal nonnegative nontrivial periodic solutions, as time tends to infinity.

1. Introduction

In this paper, we consider the following periodic degenerate parabolic equation:

∂u ∂t −Δu

m_{ua−bu, x, t∈Ω×R, 1.1}

ux, t 0, x, t∈∂Ω×R, 1.2

ux,0 u0x, x∈Ω, 1.3

wherem >1,Ωis a bounded domain inRn_{with smooth boundary∂Ω,u}

0xis a nonnegative

bounded smooth function,aax, tandbbx, tare positive continuous functions and of

T-periodicT >0with respect tot.

and fisheries management. The term Δum _{models a tendency to avoid crowding and the}

reaction termua−bumodels the contribution of the population supply due to births and deaths; see1 . The homogeneous Dirichlet boundary conditions model the inhospitality of the boundary. The time dependence of the coefficients reflects the fact that the time periodic variations of the habitat are taken into account. Reaction diffusion equations with such reaction term can be regarded as generalization of Fisher or Kolomogorv-Petrovsky-Piscunov equations which are used to model the growth of populationsee2,3 . Especially, when

m 1, the1.1is the classical Logistic equation and some related problems have attracted much attention of researcherssee4–6 , etc..

In the recent years, there are a lot of work dedicated to the existence, uniqueness, regularity, and some other qualitative properties, of weak solutions of this kind of degenerate parabolic equationssee7–9 , etc.. But to our knowledge, there is few work that has been accomplished in the literature for periodic degeneracy parabolic equation, and most of the known results so far only concerned with the existence of periodic solutions but not consider the attractionsee10,11 , etc.. So our work is not a simple extension to the previous work.

The purpose of this paper is to investigate the asymptotic behavior of nontriv-ial nonnegative solutions of the initnontriv-ial boundary value problem 1.1–1.3. Since the equation has periodic sources, it is of no meaning to consider the steady state. So we have to seek some new approaches. Our idea is to consider all nonnegative periodic solutions. We first establish the existence of nontrivial nonnegative periodic solutions by monotone iterative method. Then we establish the a priori upper bound R and a priori lower bound r according to the maximum norm for all nontrivial nonnegative periodic solutions. By which we obtain asymptotic behavior of nontrivial nonnegative solutions of the problem 1.1–1.3. That is all nontrivial nonnegative solutions will lie between a minimal and a maximal nonnegative nontrivial periodic solutions, as time tends to infinity.

The paper is organized as follows. In Section 2, we introduce some necessary preliminaries. In Section 3, we establish the existence of nontrivial nonnegative periodic solutions by monotonicity method. In Section 4, we show the asymptotic behavior of nontrivial nonnegative solutions of1.1–1.3.

2. Preliminaries

In this section, we present the definitions of weak solutions and some useful principles. Since1.1is degenerate at points whereu 0, problem1.1–1.3might not have classical solutions in general. Therefore, we focus our main efforts on the discussion of weak solutions in the sense of the following.

Definition 2.1. A nonnegative functionuis called to be a weak solution of problem1.1–1.3 inQT Ω×0, T, ifu∈L20, T;H₀1Ω∩CTQTandusatisfies

QT

−u∂ϕ ∂t ∇u

m_{· ∇ϕ−ua−buϕ}

dx dt

Ωu0ϕx,0dx−

Ωux, Tϕx, Tdx,

2.1

for any test functions ϕ ∈ C1_Q

T with ϕ|∂Ω×0,T 0, where CTQT denotes the set of

A supersolutionuresp., a subsolutionuis defined in the same way except that the “” in2.1is replaced by “≥”“≤”andϕis taken to be nonnegative.

Definition 2.2. A function uis called to be a T-periodic solution of problem 1.1-1.2 if it is a solution such thatu ∈ CTQT. A functionuis called to be aT-periodic subsolution if

it is a subsolution such thatu·,0 ≤ u·, T inΩ. A functionuis called to be aT-periodic supersolution if it is a supersolution such that u·,0 ≥ u·, T in Ω. A pair of T-periodic supersolutionuand subsolutionuis called to be ordered ifu≥uinQ_T.

Several properties of solutions of problem1.1–1.3are needed in this paper. We first show the comparison principle.

Lemma 2.3 comparison. Assume u₀x ≤ u0x, if ux, t is a subsolution of 1.1–1.3

corresponding to the initial datumu₀x, andux, tis a supersolution of 1.1–1.3corresponding to the initial datumu0x, thenux, t≤ux, t.

Proof. Without loss of generality, we might assume that ux, t, ux, t are bounded. From

Definition 2.1, we have

t

0

Ω−u

∂ϕ ∂t ∇u

m_{∇ϕ dx dt}

Ωux, tϕx, tdx−

Ωu0xϕx,0dx≥

t

0

Ωua−buϕ dx dt,

t

0

Ω−u

∂ϕ ∂t ∇u

m_{∇ϕ dx dt}

Ωux, tϕx, tdx−

Ωu0xϕx,0dx≤

t

0

Ωu

a−buϕ dx dt,

2.2

with nonnegative test functionϕand 0< t≤T. Subtracting the above inequalities, we get

Ω

ux, t−ux, tϕx, tdx≤

Ω

ux,0−ux,0ϕx,0dx

_t

0

Ω

ux, s−ux, sϕs Φx, sΔϕ

dx ds

t

0

Ωa

ux, s−ux, sϕ dx ds

−

t

0

Ωb

ux, s−ux, sϕ dx ds,

2.3

where

Φx, s≡ ₁

0

Sinceu,uandax, tare bounded onQt, it follows fromm > 1 thatΦx, s is a bounded

nonnegative function. Thus, by choosing appropriate test function ϕ exactly as12, Pages 118–123 , we obtain

Ω

ux, t−ux, tdx≤ ϕ _∞

Ω

ux,0−ux,0dxC t

0

Ω

ux, t−ux, tdx ds,

2.5

wheres max{s,0}and C > 0 is a bounded constant. Sinceux,0 ≤ ux,0, combining

with the Gronwall’s lemma, we see thatux, t≤ux, ta.e. inΩfor any 0< t≤T. The proof is completed.

Lemma 2.4global existence. For any nonnegative bounded initial valueu0x, problem1.1–

1.3admits a global nonnegative solution.

Proof. Local existence can be proved as13 . Global existence and nonnegativity follow from

Lemma 2.3by standard arguments.

Lemma 2.5regularity7 . Letux, tbe a weak solution of problem1.1–1.3, then there exist positive constantsKandβ∈0,1, such that

|ux1, t1−ux2, t2| ≤K

|x1−x2|β|t1−t2|β/2

, 2.6

for every pair of pointsx1, t1,x2, t2∈QT.

3. Existence of Periodic Solutions

In this section, we show the existence of nontrivial nonnegative periodic solutions of the problem1.1-1.2by monotonicity method. First, we introduce the following remark.

Remark 3.1 see 14 . According to Lemmas 2.3–2.5, the semiflow associated with the solutionu≡ux, t;u0of problem1.1–1.3, namely, the map

Pt:L∞Ω−→L∞Ω,

Ptu0:u·, t;u0 t >0

3.1

has the following properties:

iPtis well defined for anyt >0Lemma 2.3; iiPtis order preservingLemma 2.4;

iiiPt is compact. In fact, the family {Pru0 | u0∞≤M}r∈0,t M > 0is uniformly

bounded in L∞Ω byLemma 2.3. Then byLemma 2.5, the set {Pru0 | u0∞ ≤

Theorem 3.2. If problem 1.1-1.2 admits a pair of ordered nontrivial nonnegative T-periodic subsolutionux, tandT-periodic supersolutionux, t, then problem1.1-1.2admits a nontrivial nonnegative periodic solutions.

Proof. FromRemark 3.1, we just need to construct a pair of orderedT-periodic subsolution andT-periodic supersolution. The existence of nontrivial nonnegative periodic solutions of problem1.1-1.2will come from the similar iteration procedure as that in15 .

Letλ1, ϕ1be the first eigenvalue and its corresponding eigenfunction to the Laplacian

operator−Δon the domainΩ, μ1, φ1the first eigenvalue and its corresponding eigenfunction

to the Laplacian operator−Δon some domainΩ⊃Ω, with respect to homogeneous Dirichlet data, respectively. It is clear thatφ1x>0 for allx∈Ω. Denote

ammin QT

ax, t, aMmax QT

ax, t, bMmax QT

bx, t, _3.2

and define

urϕ1

1/m

, uRφ1

1/m

, 3.3

where

r min

1 max_Ωϕ1

am

2λ1

m/m−1

,am/2bM

m

max_Ωϕ1

, R

aM/μ1

m/m−1

min_Ωφ1

. 3.4

Clearly, u and u are the T-periodic subsolution and supersolution of 1.1 subject to the condition 1.2, respectively. Further, we may assume u ≤ u, else we may changeΩ and thenr, Rappropriately. Thus we complete the proof.

4. Asymptotic Behavior

In this section, we show the asymptotic behavior of nontrivial nonnegative solutions of the initial boundary value problem. First, we employ Moser’s technique to obtain the upper bound ofL∞norm for a nonnegative periodic solutionu.

Lemma 4.1. Letube a nontrivial nonnegative periodic solution of 1.1-1.2, then there exists a positive constantRwhich is independent ofu, such that

ut_L∞_QT< R, 4.1

whereut u·, t.

Proof. Letube a nontrivial nonnegative periodic solution of1.1-1.2, multiply the1.1by

up1 _{p≥0and integrate the resulting relation overΩ, we have}

1

p2

d dtut

p2

p2

4mp1

pm12

_∇ump1/2_t 2

2 ≤Mut

p2

whereMsup_x,t∈Q

Tax, t. Hence

d dtut

p2

p2C ∇ump1/2t 2

2≤C

p1ut_pp2₂, 4.3

whereCdenotes various positive constants independent ofpandu. Set

pk2km−3, αk

2pk2

pkm1, uku

pkm1/2_{t, k1,2, . . . , 4.4}

from4.3we have

d

dtukt

αk

αkC∇ukt

2 2≤C

pk1

uktαk

αk. 4.5

Here we appeal to the Gagliardo-Nirenberg inequality

ukt_αk ≤C∇uktθk

2 ukt 1−θk

1 , 4.6

with

θk

1− 1

αk

1

N −

1 2 1

−1

pk−m3

N

pk2

N2 ∈0,1, 4.7

whereCdenotes a positive constant independent ofkandp. From4.5,4.6, and the fact thatukt₁uk−1tααkk−1−1, we have

d

dtukt

αk

αk≤ ukt

αk

αk

−Cukt2/θk−αk

αk χ

2αk−11−1/θk k−1 C

pk1

, 4.8

with χk max{1,sup_tuktαk}. Taking the periodicity ofuktαk into account, we can

obtain

ukt_αk ≤Cpk1

χ2αk−11/θk−1

k−1

1/ck

, 4.9

whereck2/θk−αk. From the boundedness ofαkandck, we can see that

ukt_αk ≤C2kβχ2αk−11−θk/2−θkαk

whereβis a positive constant independent ofk. Noticing thatαk<2 implies

21−θkαk−1

2−θkαk

< 21−θkαk−1

2−2θk

<2, 4.11

withχk−1≥1, we get

ukt_αk ≤Cλkχ2_k−1, 4.12

whereλ2β_{>1. That is}

lnukt_αk ≤lnχk≤lnCklnλ2 lnχk−1, 4.13

and thus

lnukt_αk ≤lnC

k−2

i0

2i2k−1lnχ1

k−2

j0

k−j2jlnλ

≤2k−1−1lnC2k−1lnχ1fklnλ,

4.14

or

ut_pk2≤C2k−1χ₁2k−1λfk2/pk2, 4.15

where

fk 2k1−2k−1−k−2. 4.16

Lettingk → ∞, we get

ut_∞≤Cχ1≤C

max

1,sup

t

utm_m1

. 4.17

Now we estimateut_m1. Setpm−1 in4.3, we get

d dtut

m1

m1C∇umt22≤Cutmm11. 4.18

By H ¨older’s inequality and Sobolev’s theorem, we have

From4.18,4.19we can obtain

d dtut

m1

m1Cut2mm1≤Cutmm11. 4.20

By Young’s inequality, we get

d dtut

m1

m1Cut2mm1≤C, 4.21

whereCdenotes different positive constants independent ofu. By4.21and the periodicity ofut, we have

utm_m1≤C. 4.22

Together with4.17, we complete the proof of this lemma.

Lemma 4.2. There exists a constantrwith0< r < R, such that no nontrivial nonnegative periodic solutionsuof problem1.1-1.2satisfies

0<u_L∞_QT≤r. 4.23

Proof. To arrive at a contradiction, we assume that problem 1.1-1.2 admits a nontrivial nonnegative periodic solutionusatisfying 0 < uL∞_QT ≤ r. For any given φx ∈C∞₀ Ω,

we can chooseφ2_{/uas a test function. Multiplying1.1byφ}2_{/uand integrating overQ}

T, we

obtain

QT

φ2 u

∂u ∂tdt dx

QT

mum−1∇u∇

φ2 u

dt dx

QT

φ2a−budt dx. 4.24

By the periodicity ofu, the first term of the left-hand side in4.24satisfies

QT

φ2 u

∂u ∂tdt dx

Ωφ

2

_T

0 ∂lnu

The second term of the left-hand side in4.24can be rewritten as

QT

mum−1∇u∇ φ2 u dt dx QT

mum−1∇u∇

φ·φ

u dt dx QT

mum−1∇u _φ

u∇φφ∇ _φ u dt dx QT

mum−1 _∇

φ u

φ∇u dt dx

QT

mum−1φ∇u∇ φ u dt dx QT

mum−1 _∇φ

u

u∇φ−u2∇ _φ u dt dx QT

mum−1φ∇u∇ _φ u dt dx QT

mum−1∇φ2dt dx−

QT

mum−1u∇φ−φ∇u∇ _φ u dt dx QT

mum−1_∇φ2_{dt dx−}

QT

mum−1_u2_∇

φ u

2dt dx.

4.26

Thus

QT

mum−1∇u∇

φ2 u

dt dx≤

QT

mum−1∇φ2dt dx. 4.27

Combining4.24with4.25and4.27, we obtain

QT

φ2a−budt dx≤

QT

mum−1∇φ2dt dx. 4.28

By an approximating process, we can chooseφφ1withμ1 is the first eigenvalue andφ1is

its corresponding eigenfunction to the eigenvalue problem

−Δuμu, x∈Ω⊃Ω,

Thenμ1>0 andφ1xis strictly positive inΩ. From4.28we have

0≤

QT

mum−1∇φ12−φ2₁a−bu

dt dx

≤

QT

−mrm−1φ1Δφ1−φ12a−bu

dt dx

QT

φ₁2μ1mrm−1−a−bu

dt dx

Ωφ

2 1

T

0

μ1mrm−1−abu

dt dx.

4.30

Thus there existsy0∈Ωsuch thatgy0

_T

0μ1mrm−1−ay0, t by0, tuy0, tdt≥0, then

1

T _T

0 ay0, t

dt≤μ1mrm−1b

y0, t

r. 4.31

Obviously, we can choose suitable smallr0< R, such that for anyr < r0, the above inequality

does not hold. It is a contradiction. The proof is completed.

In the following, we will make use of the a priori boundedness of all nontrivial nonnegative periodic solutions to show the asymptotic behavior of nontrivial nonnegative solutions of the initial boundary value problem1.1–1.3.

Theorem 4.3. Problem1.1-1.2admits a minimal and a maximal nonnegative nontrivial periodic

solutionsu∗x, tandu∗x, t. Moreover, ifux, tis the solution of the initial boundary value problem

1.1–1.3with initial valueu0x>0, then for anyε >0, there existst0depending onu0xandε,

such that

u_∗x, t−ε≤ux, t≤u∗x, t ε, forx∈Ω, t≥t0. 4.32

Proof. First, we show the existence of the maximal periodic solutionu∗x, t. Define a Poincar´e map

PT:C

Ω−→CΩ, withPTu0x ux, T, 4.33

whereux, tis the solution of1.1–1.3with initial valueu0x. ByRemark 3.1, the mapPT

is well defined. Letunx, tbe the solution of1.1–1.3with initial valueu0x P_Tn−1ux,

whereux Kφ1₁/mxandKis a positive constant satisfying

Km−1μ1min

x∈Ω φ m−1/m

1 x≥ ax, tCQT, 4.34

argument, we conclude that there exist a function u∗x ∈ CΩ and a subsequence of

{Pn

Tux}, denoted by itself for simplicity, such that

u∗x lim

n→ ∞P

n

Tux. 4.35

Similar to the proof of Theorem 4.1 in4 , we can prove that u∗x, t, which is the even extension of the solution of the initial boundary value problem1.1–1.3with initial value

u∗x, is a periodic solution of the problem1.1-1.2. Moreover,Lemma 4.1shows that any nonnegative periodic solutionux, tof 1.1-1.2must satisfyux, t ≤ R forx, t ∈ QT.

Therefore, if we take K is larger than R/min_x∈Ωφ1₁/mx, by the comparison principle we haveu∗x ≥ ux,0and thus u∗x, t ≥ ux, t, which means thatu∗x, t is the maximal periodic solution of problem1.1-1.2. The existence of the minimal periodic solution can be obtained with the same method.

Letux, tbe the solution of the initial boundary value problem1.1–1.3with any given nonnegative initial valueu0x, and letvx, tbe the solution of1.1–1.3with initial

valuevx,0 Kφ1₁/mx, whereKis a positive constant satisfying

K≥max

⎧ ⎨ ⎩

u0_L∞_Ω

min_x∈Ωφ1₁/mx,

1 min_x∈Ωφ1₁/mx

ax, t_CQT μ1

1/m−1⎫_⎬

⎭, 4.36

then for anyx, t∈Q_T, we have

ux, tkT≤vx, tkT, k0,1,2, . . . . 4.37

A similar argument as4 shows thatv∗x, t limk→ ∞vx, tkT, andv∗x, tis a nontrivial

nonnegative periodic solution of1.1-1.2. Therefore, there existsk0such that

ux, tkT≤v∗x, t≤u∗x, t, 4.38

for anyk≥k0andx, t∈QT. Provided that the periods ofv∗x, tandu∗x,tare taken into

account, for anyε >0, there existst0depending onu0xandεsuch that

ux, t≤u∗x, t ε, forx∈Ω, t≥t0. 4.39

By the similar way andLemma 4.2, we can obtain

u∗x, t−ε≤ux, t, forx∈Ω, t≥t0. 4.40

Acknowledgments

The authors express their thanks to the referees for their very helpful suggestions to improve some results in this paper. This work is supported by NSFHA200909, NSFC10801061, and ProjectHIT. NSRIF. 2009049. It was also supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology.

References

1 A. Okubo,Diffusion and Ecological Problems: Mathematical Models, vol. 10 ofBiomathematics, Springer, Berlin, Germany, 1980.

2 R. A. Fisher, “The advance of advantageous genes,”Annals of Eugenics, vol. 7, pp. 355–369, 1937.

3 E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, “Partial differential equations in ecology: spatial interactions and population dynamics,”Ecology, vol. 75, no. 1, pp. 17–29, 1994.

4 H. Amann, “Periodic solutions of semilinear parabolic equations,” inNonlinear Analysis (Collection of Papers in Honor of Erich H. R¨othe), pp. 1–29, Academic Press, New York, NY, USA, 1978.

5 P. Hess, “On the asymptotically periodic Fisher equation,” inProgress in Partial Differential Equations: Elliptic and Parabolic Problems (Pont-`a-Mousson, 1991), vol. 266 ofPitman Research Notes in Mathematics Series, pp. 24–33, Longman Science and Technology, Harlow, UK, 1992.

6 S. Ahmad and A. C. Lazer, “Asymptotic behaviour of solutions of periodic competition diffusion system,”Nonlinear Analysis: Theory, Methods & Applications, vol. 13, no. 3, pp. 263–284, 1989.

7 E. DiBenedetto, “Continuity of weak solutions to a general porous medium equation,” Indiana University Mathematics Journal, vol. 32, no. 1, pp. 83–118, 1983.

8 I. S. Pop, M. Sep ´ulveda, F. A. Radu, and O. P. Vera Villagr´an, “Error estimates for the finite volume discretization for the porous medium equation,”Journal of Computational and Applied Mathematics. In press.

9 L. Ma, L. Zhao, and X. Song, “Gradient estimate for the degenerate parabolic equationut ΔFu

Huon manifolds,”Journal of Differential Equations, vol. 244, no. 5, pp. 1157–1177, 2008.

10 R. Huang, Y. Wang, and Y. Ke, “Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms,”Discrete and Continuous Dynamical Systems. Series B, vol. 5, no. 4, pp. 1005–1014, 2005.

11 Y. Ke, R. Huang, and J. Sun, “Periodic solutions for a degenerate parabolic equation,” Applied Mathematics Letters, vol. 22, no. 6, pp. 910–915, 2009.

12 J. R. Anderson, “Local existence and uniqueness of solutions of degenerate parabolic equations,”

Communications in Partial Differential Equations, vol. 16, no. 1, pp. 105–143, 1991.

13 Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li,Nonlinear Diffusion Equation, Singapore World Science and Technology Press, Singapore, 2001.

14 P. Hess, M. A. Pozio, and A. Tesei, “Time periodic solutions for a class of degenerate parabolic problems,”Houston Journal of Mathematics, vol. 21, no. 2, pp. 367–394, 1995.