Volume 2009, Article ID 827087,12pages doi:10.1155/2009/827087
Research Article
On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation
Zongqi Liang and Huashui Zhan
School of Science, Jimei University, Xiamen 361021, China
Correspondence should be addressed to Zongqi Liang,[email protected] Received 21 April 2009; Accepted 25 August 2009
Recommended by Guang Zhang
By Oleinik’s line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in0, T×R2:∂xxuu∂yu−∂tuf·, u, provided that Tis suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.
Copyrightq2009 Z. Liang and H. Zhan. 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.
1. Introduction
We consider the following Cauchy problem:
∂xxuu∂yu−∂tuf·, u, t, x, y
∈0, T×R2, 1.1 u0,· u0
x, y ,
x, y
∈R2. 1.2
This problem arises in financial mathematics recently; more and more mathematicians have been interested in it. In1, Antonelli et al. introduced a new model for agents’ decision under risk, in which the utility function is the solution to1.1-1.2; they also proved, by means of probability methods, the existence of a continuous viscosity solution of 1.1-1.2, which satisfies
u x, y, t
−u
ξ, η, τ≤CT
|x−ξ|y−η 1.3
for everyx, y,ξ, η ∈ R2,t ∈ 0, T, under the assumption that f is uniformly Lipschitz continuous function. In 2, Citti et al. studied the interior regularity properties of this problem; they proved that the viscosity solutions are indeed classical solutions. On the
other hand, Antonelli and Pascucci3showed that the solutionufound in1can be also considered as a distributional solution.
However, all the above results are obtained whenTis suitably small; say, the solution is local. The global weak solutions of the Cauchy problem for a more general class of equations, that contains1.1, are obtained in4–7, and so forth. This kind of solutions, however, is few regular and does not satisfy condition1.3in general.
In this paper, we will solve the Cauchy problem1.1-1.2in another simpler way and get the result as2again. Moreover, some examples are provided by numerical computation.
The results of computation show that the strong solutions of the above equation may blow-up in finite time, though there exist the global weak solutions.
2. Line Method
In order to describe our method, we have to quote the well-known Prandtl system for a nonstationary boundary layer arising in an axially symmetric incompressible flow past a solid body, it has the form
∂tuu∂xuv∂yu∂tUU∂xU∂2yu,
∂xru ∂yrv 0
2.1
in a domainD {0 < t < T, 0 < x < X, 0 < y < ∞}, whereUt, xand rxare given functions. If we introduce the Crocco variables:
τt, ξx, η ut, x
Ut, x, 2.2
we obtain the following equation forwτ, ξ, η ∂yu/U:
w2wηη−wτ−ηUwξAwηBw0. 2.3
Oleinik and Samokhin 8 had done excellent work in the boundary theory by the line method. Comparing this equation with1.1, it is natural to conjecture that we are able to solve problem1.1-1.2by Qleinik’s method.
Consider the following initial boundary problem:
wηη−wτwwξ f
τ, ξ, η, w
, 2.4
w 0, ξ, η
u0 ξ, η
, 2.5
whereu0∈C2R2; its first-order derivatives andu0ηηare all bounded.
Definition 2.1. A function wτ, ξ, ηis said to be a solution of problem 2.4-2.5 if w has first-order derivatives in 2.4 which is continuous in0, T×R2, and its derivative wηη is continuous;wsatisfies2.4in0, T×R2, together with condition2.5.
The solution of problem2.4-2.5will be constructed as the limit of a sequencewn, n → ∞, which consists of solutions of the equations
Lnwn wnηη−wnτwn−1wξn−f·, wn 0, 2.6 wn
0, ξ, η w0
ξ, η
. 2.7
Asw0τ, ξ, ηwe take a function which is smooth in0, T×R2. Suppose that for some nonnegative numberp
f·, v≤c
1|v|p
, 2.8
and whenv1−v2≥0,
c1v1−v2≥f·, v1−f·, v2≥c2v1−v2, max
∂f
∂τ ,
∂f
∂ξ ,
∂f
∂η
∂2f
∂v2
≤c. 2.9
Lemma 2.2. LetVbe a smooth function such thatLnV≥0 in0, T×R2,V ≤wnforτ 0. Then V ≤ wn everywhere0, T×R2. LetV1be a smooth function such that LnV1 ≤ 0 in0, T×R2, V ≥wnforτ 0. ThenV1≥wneverywhere in0, T×R2.
Proof. Let us prove the first statement ofLemma 2.2. The differencezn wn−V satisfies the inequality
0≥Lnzn Lnwn−LnV znηη−znτwn−1znξ −
f·, wn−f·, V
. 2.10
Letzn1 e−ατzn. Then
0≥zn1ηη−zn1ταzn1wn−1zn1ξ−e−ατ
f·, wn−f·, V
≥zn1ηη−zn1τ−αzn1wn−1zn1ξ−c1zn1.
2.11
If we chooseα large enough, by the maximal principle, we knowV ≤ wn everywhere in 0, T×R2.
Let us construct functions satisfying the conditions of Lemma 2.2. To this end, we define a twice continuously differentiable even function such that V1 1−e−β|η|eβτ for
|η|>1,V1ϕηeβτfor|η| ≤1, whereϕηis aC2function,|ϕηη| ≤c.
When|η|>1,
LnV1 V1ηηn −V1τ−wn−1V1ξn −f·, V1 −β21e−β1|η|eβτ−β
1−e−β1|η|
eβτ−f·, V1
≤ −β21e−β1|η|eβτ−β
1−e−β1|η|
eβτc
1−e−β1|η|p
epβτ c <0
2.12
if we choseβlarge enough andβτ≤T0small enough.
When|η| ≤1,
LnV1 ϕηηeβτ−βϕeβτ−f·, V1
≤ϕηηeβτ−βϕeβτc
1ϕpeβτp
<0
2.13
by the same reason.
LetV ψηe−ατ,α1> ψη≥α0>0,|ψηη| ≤c. Then
LnV ψηηe−αταψeατ−f·, V
≥ψηηe−αταψeατ−c
1ψpeατp
≥0 2.14
if we choseαlarge enough andατ≤T0small enough.
Similarly, we are able to prove the second statement ofLemma 2.2.
Thus we have the following.
Lemma 2.3. Suppose thatfsatisfies2.9andV0, ξ, η≤w0≤V10, ξ, η, then
V ≤wn≤V1. 2.15
The smooth functionsV,V1can be constructed as in [8], and we omit details here.
Let
Φn Φ unτ2 unξ2
unη2
, 2.16
where un wn. We will show that there exist positive constants M and T such that the conditionsΦμ≤Mforτ≤T,μ≤n−1, imply thatΦn≤Mforτ≤T.
First, we rewrite2.6as
unηη−uτun−1unξ −f·, un 0, τ, ξ, η
∈0, T×R2. 2.17
Applying the operator 2unτ∂/∂τ 2unξ∂/∂ξ 2unη∂/∂ηto2.17,
2unτunτηη2unτ
un−1τ unξ un−1unξτ
−2unτunττ−2∂f
∂uunτ2−2unτ∂f
∂τ, 2unξunξηη2unξ
un−1ξ unξ un−1unξξ
−2unξunτξ−2∂f
∂u
unξ2
−2unξ∂f
∂ξ, 2unηunηηη2unη
wηn−1unξ wn−1unξη
−2unηunτη−2∂f
∂u
unη2
−2unη∂f
∂ξ 0,
2.18
then
un−1Φξ
2unτunτξ2unξunξξ2unηunηξ un−1,
−Φτ −2unτunττ−2unξunξτ−2unηunητ, Φηη2
uτη2
2unτunτηη2 uξη2
2unξunξηη2 uηη2
2unηunηηη, Φηηun−1Φξ−Φτ−2∂f
∂uΦ2unτunξun−1ξ 2 unξ2
un−1ξ 2unηunξun−1η −2unτ∂f
∂τ−2unξ∂f
∂ξ−2unη∂f
∂ξ0.
2.19
By2.9,2.15, and Cauchy inequality, we are able to get
Φηηun−1Φξ−ΦτRnΦ≥0, 2.20
whereRn depends onun−1 and its derivatives are up to the second. LetΦ1 Φe−γτ with a positive constantγto be chosen later. Then
Φ1ηηun−1Φ1ξ−Φ1τ Rn−γ
Φ≥0 2.21
if we chooseγaccording toMsuch thatRn−γ ≤ −1. IfΦ1 attains its positive maximum at τ 0, then
Φ1|τ0 Φe−γτ|τ0 Φ|τ0≤c, 2.22
where the constantcdoes not depend onn. At the same time, the positive maximum ofΦ1in 0, T×R2cannot be attained by maximal principle. Thus we have
Φ1≤c. 2.23
So, if we letT1≤T small enough such thateγT12 and setM2c,then
Φ≤ceγT1 M. 2.24
In order to estimate the second derivatives ofunin0, T1×R2, consider the function
F unττ2 unξξ2
unηη2
unτξ2
unξη2
unτη2
. 2.25
Applying the operator
P2unττ ∂2
∂τ2 2unξξ ∂2
∂ξ2 2unηη ∂2
∂η2 2unτξ ∂2
∂τ∂ξ2unτη ∂2
∂τ∂η 2unξη ∂2
∂ξ∂η 2.26
to both sides of2.17, we find that 02unττunηηττ2unττ
un−1ττ unξ 2un−1τ unξτun−1unξττ
−2unττunτττ−2unττ ∂2f
∂u2unτ2∂f
∂uunττ
2unξξunηηξξ2unξξ
un−1ξξ unξ 2un−1ξ unξξun−1unξξξ
−2unξξunτξξ−2unξξ ∂2f
∂u2
unξ2 ∂f
∂uunξξ
2unηηunηηηη2unηη
un−1ηη unξ 2un−1η unξηun−1unξηη
−2unηηunτξξ−2unηη ∂2f
∂u2
unη2
−∂f
∂uunηη
2unτξunηητξ2unτξ
un−1τξ unξ un−1τ unξξun−1ξ unξτun−1unξξτ
−2unτξunττξ−2unτξ ∂2f
∂u2unτunξ ∂f
∂uunτξ
2unξηunηηξη2unξη
un−1ξη unξ un−1ξ unξηun−1η unξξun−1unξξη
−2unξηunττξ−2unξη ∂2f
∂u2unηunξ ∂f
∂uunξη
2unτηunηητη2unτη
un−1τη unξ un−1τ unξηun−1η unξτun−1unξτη
−2unτηunττη
−2unτη ∂2f
∂u2unηunτ ∂f
∂uunτη
.
2.27
At the same time, we can calculate that
Fη2unττunττη2unξξunξξη2unηηunηηη2unτξunτξη2unξηunξηη2unτηunτηη, Fηη2
unττη2
2unττunττηη2 unξξη2
2unξξunηηξξ2 unηηη2
2unηηunηηηη 2
unτξη2
2unτξunτξηη2 unηξξ2
2unξηunξηηη2 unτηη2
2unτηunτηηη, un−1Fξ un−1
2unττunττξ2unξξunξξξ2unηηunηηξ2unτξunτξξ2unξηunξηξ2unτηunτηξ ,
−Fτ −
2unττunτττ2unξξunξξτ 2unηηunηητ2unτξunτξτ2unξηunξητ2unτηunτητ ,
2.28
and so we have
Fηηun−1Fξ−Fτ−2∂f
∂uF−2unττ
wn−1ττ unξ 2wn−1τ unξτ
−2unξξ
un−1ξξ unξ 2un−1ξ unξξ
−2unηη
un−1ηη unξ 2wn−1η unξη
−2unτξ
un−1τξ unξ un−1τ unξξun−1ξ unξτ
−2unξη
un−1ξη unξ un−1ξ unξηun−1η unξξ
−2unτη
un−1τη unξ un−1τ unξηun−1η unξτ
−2unττ∂2f
∂u2unτ2−2unξξ∂2f
∂u2
unξ2
−2unηη∂2f
∂u2
unη2
−2unτξ∂2f
∂u2unτunξ −2unξη∂2f
∂u2unηunξ −2unτη∂2f
∂u2unηunτ
−2∂f
∂u
unτ2 unξ2
unη2
0.
2.29 By the introduced assumption that the first-order and second-order derivatives of un−1,
∂f/∂u, and∂2f/∂u2 are all bounded and using Cauchy inequality, we can get from2.29 that
Fηη−2αFη−un−1Fξ−FτRn1F ≥0. 2.30 By the transformationF1 Fe−γτ, if we choseγlarge enough, we are able to show that there exist positive constantsMandTsuch that the conditionsFμ≤Mforτ ≤T,μ≤n−1, imply thatFn≤Mforτ≤T. Thus we have the following.
Theorem 2.4. Letwnbe the solutions of problems2.4-2.5, then the derivatives ofwn up to the second-order are uniformly bounded with respect tonin the domain0, T×R2with a small positive numberT.
Now let us establish uniform convergence ofwnunin0, T×R2. Forvnwn−wn−1 we obtain the following equation from2.6:
vηηn −vτnwn−1vξn−vn−1wn−1ξ −
f·, wn−f
·, wn−1 0, vn
0, ξ, η 0.
2.31
Letvneατvη1. Then
v1ηηn −v1τn wn−1vn1ξ−vn−11 wn−1ξ −αvn1−e−ατ
f·, wn−f
·, wn−1 0, v1ηηn −v1τn wn−1vn1ξ−vn−11 wn−1ξ
αv1ne−ατ
f·, wn−f
·, wn−1
αv1ne−ατ ∂f
∂wvn1 ≥α−cvn1,
2.32
where we have chosenτ ≤Tsmall enough such thate−ατ 2,and 2∂f/∂w≥ −c.
Ifv1 attains its positive maximal value in0, T×R2,we can chooseαlarge enough such that
wξn−1 α−c
<1, 2.33
and then at the maximal point we have
α−cvn1 ≤ −vn−11 wξn−1. 2.34
Ifv1nattains its negative minimal value in0, T×R2, we have
α−c
−v1n
≤ −vn−11 wn−1ξ . 2.35
Notice that atτ 0, vn1 vn0. By2.34and2.35,
maxv1n≤qmaxvn−11 , q <1, 2.36
which means that the seriesv11v12· · ·v1n· · ·, whose sum has the formwne−ατ, is majorized by a geometrical progression and, therefore, is uniformly convergent. The fact thatwnand its derivatives up to the second-order are bounded implies that the first derivatives ofwnare uniformly convergent asn → ∞.
It follows from2.6thatwnηηare also uniformly convergent asn → ∞.
Now, we can takew−1w0w0; then by the above discussion, we have the following theorem.
Theorem 2.5. Suppose thatV0, ξ, η≤w0≤V10, ξ, ηandfsatisfies2.9and is suitable smooth, then there exists a small positive numberTsuch that the Cauchy problem2.4has a classical solution.
By the way, it is easy to prove the uniqueness of the solution for the Cauchy problem 2.4, and we omit the details here.
3. Computational Examples
In this section, a numerical simulate is made for the equations by differential method.
Numerical computation of these examples shows that the strong solutions for the corresponding Cauchy problem of1.1-1.2will blow-up in finite time.
LetΩ 0, Lx×0, Lyandux, y,0 u0x, y,x, y∈Ω, butux, y,0 0,x, y∈ R2/Ω. Then instead of studying the Cauchy Problem1.1-1.2, we can study the following
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
uπ,y,t
0 1 2 3 4 5 6 7
y t0.02
t0.04 t0.042
Figure 1:f·, u u.
−5
−4
−3
−2
−1 0 1 2 3 4
uπ,y,t
0 1 2 3 4 5 6 7
y t0.2
t0.4 t0.6
Figure 2:f·, u sinu.
initial boundary problem:
∂xxuu∂yu−∂tuf·, u, x, y, t
∈Ω×0, T, u
x, y,0 u0
x, y ,
x, y
∈Ω, u|∂Ω 0.
3.1
−1
−0.5 0 0.5 1
ux,y,t
8 6
4 2
0 0 2
4 6
8
x y
a
−5 0 5
ux,y,t
0 1 2 3y 4 5 6 7 8
6 4
2 0
x
b
Figure 3: The numerical results inaatt0 and inbatt0.046 whenf·, u u2.
Iff·,0 0, it is clear that ifux, y, tis a classical solution of3.1, thenux, y, tis a strong solution of the Cauchy problem1.1-1.2.
To dissect domainΩ, suppose thatLx Ly 2πandhx 2π/N,hy 2π/Mstands for the space step-length in the axisxand axisy, andkT/Jstands for the time step-length.
LetΩh {ihx, jhy|0 ≤i≤N; 0≤j ≤M}and defineunij uihx, jhy, nk. The differential scheme of the original equation is to ensure numerical stability, here we apply arithmetic averages in order to avoid “oscillation” and “shifting” of the numerical solution
uni1,j−2uni,juni−1,j h2x
uni1,juni,j uni−1,juni,j1uni,juni,j−1 6
uni,j1−uni,j−1 2hy
−un1i,j −1/4
uni−1,juni1,juni,j1uni,j−1 k
f ihx, jhy, nk,uni1,juni,juni−1,juni,j1uni,juni,j−1 6
, un|∂Ωh 0, n1,2, . . ., u0i,j u0
ihx, jhy .
3.2
−30
−20
−10 0 10 20 30
uπ,y,t
0 1 2 3 4 5 6 7
y t0.008
t0.01 t0.015
Figure 4:f·, u u.
So we get the following explicit formula:
un1i,j 1 4
uni−1,juni1,juni,j1uni,j−1 k
h2x
uni1,j−2uni,juni−1,j
k 12hy
uni1,j2uni,juni−1,juni,j1uni,j−1
uni,j1−uni,j−1
−kf ihx, jhy, nk,uni1,j2uni,j uni−1,juni,j1uni,j−1 6
.
3.3
Experiment 1. SupposeΩ 0,2π×0,2π,hxhy2π/40,k0.001,u0x, y sinxsiny which itself does not satisfy1.1; we get the graphssee Figures1–3whereux, y, tchanges according to the changes oftwhen different functions are given tof·, u.
Figure 1 shows that whenf·, u u, att 0.04, the numerical solutions become oscillatory; att 0.042,the bifurcation of solutions occurs; whent > 0.042,the solutions will blow-up. SimilarlyFigure 2shows that whenf·, u sinu, att 0.6,the bifurcation of solutions occurs; whent >0.6,the solutions will blow-up.Figure 3is the spatiotemporal graphs of solutions whenf·, u u2att0initial valueandt0.0046. Whent >0.0046, the solutions will blow-up.
Experiment 2. The initial value is unknown in the general situation; so we use random numbers−0.01,0.01 as the initial value and draw graphs see Figures4 and 5 where ux, y, tchanges astchanges when different functions are given tof·, u.
Figures4and5show that even though the initial value is sufficiently small, the blow- up will appear in finite time for the different functions.
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02
u2π/3,t
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
t Figure 5:f·, u 1−sinu.
The numerical result shows that there is a locality solution of the equation. Whent becomes larger, the bifurcation of solutions occurs in finite time and blow-up appears. For this problem, it is essential to have a further research.
Acknowledgments
The research was supported by the Fujian National Science Foundation of China Grant 2008J0198, 2009J1009. The authors would like to thank Professor Zhao Junning for insightful discussions and Professor Xu Chuanju for helpful suggestions and comments.
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