Mathematical Analysis on Parabolic
System with Strong
Cross-Diffusion
(joint work with Prof. Ansgar Juengel)
CHEN, Li （陈丽）
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universitat,
Mainz, Germany.
Department of Mathematical Science,Tsinghua University,
Beijing, P. R. China.
http://faculty.math.tsinghua.edu.cn/faculty/~lchen/
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
1
Parabolic System
Diffusion Matrix and Potential
We will focus on the dicussion on the influences from the diffusion
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
2
Population dynamics
tu- d1Mu-r11Mu2- r12M(uv) u(a1- b1u- c1v)
tv - d2Mv- r21M(uv)- r22Mv2 v(a2- b2u- c2v)
Kinetic
(ODE)
! Diffusion ! Cross-Diffusion
Lotka-Volterra competition system
(PDE)
(strong coupled PDE)
Shigesada et al.Theor. Biol.,1979,
ai¸0 : the intrinsic growth rate of the i-th species
b1,c2¸ 0 : the coefficients of intra-specific competition
b2,c1¸ 0 : the coefficients of inter-specific competition
The solutions of Kinetic system depend on the constants A,B and C in various cases.
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
3
Boundary and Initial conditions for PDE system
Diffusion
(Heat equation)
-t u+M u=0, in W£ (0,1)
r¢g=0 on W
u(x,0)=y(x) in W
u(x,t) is smooth for t>0 and
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
4
• 9 ! global smooth non-negative solution; (well known)
• Long time behavior, P. N. Brown, P. de Mottonim (1980’s).




A>max{B,C}; (u,v)! (a1/b1,0)
A<min{B,C}; (u,v)! (0,a2/c2)
B>A>C (weak competition); (u,v)! (u*,v*)=
B<A<C (strong competition).
 (a1/b1,0) and (0,a2/c2) are both locally stable
 (u*,v*) is unstable
 W convex,

no stable positive steady-state solution (Kishimoto, Weinberger,1985)
W dumb-bell type, at least one stable positive… (Matano, Mimura,1983)

……. (Y. Kan-on, E. Yanagida, M. Mimura, S. I. Ei, Q. Fang, H. Ninomiya……)
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
5
PDE with strong cross-diffusion and self-diffusion
Some known results on time dependent case
 r11=r22=0 and no cross-diffusion for the second species
(global exis. & qualitative behavior)
•
Pozio & Tesei, 1990, Nonlin. Anal.;
•
Lou, Ni, & Wu, 1996, Adv. Math., Beijing;
•
Redlinger, 1995, J. Diff. Eqs.;
•
Choi, Lui, & Yamada, 2003, Discrete Contin. Dyn. Syst.
•
……
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
6
 r11=r22=0 and “small” cross-diffusion
• (d1=d2=0,1-D, global exis.) Kim, 1984, Nonlin. Anal.;
• (Global exis.) Deuring, 1987, Math. Z;
 xTA(u,v)x¸min{d1,d2}|x|2
•
(2-D, Global exis.) Yagi, 1993, Nonlin. Anal.;
•
(Global exis.) Galiano, Garzon &Juengel, 2001, Rev.
Real Acad. Ciencias, Serie A. Mat .
 For any di>0,rij>0
•
(1-D, Global exis.) Galiano, Garzon &Juengel, 2003,
Num. Math.;
•
(Multi-D. Global exis.) L, Chen & Juengel, 2004, SIAM J.
Math. Anal. (Idea will be introduced later)
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
7
Diffusion Matrix
Main Features
• Nonsymetric
• Nonpositive definite
• Degenerate (d1=d2=0)
It is hard to use classical techenics (such as maximum principle to get a priori
estimates) on such system due to strong cross-diffusion.
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
8
Idea (Approximation+A priori estimates)
• Exponential transformation u=exp{fu} , v=exp{fv},
Symetric and non-negative definite.
New difficulties: time derivatives
Plasma-Cargese, 26/10/04-29/29/04
t[exp{fu}], t[exp{fv}].
strong cross-diffusion parabolic system
9
• Relative entropy
y(x)=x(ln x-1)+1
Entropy inequality
It can be formally derived by using ln u and ln v as the test function in
the weak formulation of the problem.
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
10
• Compactness Argument (omitted)
• Approximation

Semi-discretization in time difficulty: cross-diffusion
M(uv)
Approximated by finite difference
Besides these, we need other regularizations, such as

Fully discretization both in time and space

Finite difference in time
Decomposition: (0,T]=[ k=1K((k-1)t,kt], t=T/K

Galerkin method in space
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
11
Existence of the weak solution
Asumptions
W2 C0,1 , N¸ 1, di¸ 0, rij>0, ai,bi,ci¸ 0, i,j=1,2.
u0,v02 LY(W), u0,v0¸ 0.
The existence can be also obtained in the case without self-diffusion, which is useful
to study the pattern formation. (introduce later)
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
12
Q: Long time behavior of the solution
 r21=0, (trianglular cross-diffusion case) D. Le, L. Nguyen, T. Neuyen, (2002,2003)
 Few other results until now…
Steady state solution
(Y.Lou, W.Ni, H. Matano, M. Mimura, Y. Nishiura, A. Tesei, T. Tsujikawa,Y.Kan-on…)
Large di (diffusion) or rii (self-diffusion)
no non-constant steady state solution (NCSS)
 In weak competition case, if r12, r21 (cross-diffusion) are samll,
NCSS
 In weak competition case, if r12 or r21 (cross-diffusion) large,
NCSS exists

No
 Do nonconstant steady solution exist if both cross-diffusion r12
and r21 are large and qualitatively similar?
There are also some results on steady state solution and some stability results with
vanishing Dirichlet boundary data.
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
13
• Entropy-entropy production method for long time behavior
It holds that without self-diffusion(more reasonable)
This inequality can be obtained directly from the approximate problem
by choosing appropriate test function.
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
14
• logarithmic Sobolev inequality
)
• Csiszar-Kullback inequality for logarithmic relative entropy
)
• Discussion on Steady state solution
)
No non-constant steady state solution
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
15
Long time behavior of the weak solution
Steady state solution
a1 ,a2 ,b1 ,c2 > 0, c1= b2= 0
No Non-constant steady state solution
 The only possible steady state solution is

(u*,v*)=(a1/b1,a2/c2).
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
16
Conclusion

Existence

No restriction on the diffusion coefficients di and rij

The global existence result holds in any space dimension

The method provides the existence of non-negative solution


The degenerate case di=0 and no self-diffusion case rii can
be also treated
Long time behavior
Give some convergence rate of the entropy
 No NCSS exist even with strong cross-diffusion in the
case of vanishing source terms or vanishing inter-specific
competition

Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
17
Future problems
• Uniqueness and regularity of the weak
solution
• Long time behavior in more general cases
• ……
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
18
Thank you!
谢谢！
Plasma-Cargese, 26/10/04-29/29/04
strong cross-diffusion parabolic system
19