CHEN, Li （ 陈丽 ） Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universitat,

1. 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/ strong cross-diffusion parabolic system

2. Parabolic System Diffusion Matrix and Potential We will focus on the dicussion on the influences from the diffusion strong cross-diffusion parabolic system

3. Population dynamics ai¸0 : the intrinsic growth rate of the i-th species b1,c2¸ 0 : the coefficients of intra-specific competitionb2,c1¸ 0 : the coefficients of inter-specific competition ¶tu ¶tv • r11Mu2- r12M(uv) • - r21M(uv)- r22Mv2 • u(a1- b1u- c1v) • = v(a2- b2u- c2v) • - d1Mu • - d2Mv Kinetic (ODE) Lotka-Volterra competition system ! Cross-Diffusion (strong coupled PDE) Shigesada et al.Theor. Biol.,1979, ! Diffusion (PDE) The solutions of Kinetic system depend on the constantsA,BandCin various cases. strong cross-diffusion parabolic system

4. 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 strong cross-diffusion parabolic system

5. 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 locallystable • (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……) strong cross-diffusion parabolic system

6. 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. • … … strong cross-diffusion parabolic system

7. r11=r22=0and “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) strong cross-diffusion parabolic system

8. 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. strong cross-diffusion parabolic system

9. Idea(Approximation+A priori estimates) • Exponential transformationu=exp{fu} , v=exp{fv}, Symetric and non-negative definite. New difficulties: time derivatives¶t[exp{fu}], ¶t[exp{fv}]. strong cross-diffusion parabolic system

10. 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. strong cross-diffusion parabolic system

11. Compactness Argument(omitted) • Approximation • Semi-discretization in time difficulty: cross-diffusionM(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 strong cross-diffusion parabolic system

12. 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 obtainedin the case without self-diffusion, which is useful to study the pattern formation. (introduce later) strong cross-diffusion parabolic system

13. Q: Long time behavior of the solution Steady state solution (Y.Lou, W.Ni, H. Matano, M. Mimura, Y. Nishiura, A. Tesei, T. Tsujikawa,Y.Kan-on…) • r21=0, (trianglular cross-diffusion case) D. Le, L. Nguyen, T. Neuyen, (2002,2003) • Few other results until now… • 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, No NCSS • In weak competition case, if r12 or r21(cross-diffusion) large, NCSS exists • 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. strong cross-diffusion parabolic system

14. Entropy-entropy production methodfor long time behavior It holds thatwithout self-diffusion(more reasonable) This inequality can be obtained directly from the approximate problem by choosing appropriate test function. strong cross-diffusion parabolic system

15. logarithmic Sobolev inequality ) • Csiszar-Kullback inequality for logarithmic relative entropy ) • Discussion on Steady state solution ) No non-constant steady state solution strong cross-diffusion parabolic system

16. 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). strong cross-diffusion parabolic system

17. Conclusion • Existence • No restriction on the diffusion coefficients diand 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 riican 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 strong cross-diffusion parabolic system

18. Future problems • Uniqueness and regularity of the weak solution • Long time behavior in more general cases • …… strong cross-diffusion parabolic system

19. Thank you! 谢谢！ strong cross-diffusion parabolic system