January  2006, 14(1): i-iii. doi: 10.3934/dcds.2006.14.1i



School of Mathematics, University of Minnesota, Minneapolis, MN 55455


Mathematical Institute Tohoku University, 6-3Aoba, Aramaki, Aoba-ku, Sendai-shi, 980-8578

Published  October 2005

Interaction between nonlinearity and diffusion is a fascinating subject. Many interesting phenomena are known to result from such interaction, for example, generation of interfaces and singularities, formation of spatial and temporal patterns, propagation of waves, symmetrization and symmetry-breaking of solutions, and so on. Nonlinear diffusive systems have undergone a thorough investigation aimed at mathematical understanding of the mechanisms behind the phenomena.
The analysis serving this aim uses various classical and newly developed techniques relying on results from the bifurcation theory, singular perturbation theory, variational method, and the theory of finite and infinite dimensional dynamical systems. The qualitative theory of parabolic and elliptic equations has been developing extensively in the last few decades, and a lot of important, interesting and beautiful results have been obtained concerning the dynamics of solutions and qualitative description of steady states.

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