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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