A Class of Pattern-Forming Models

Summary.

A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear operator contains a parameter ρ which could be interpreted as a measure of the pattern-forming tendency for the equation. Examples in this class of equations are an integrodifferential equation studied by Goldstein, Muraki, and Petrich and others in an activator-inhibitor context, and a class of fourth-order parabolic PDE's appearing in the literature in various physical connections and investigated rigorously by Coleman, Leizarowitz, Marcus, Mizel, Peletier, Troy, Zaslavskii, and others. The former example reduces to the real Ginzburg-Landau equation when ρ = 0 .

The most complete results, including threshold results for the appearance of globally minimizing patterns and many other properties of the patterns themselves, are given for complex-valued solutions in one space variable. A complete linear stability analysis for all such sinusoidal solutions is also given; it extends the set of stable solutions considerably beyond the global minimizers.

Other results, including threshold results and the existence of large amplitude patterns as well as of bifurcating solutions, are provided for real-valued solutions; these results are relatively independent of the number of space variables. Finally, a slightly different class of evolution equations is given for which no patterned global minimizer exists, but a sequence of patterned solutions exist whose instabilities (if they are unstable) become ever weaker and the fineness of the oscillation becomes ever more pronounced.