Enhanced Group Analysis of Variable Coefficient Semilinear Diffusion Equations with a Power Source

Abstract:
The problem of group classification is solved for the class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form
f(x)u_t=(g(x)u_x)_x+h(x)u^m. This class is a subclass of a wider class of quasi-linear reaction--diffusion equations with power nonlinearities, investigated in [1]. The semilinear class is singular with respect to symmetry properties. This is why it was excluded from consideration in [1].

Solving group classification problem for this class demands a development of new tools. Its generalized extended equivalence group is constructed and is applied to gauge arbitrary elements. The used gauge is multi-step one. At first, we put g=f with usual equivalence transformations. This step is not obvious since the gauge g=1 seems simpler than g=f but this is not the case. The next step is to gauge arbitrary elements with a mapping of the gauged class g=f to another class, having the equivalence group of a simpler structure. Since the case m=2 possesses an extension of the (conditional) equivalence group in comparison with the general values of m, an additional mapping to one more class is necessary in this case. The mappings are not one-to-one but the pre-images of the same equations are equivalent within the initial class. This allows us at first to perform the group classification of the image-classes and then to derive the group classification of the initial class.

The additional equivalence transformations between cases of Lie symmetry extensions are found that gives the group classification in the class under consideration with respect to all point transformations. The structure of the set of admissible transformations in this class is described exhaustively. Wide families of exact solutions are constructed by the classical Lie method or generated from known ones by additional equivalence transformations.

[1] Vaneeva O.O., Johnpillai A.G., Popovych R.O. and Sophocleous C., Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl., 2007, V.330, 1363-1386; math-ph/0605081.

Joint work with Roman Popovych and Christodoulos Sophocleous.