Second-order differential invariants of a family of diffusion equations. (English) Zbl 1078.35051

The nonlinear diffusion equation \(u_t-u_{xx}=f(u,u_x)\) is considered. By means of the Lie criterion of infinitesimal invariance the authors construct the Lie algebra \(L_{\varepsilon}\) of the equivalence transformations. Second order differential invariants with respect to the equivalence transformations of the equation are calculated. On this base the characterization of the subclass of nonlinear diffusion equations is given, which can be linearized through an equivalence transformation.


35K55 Nonlinear parabolic equations
76R50 Diffusion
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
58J70 Invariance and symmetry properties for PDEs on manifolds
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