A group classification of heat equations with nonlinear source (Q2740282)

The authors describe a new approach to group classification of the equation NEWLINE\[NEWLINEu_t=u_{xx}+F(t,x,u,u_x)\tag{1}NEWLINE\]NEWLINE with an arbitrary function \(F:\mathbb{R}^4 \mapsto \mathbb{R}\) (arbitrary element).NEWLINENEWLINENEWLINEFirst the most general form of infinitesimal operators which generates the invariance group of (1) is found. Then the authors announce the result from which it follows that there exists 28 non-equivalent realizations of 3-dimensional invariance Lie algebras of the equation (1) with arbitrary element depending on a single argument. Finally the list of 12 non-equivalent realizations of 4-dimensional invariance Lie algebras and corresponding invariant equations are given. The authors also analyze the problem of completeness of the classification carried out.