On nonlinear diffusion equations with \(x\)-dependent convection and absorption (Q1879690)

The \(1+1\)-dimensional nonlinear diffusion equation with \(x\)-dependent convection and source terms \(x_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u)\) is investigated by the methods of group analysis of differential equations. Here \(D(u)\) is the given diffusion coefficient, \(Q(x,u)\) and \(P(x,u)\) are respectively the convection and source terms. The authors formulate conditions which admit the second order generalized conditional symmetries (conditional Lie-Bäcklund symmetries) and first order sign-invariants for this equation. A number of new equations with \(x\)-dependent convection and source terms, in which the diffusion terms offer different possibilities, such as power-law, exponential, Mullin, Fujita and others, are obtained together with construction of their explicit exact solutions with indicated symmetries. remarks: For the case of \(x\)-dependent \(Q\) and \(P\) terms there are comparatively few works on symmetry-group analysis for nonlinear diffusion equation.