\(Q\)-conditional symmetry of a nonlinear two-dimensional heat-conduction equation (Q2740350)

This paper deals with the following equationNEWLINE\[NEWLINEH(u)u_t+\Delta u=F(u)\tag{1}NEWLINE\]NEWLINE where \(H(u), F(u)\) are nonlinear functions, \(\Delta u=\partial^2u/\partial x_1^2+\partial^2u/\partial x_2^2\). The operator NEWLINE\[NEWLINEQ=A(x,u)\partial_t+B^a(x,u)\partial _{x_a}+C(x,u)\partial_u\tag{2}NEWLINE\]NEWLINE generates \(Q\)-conditional symmetry of (1) if this equation is compatible with the following one: \(A(x,u)u_t+B^a(x,u)u_{x_a}=C(x,u)\) (see the book of \textit{W. I. Fushchich, W. M. Shtelen} and \textit{N. I. Serov} [Symmetry analysis and exact solutions of equations of nonlinear mathematical physics, Mathematics and its Applications, 246, Dordrecht: Kluwer Academic Publishers (1993; Zbl 0838.58043)]). Compatibility conditions involving coefficients \(H, B^a\), and \(C\) are found in the cases where \(A=1\), \(A=0\) and \(B^1=1\).NEWLINENEWLINENEWLINENext the compatibility of (1) with the pair of operators of the form (2) is analyzed. E.g., the case where \(Q_1=\alpha(u)\partial_t+\partial _{x_1}+G(u)\partial_u, Q_2=\partial_t+\partial _{x_2}\) is studied. NEWLINENEWLINENEWLINEThe authors construct Ansätze which reduce (1) to ODEs and find several new exact solutions.