Lie analysis of a $$(2+1)$$-dimensional reaction–diffusion equation with time-dependent diffusion coefficient and arbitrary source

Abstract The paper proposes the analysis of a rather general reaction–diffusion equation in $$(2+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensions with time-dependent coefficients. The novelty brought by our approach refers to the analysis of the equation with the diffusion coefficient as an explicit function of time and the dependent variable. The free derivative term called source will be considered here to be both linear and quadratic in the main variable. The reported results are of twofold importance: (i) for modeling real diffusion phenomena, such as the spread of tumor cells throughout brain tissue; (ii) for investigating mathematical models described by PDEs with variable coefficients in $$(2+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensions, insufficiently studied until now.