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Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations. (English) Zbl 1024.35042
Summary: We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form $f\left(x\right){u}_{t}={\left[g\left(x\right){u}^{n}{u}_{x}\right]}_{x}$. We present a complete classification of Lie symmetries and form-preserving point transformations in the case where $f\left(x\right)=1$ which is equivalent to the original equation. We also introduce certain nonlocal transformations. When $f\left(x\right)={x}^{p}$ and $g\left(x\right)={x}^{q}$ we have the most known form of this class of equations. If certain conditions are satisfied, then this latter equation can be transformed into a constant coefficient equation. It is also proved that the only equations from this class of partial differential equations that admit Lie-Bäcklund symmetries is the well-known nonlinear equation ${u}_{t}={\left[{u}^{-2}{u}_{x}\right]}_{x}$ and an equivalent equation. Finally, two examples of new exact solutions are given.
##### MSC:
 35K55 Nonlinear parabolic equations 58J72 Correspondences and other transformation methods (PDE on manifolds) 58J70 Invariance and symmetry properties 35C05 Solutions of PDE in closed form