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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 $$f(x)u_t=[g(x)u^nu_x]_x$$. We present a complete classification of Lie symmetries and form-preserving point transformations in the case where $$f(x)=1$$ which is equivalent to the original equation. We also introduce certain nonlocal transformations. When $$f(x)=x^p$$ and $$g(x)=x^q$$ 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 $$u_t=[u^{-2}u_x]_x$$ and an equivalent equation. Finally, two examples of new exact solutions are given.

##### MSC:
 35K55 Nonlinear parabolic equations 58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds 58J70 Invariance and symmetry properties for PDEs on manifolds 35C05 Solutions to PDEs in closed form
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