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Properties of the fast diffusion equation and its multidimensional exact solutions. (Russian, English) Zbl 1035.35057

Sib. Mat. Zh. 44, No. 4, 862-869 (2003); translation in Sib. Math. J. 44, No. 4, 680-685 (2003).
The author studies invariant transformations of the fast diffusion equation \[ u_t = \Delta\ln u,\quad u = u(x,y,t), \] which arises in many important natural phenomena such as the spreading hyperfine film and others. It is well known that this equation admits an infinite dimensional algebra of point symmetries which manifests a peculiarity of this kind of equations.
The author starts with proving the invariance of the equation under certain harmonic transformation, obtaining the one-dimensional reduced equation with respect to the spatial variable. From the knowledge of the symmetry, new exact multidimensional solutions are derived. The solutions obtained depend on arbitrary harmonic functions. In particular, the author exposes new exact solutions of the Liouville equation which a stationary analog of a fast diffusion equation with a source. Finally, the author discusses an expansion of the approach presented on quasilinear parabolic systems.

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35C05 Solutions to PDEs in closed form
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