Radial equivalence and applications to the qualitative theory for a
class of non-homogeneous reaction-diffusion equations
Some transformations acting on radially symmetric solutions to the
following class of non-homogeneous reaction-diffusion equations
| x | σ 1 ∂ t u = ∆ u m + | x | σ 2
u p , ( x , t ) ∈ R N × ( 0 , ∞ ) , which has been proposed in a number
of previous mathematical works as well as in several physical models,
are introduced. We consider here m≥1, p≥1, N≥1 and
σ 1 , σ 2 real exponents. We apply these transformations in connection
to previous results on the one hand to deduce general qualitative
properties of radially symmetric solutions and on the other hand to
construct self-similar solutions which are expected to be patterns for
the dynamics of the equations, strongly improving the existing theory.
We also introduce mappings between solutions which work in the
semilinear case m=1.