Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

We study the dynamics of the following porous medium equation with strong absorption partial derivative(t)u =Delta u(m) - vertical bar x vertical bar(sigma)u(q), posed for (t, x)is an element of(0,infinity) x R-N, with m > 1, q is an element of (0, 1) and sigma > 2(1 - q)/(m - 1). Considering the Cauchy problem with non-negative initial condition u(0) is an element of L-infinity(R-N), instantaneous shrinking and localization of supports for the solution u(t) at any t > 0 are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.