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

Abstract

We study the dynamics of the following porous medium equation with strong absorption [Formula: see text] posed for [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text]. Considering the Cauchy problem with non-negative initial condition [Formula: see text], instantaneous shrinking and localization of supports for the solution [Formula: see text] at any [Formula: see text] 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.

Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source

“…Let us point out that, in strong contrast with the range σ > 2(1 − q)/(m − 1) analyzed in [21] and where the self-similarity exponents were uniquely determined, in the present case we have two free parameters for the shooting technique: both the initial value of the solution at x = 0 and the self-similar exponent β. Thus, in order to have uniqueness, we need to fix this initial value in view of the rescaling (1.8), as explained above.…”

“…More precisely, the analysis performed by Belaud and collaborators [3][4][5], along with the instantaneous shrinking of supports for bounded solutions to Eq. (1.1) proved in [21], shows that, for 0 < σ < σ c , any non-negative solution to Eq. (1.1) with bounded initial condition vanishes in finite time.…”

“…On the contrary, after developing the general theory of well-posedness for Eq. (1.1), we have focused on the range σ > σ c in our previous work [21] and proved that, in the latter, finite time extinction depends strongly on how concentrated is the initial condition in a neighborhood of the origin. More precisely, initial data which are positive in a ball B(0, δ) give rise to solutions with a non-empty positivity set for all times, {x ∈ R N : u(t, x) > 0} = ∅ for all t > 0, (…”

“…The proof of Theorem 1.1 is based on a shooting method with respect to the free exponent β and follows the same strategy as [21,Section 4]. However, a number of preparatory results are proved in a different way and the analysis near the interface requires to be improved in some cases with the help of a phase space analysis.…”

“…where ζ = ξ √ β. Noticing that in the limit β → ∞ the last term in (2.6a) vanishes, we proceed exactly as in the proof of [21,Lemma 4.4] (see also the proof of [36,Theorem 2] from where the idea comes) to conclude that there exists β u > 0 such that (β u , ∞) ⊆ A. We omit here the details as they are totally similar to the ones in the quoted references.…”