## Abstract

We study the long-time behavior of nonnegative solutions to the Cauchy problem {ρ(x) ∂^{t}u = Δ_{u}^{m} in Q := ℝ^{N} x R_{+} u(x, 0) = u_{o} in ℝ^{N} in dimensions N ≥ 3. We assume that m > 1 and ρ(x) is positive and bounded with ρ(x) ≤ Clxl^{-γ} as |x| → ∞ with γ ≥ 2. The initial data uo we nonnegative and have finite energy, i. e., f ρ(x)u_{o} dx < ∞. We show that in this case nontrivial solutions to the problem have a longtime universal behavior in separate variables of the form u(x,t)∼i^{∼1/(m-1)}W(x), where V = W^{m} is the unique bounded, positive solution of the sublinear elliptic equation-ΔV = cp(x)V^{1}/^{m} in ℝ^{N} vanishing as |x| → ∞; c = l/(m - 1). Such a behavior of u is typical of Dirichlet problems on bounded domains with zero boundary data. It strongly departs from the behavior in the case of slowly decaying densities, ρ(x) ∼ |x|^{-γ} as |x| → ∞ with 0 ≤ γ ≤ 2, previously studied by the authors. If ρ(x) has an intermediate decay, ρ ∼ |x|^{-γ} as |x| → ∞ with 2 < γ < γ2 := N - (N - 2)/m, solutions still enjoy the finite propagation property (as in the case of lower γ). In this range a more precise description may be given at the diffusive scale in terms of source-type solutions U(x, t) of the related singular equation |x| ^{-γ}u_{t} = Δu^{m}. Thus in this range we have two different space-time scales in which the behavior of solutions is non-trivial. The corresponding results complement each other and agree in the intermediate region where both apply, thus providing an example of matched asymptotics.

Original language | English |
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Pages (from-to) | 521-549 |

Number of pages | 29 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2010 |

## Keywords

- Inhomogeneous porous media
- Matched asymptotics
- Scaling methods