### Abstract

We consider the following reaction-diffusion equation: (KS){ u_{t} = Δ · (Δ_{u}^{m} - u^{q-1} Δ_{v}), x ∈ ℝ^{N}, 0 < t < ∞, 0 = Δ_{v} - v + u, x ∈ ℝ, 0 < t < ∞, (u(x,0}=u_{0}(x), x ∈ ℝ^{N}, where N ≥ 1, m > 1, q ≥ max{m + 2/N, 2}. In [Sugiyama, Nonlinear Anal. 63 (2005) 1051-1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of q ≥ max{m + 2/N, 2}, the above problem (KS) is solvable globally in time for "small L^{N(q-m)/2} data". Moreover, the decay of the solution (u, v) in L^{p}(ℝ^{N}) was proved. In this paper, we consider the case of "q ≥ max{m+ 2/N, 2} and small L^{ℓ} data" with any fixed ℓ ≥ ^{N(q-m)/2} and show that (i) there exists a time global solution (u, v) of (KS) and it decays to 0 as t tends to ∞ and (ii) a solution u of the first equation in (KS) behaves like the Barenblatt solution asymptotically as t tends to ∞, where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation u_{t} = Δ_{u}^{m} with m > 1.

Original language | English |
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Pages (from-to) | 597-621 |

Number of pages | 25 |

Journal | Mathematical Modelling and Numerical Analysis |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 28 2006 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Modelling and Simulation
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematical Modelling and Numerical Analysis*,

*40*(3), 597-621. https://doi.org/10.1051/m2an:2006025