## Abstract

We consider the following reaction-diffusion equation: KS = {(u_{t} = ▽ · (▽u^{m} - u^{q-1} ▽v), x ∈ ℝ^{N}, 0 < t < ∞, 0 = ▽v - v + u, x ∈ ℝ^{N}, 0 < t < ∞, u(x, 0) = u_{0}(x), x ∈ ℝ^{N}, where N ≥ ≥ 1, m ≥ 1, and q ≥ m + 2/N with q > 3/2. In our previous work [14], in the case of m > 1, q ≥ 2, q > m + 2/N, we showed that a solution u to the first equation in (KS) behaves like "the Barenblatt solution" asymptotically as t → ∞, where the Barenblatt solution is well known as the exact solution to u_{t} = Δu^{m} (m > 1). In this paper, we improve the result obtained in [14] and establish the optimal convergence rate for the asymptotic profile. In particular, our new result covers the critical case when We also consider the semilinear case of m = 1 and prove that u behaves like "the heat kernel" asymptotically as t → ∞ when q ≥1 + 2/N. Indiana University Mathematics Journal

Original language | English |
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Pages (from-to) | 1279-1297 |

Number of pages | 19 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)