Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases
Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases
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DOI：
10.1512/iumj.2007.56.2977
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发表时间：
2007-08
影响因子：
1.1
通讯作者：
S. Luckhaus;Y. Sugiyama
中科院分区：
文献类型：
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作者：
S. Luckhaus;Y. Sugiyama
我们考虑下面的反应扩散方程：{ut=V(∇um-u q-1∇v)，x∈RN，0 32.在我们以前的工作[14]中，对于m>；1，q>；2，q>；m+2/N，我们证明了(KS)中第一个方程的解M渐近地表现为t→∞，其中Barenblatt解是众所周知的u t=Δu m(m>；1)的精确解.本文改进了文[14]中的结果，建立了渐近轮廓的最优收敛速度。特别地，我们的新结果涵盖了当Q=m+2Ν时的临界情况。我们还考虑了m=1的半线性情形，证明了当q→∞1+2/N时，u的行为类似于t≥。
We consider the following reaction-diffusion equation: {u t = V (∇u m - u q-1 ∇v), x ∈ R N , 0 3 2. In our previous work [14], in the case of m > 1, q > 2, q > m + 2/N, we showed that a solution M 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 q=m + 2 Ν. 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.