The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(\mathbb{R}^n)$ and its application to self-similar solutions

• Autores: Hideo Kozono, Yoshie Sugiyama
• Localización: Indiana university mathematics journal, ISSN 0022-2518, Vol. 57, Nº 4, 2008, págs. 1467-1500
• Idioma: inglés
• DOI: 10.1512/iumj.2008.57.3316
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• Resumen

  • We shall show the existence of a global strong solution to the semilinear Keller-Segel system in Rn , n=3 of parabolic-parabolic type with small initial data u0?Lwn/2(Rn) and v0?BMO . Our method is based on the perturbation of linearization together with the Lp-Lq -estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.