Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction

✍ Scribed by Shingo Takeuchi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 465 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00304-2

✦ Synopsis

Global existence and asymptotic behavior of solutions for degenerate parabolic equations including (u_{t}=\lambda \operatorname{div}\left(|\nabla u|^{p-2} \nabla u\right)+|u|^{q-2} u\left(1-|u|^{\gamma}\right)) are studied, where (\lambda) is a positive parameter; (p>2, q \geq 2) and (r>0) are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if (p \geq q) and stable if (p<q). For the latter case, some remarks on the attractivity of maximal stationary solution are also given.