Advances in Difference Equations
Volume 2010 (2010), Article ID 451619, 11 pages
doi:10.1155/2010/451619
Research Article

Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms

Yasumaro Kobayashi

Faculty of Urban Liberal Arts, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Received 20 April 2010; Accepted 14 October 2010

Academic Editor: Abdelkader Boucherif

Copyright © 2010 Yasumaro Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the quasilinear parabolic equation with inhomogeneous term 𝑢 𝑡 = Δ 𝑢 𝑚 + ⟨ 𝑥 ⟩ 𝜎 𝑢 𝑝 + 𝑓 ( 𝑥 ) , 𝑢 ( 𝑥 , 0 ) = 𝑢 0 ( 𝑥 ) , where 0 ≤ 𝑓 ( 𝑥 ) , 𝑢 0 ( 𝑥 ) ∈ 𝐶 ( ℝ 𝑁 ) , 𝑚 > 0 , 𝑝 > m a x { 1 , 𝑚 } , and 𝜎 > − 2 , ⟨ 𝑥 ⟩ ∶ = ( | 𝑥 | 2 + 1 ) 1 / 2 . In this paper, we investigate the critical exponents of this equation.