Abstract
We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.
Highlights
We deal with the following degenerate parabolic system: ut Δum[1] up1 vq[1 ], vt Δvm[2] vp2 uq[2 ], x ∈ Ω, t > 0
∞, 1.4 while we say that u x, t, v x, t exists globally if sup u ·, t L∞ Ω v ·, t L∞ Ω < ∞ for any T ∈ 0, ∞
The system turns out to be degenerate if mi > 1 i 1, 2 ; for example, in 12, 13, Galaktionov et al studied the following degenerate parabolic equations: ut Δum[1] vq1, vt Δvm[2] up2, x, t ∈ Ω × 0, T, u x, t v x, t 0, x, t ∈ ∂Ω × 0, T, 1.6 u x, 0 u0 x, v x, 0 v0 x, x ∈ Ω
Summary
We deal with the following degenerate parabolic system: ut Δum[1] up vq[1 ], vt Δvm[2] vp uq[2 ], x ∈ Ω, t > 0. There are a number of important phenomena modeled by parabolic equations coupled with nonlocal boundary condition of form 1.2 In this case, the solution could be used to describe the entropy per volume of the material see 21–23. U x, 0 u0, v x, 0 v0, x ∈ Ω, where p and q are positive parameters They gave the criteria for finite time blowup or global existence, and established blow-up rate estimate.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
