Abstract
This paper deals with the properties of positive solutions to a semilinear 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 blowup rate estimate for small weighted nonlocal boundary.
Highlights
We devote our attention to the singularity analysis of the following semilinear parabolic system: ut − Δu vp, vt − Δv uq, x ∈ Ω, t > 0
Most physical settings lead to the default assumption that the functions f x, y, g x, y defined for x ∈ ∂Ω, y ∈ Ω are nonnegative and continuous, and that the initial data u0 x, v0 x ∈ C1 Ω are nonnegative, which are mathematically convenient and currently followed throughout this paper
Proposition 1.1. i All solutions are global if pq ≤ 1, while there exist both global solutions and finite time blowup solutions depending on the size of initial data when pq > 1 (See [4]). ii The asymptotic behavior near the blowup time is characterized by
Summary
We devote our attention to the singularity analysis of the following semilinear parabolic system: ut − Δu vp, vt − Δv uq, x ∈ Ω, t > 0. H There exists a constant 0 < δ < 1, such that Δu0 1 − δ v0p ≥ 0, Δv0 1 − δ uq0 ≥ 0 It seems that such an assumption is necessary to obtain the estimates of type 1.5 or 1.10 unless some additional restrictions on parameters p, q are imposed for the related problem, we refer to the recent work of Matano and Merle 25. By comparing with Proposition 1.1 ii , Theorem 1.4 could be explained as the small perturbation of homogeneous Dirichlet boundary, which leads to the appearance of blowup, does not influence the precise asymptotic behavior of solutions near the blowup time and the blowup rate exponents p 1 / pq − 1 and q 1 / pq − 1 are just determined by the corresponding ODE system ut vp, vt uq.
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.
