期刊
JOURNAL OF DIFFERENTIAL EQUATIONS
卷 336, 期 -, 页码 73-125出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2022.07.010
关键词
Blowup profile; Stability; Semilinear heat equation; Nonlocal equation; Gierer-Meinhart system; Shadow limit model
类别
资金
- International Center for Research and Postgraduate Training in Mathematics -Institute of Mathematics Vietnam Academy of Science and Technology [ICRTM04_2021.05]
In this paper, we consider a nonlocal parabolic PDE and study the construction and asymptotic behavior of the blow-up solution in the critical parameter regime. Using a formal and rigorous approach, we find an approximate solution, linearize the equation, and reduce the problem to a finite-dimensional one. By applying index theory, we solve the finite-dimensional problem and obtain the exact solution to the full problem.
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. Unlike the earlier work in [13] where we consider a subcritical regime of parameters, we focus here on the critical regime, which is much more complicated. Our main result concerns the construction of a blow-up solution with the description of its asymptotic behavior. Our method relies on a formal approach, where we find an approximate solution. Then, adopting a rigorous approach, we linearize the equation around that approximate solution, and reduce the question to a finite dimensional problem. Using an argument based on index theory, we solve that finite-dimensional problem, and derive an exact solution to the full problem. We would like to point out that our constructed solution has a new blowup speed with a log correction term, which makes it different from the speed in the subcritical range of parameters and the standard heat equation. (C) 2022 Elsevier Inc. All rights reserved.
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