Invariant sets and the blow up threshold for coupled systems of reaction–diffusion

Liu, Baiyu; Si, Xinhui

doi:10.1080/00036811.2014.898274

Cited by 1 publication

(5 citation statements)

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“…Combining the comparison principle and the potential well method, Gazzola and Weth analyze the behavior of the solutions for the single parabolic equation when the initial data vary in the phase space

H_{0}^{1} false(normalΩ false)

, especially the initial energy above the mountain pass level. Extending the above corresponding results, we perfect the assumptions about f ( u , v ) and g ( u , v ) in the study of Liu and Si and resolve the problem . At low and critical initial energy level, we introduce a family of potential wells to estimate the longtime behavior of the global solution.…”

Section: Introductionmentioning

confidence: 71%

“…s)ds ≥ 0, which was proposed in the previous study 30 . Compared with the assumption (1), which loses sight of problem (1.4), new assumption (a) not only contains the coupled source terms, which are in problem (1.3) and problem (1.4), but also contains types like f (u, v)…”

Section: Introductionmentioning

confidence: 83%

“…Throughout above papers, the comparison principle plays an important role in studying the reaction‐diffusion systems. However, Liu and Si avoided the comparison principle and used the potential well method to derive a threshold of blow‐up and global existence for solution of problem . Under the conditions of low‐energy level ( J ( u 0 , v 0 ) ≤ d ), the initial data are divided into two invariants, ie, stable set and unstable set .…”

Section: Introductionmentioning

confidence: 99%

“…(1)

f, g \in C^{1} false(R^{2}, double-struckR false)

and

F false(u, v false) = \int_{0}^{u} f false(s, v false) d s = \int_{0}^{v} g false(u, s false) d s \geq 0

, which was proposed in the previous study. Compared with the assumption (1), which loses sight of problem , new assumption (a) not only contains the coupled source terms, which are in problem and problem , but also contains types like f ( u , v ) = | u + v | p − 1 ( u + v ) + | u − v | p − 1 ( u − v ) and g ( u + v ) = | u + v | p − 1 ( u + v ) − | u − v | p − 1 ( u − v ).…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, one of the purpose is to change the assumption of coupled terms in the study of Liu and Si . The new assumption…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Global existence and nonexistence of generalized coupled reaction‐diffusion systems

Niu

2019

Math Methods in App Sciences

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In this paper, we consider the initial boundary value problem for a class of reaction‐diffusion systems with generalized coupled source terms. The assumption on the coupled source terms refers to the single equations and includes many kinds of polynomial growth cases. Under this assumption, the reaction‐diffusion systems have a variational structure, which is the foundation of constructing the potential wells to classify the initial data. In subcritical energy level and critical energy level, which are divided from potential well theory, the global existence solution, blow‐up in finite time solution, and asymptotic behavior of solution are obtained, respectively. Furthermore, we show the sufficient conditions of global well posedness with supercritical energy level by combining with comparison principle and semigroup theory.

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