The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffusion equations☆

By Chengkui Zhong, Meiling Yang, Chunyou Sun. 2006

283 citations recorded by the source corpus.

Abstract In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction–diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L p ( Ω ) , H 0 1 ( Ω ) and H 2 ( Ω ) ∩ H 0 1 ( Ω ) , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L p ( Ω ) , H 0 1 ( Ω ) and H 2 ( Ω ) ∩ H 0 1 ( Ω ) , the results in this part are new and appear to be optimal.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/6289b7e9df86facb5d1b5ca34a78a3d1505a894c

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In Applied Mathematics

1. Bixiang Wang
2. Yangrong Li
3. Kening Lu
4. Anhui Gu
5. Jia Li
6. Yejuan Wang
7. Peter W. Bates
8. Shengfan Zhou

In Computational Theory and Mathematics

1. Bixiang Wang
2. Shengfan Zhou
3. Anhui Gu
4. Yangrong Li
5. Kening Lu
6. Jia Li
7. Peter W. Bates
8. Yejuan Wang

In Control and Systems Engineering

1. Bixiang Wang
2. Shengfan Zhou
3. Anhui Gu
4. Yangrong Li
5. Kening Lu
6. Peter W. Bates
7. Jia Li
8. Yejuan Wang

Citations in the corpus

• Existence of strong solutions and global attractors for the suspension bridge equations (2007, unknown): “A reader interested in further literature on global attractors could see [7–14] and references therein.”
• Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations (2015, methodology): “(a)–(e) can be founded in [25] or [40].”
• Random attractors for stochastic reaction–diffusion equations on unbounded domains (2009, background): “In the case of deterministic equation, by differentiating the reaction-diffusion equation with respect to t, one can prove the existence of (L2(Rn),H1(Rn)) or (L2(D),H1 0 (D)) (D ⊂ Rn is bounded) attractors; see [24, 27, 29, 32] for autonomous equations and [23, 26, 33] for non-autonomous equations.”
• Pullback attractors for a class of non-autonomous nonclassical diffusion equations (2010, unknown): “1 ([18]).”
• Kernel sections and uniform attractors of multi-valued semiprocesses☆ (2007, background): “[13,25,35] The Kuratowski measure of noncompactness k(A) on a complete metric space M satisfies the following properties: (1) k(A) = 0 if and only if A is compact, where A is the closure of A; (2) if A1 ⊂ A2, then k(A1) k(A2); (3) k(A1 ∪ A2) max{k(A1), k(A2)}; (4) k(A) = k(A); (5) if At is a family ”
• Asymptotic regularity for some dissipative equations (2010, background): “The authors in [34] have proven the following dissipative results: Lemma 3.”