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<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

In this paper, we study the long time dynamical behavior of strong damped Kirchhoff-type suspension bridge equations by the decomposing technique of operator semigroup. Under the weaker condition of the nonlinearity, we first verify the asymptotic compactness of solution semigroup in two spaces. Then we use the decomposing technique of operator semigroup to obtain the existence of exponential attractors for the strong damped Kirchhoff-type suspension bridge equations.

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In the article, we aim to investigate the well-posedness of solution and the regularity of the global attractor for the couple stress fluid in saturated porous media with the local thermal non-equilibrium effect. To be more specific, we firstly show the existence and uniqueness of global weak solution to the model by making use of the standard Galerkin method. Second, relying on verifying the uniformly compact condition required, we prove the existence of the global attractor of the model in the space where the weak solution resides. Finally, we improve the regularity of global attractor by uniformly compact condition and obtain the C ∞ attractor for the model.