This article mainly investigates the existence of global strong solution of a class of fully nonlinear evolution equation and the strong solution of its steady-state equation. By using the T-compulsorily weakly continuous operator theory, the existence of the global strong solution of the fully nonlinear evolution equation is obtained. In addition, based on the acute angle principle, the W 2,p -strong solution for the corresponding stationary equation is also derived. KEYWORDS acute angle principle, fully nonlinear equation, strong solution, T-compulsorily weakly continuous operator theory MSC CLASSIFICATION 35A01; 35D35; 35D30; 47S99 Math Meth Appl Sci. 2020;43:1530-1542. wileyonlinelibrary.com/journal/mma

In this paper, we investigate the well-posedness of a fully nonlinear elliptic problem. By using the acute angle principle for the weakly continuous operator, we obtained the W 2,p -strong solution of the fully nonlinear elliptic problem.

This study mainly aims to explore the existence of global attractor for a modified Swift-Hohenberg equation. The method we use was the classical existence theorem of global attractors and the theory of semigroups. Use this method we prove that the equation exist a global attractor in H 1 2 space, and the global attractor attracts whatever bounded subset of H 1 2 in the H 1 2 -norm.

A dynamic bifurcation analysis on a three-species cooperating model was presented and it was proved that the problem bifurcated an attractor as the parameter λ crossed the critical value λ0. The analysis was based on the attractor bifurcation theory together with the central manifold reduction.

In this paper, the dynamic bifurcation of the three-species cooperating model is considered. It worth noting that the main theory of this paper is the Center manifold reduction and attractor bifurcation theory, which is developed by Ma [1,2]. The main work of this paper shows that if the algebraic multiplicity of the first eigenvalue is 2, there exists an S1 attractor bifurcation, and the number of its singular points can only be eight. Besides, we show that the simplified governing equations bifurcate to an S1 attractor, when the Control parameter λ crosses a critical value λ 0.