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In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

$ \nu $u_{tt}$+u_t=$Δ$u+f(u)+$ε$\dot{W}$

on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu, $ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation

$u_t=$Δ$u+f(u).$

is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.

This work is concerned with the solvability of a Navier-Stokes/*Q*-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system $u_t=Δ u-\nabla (u^α \nabla v)$, $v_t=Δ v-v+u$ under non-flux boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^{n}$. The case of $α≥ \max\{1,\frac{2}{n}\}$ with $n≥1$ was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case $α∈(\frac{2}{3},1)$ that if $\|u_0\|_{L^\frac{nα}{2}(Ω)}$ and $\|\nabla v_0\|_{L^{nα}(Ω)}$ are small enough with $n≥q3$, then the solutions are globally bounded with both $u$ and $v$ decaying to the same constant steady state $\bar{u}_0=\frac{1}{|Ω|}∈t_Ω u_0(x) dx$ exponentially in the $L^∞$-norm as $t? ∞$. Moreover, the above conclusions still hold for all $α≥q2$ and $n≥q1$, provided $\|u_0\|_{L^{nα-n}(Ω)}$ and $\|\nabla v_0\|_{L^{∞}(Ω)}$ sufficiently small.

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