We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems

\dot{x}=y, \quad \dot{y}=(ax+b)y+cx^2+dx+e,

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

In this article, we prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to $\Gamma ^{0,{\rm Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $\Gamma ^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents:

\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]

$\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝn, g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$, b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

This work, motivated by the rapid developments in Micro-Electro-Mechanical Systems (MEMS) structures, especially actuators and grippers, analyses the dynamics of a thermo-mechanical system consisting of a horizontal beam joined at one end to a vertical rod. As a result of thermal expansion or vibration of the rod, the other end may come into contact with another part of the MEMS device and that closes an electrical circuit, which is the actuating or switching function of such a beam–rod system. The interaction between the rod's contacting end and the supporting device is described by a normal compliance contact law for the displacements and by an inclusion-type Barber's heat exchange condition for the temperature. The heat-exchange coefficient is a multi-function taking into account the air resistance in the gap when there is no contact and the contact pressure when contact occurs. The model consists of a nonlinear variational inclusion for the temperature coupled with a nonlinear variational equation for the displacements. The existence of a weak solution to the problem is proved by using the Galerkin method, a regularization of Barber's condition with the Yosida approximation of a maximal monotone operator, and a priori estimates.

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a Lp–Ls type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).

The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

In the present paper, we consider the following Kirchhoff type problem

$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$

$V\in C(\mathbb R^3,\mathbb R)$

$\inf _{\mathbb R^N} V=0$

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds Gk(ℝn) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝn. In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of Gk(ℝn) as a function of n.

We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

For a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$, t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.