Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping

<p style='text-indent:20px;'>This paper is concerned with the tempered pullback attractors for 3D incompressible Navier-Stokes model with a double time-delays and a damping term. The delays are in the convective term and external force, which originate from the control in engineer and application. Based on the existence of weak and strong solutions for three dimensional hydrodynamical model with subcritical nonlinearity, we proved the existence of minimal family for pullback attractors with respect to tempered universes for the non-autonomous dynamical systems.</p>

Breakdown of Regularity of Scattering for Mass-Subcritical NLS

We study the scattering problem for the nonlinear Schrödinger equation $i\partial _t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $\Sigma$ holds and the wave operator is well defined on $\Sigma$. We show that there exists $0&lt;\beta &lt;p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta }$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.

Periodic solutions for a fractional asymptotically linear problem

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN

We consider functionals of the formJ(u)=\frac{1}{2}\int_{{\mathbb{R}}^{N}}|\nabla u|^{2}-\int_{{\mathbb{R}}^{N}}b% (x)G(u)on a {C^{1}}-submanifold M of {\mathcal{D}^{1,2}({\mathbb{R}}^{N})}, {N\geq 3}, where G is the primitive of some “zero-mass” nonlinearity g (i.e., {g^{\prime}(0)=0}), and the weight function {b:{\mathbb{R}}^{N}\to{\mathbb{R}}} is merely supposed to belong to {L^{1}({\mathbb{R}}^{N})\cap L^{\frac{2^{*}}{2^{*}-p}}({\mathbb{R}}^{N})} for some {2<p<2^{*}}, and to possess a certain decay behavior. Let V be the subspace of {\mathcal{D}^{1,2}({\mathbb{R}}^{N})} given by {V:=\{v\in\mathcal{D}^{1,2}({\mathbb{R}}^{N}):v\in C({\mathbb{R}}^{N})\mbox{ % with }\sup_{x\in{\mathbb{R}}^{N}}(1+|x|^{N-2})|v(x)|<\infty\}}. We prove that a local minimizer of the constrained functional {J|_{M}} with respect to the V-topology must be a local minimizer with respect to the “bigger” {\mathcal{D}^{1,2}({\mathbb{R}}^{N})}-topology. This result allows us to prove the existence of multiple nontrivial solutions of the zero-mass equation {-\Delta u=b(x)g(u)} in {{\mathbb{R}}^{N}}, where {g:R\to{\mathbb{R}}} is a subcritical nonlinearity, which is superlinear at zero and at {\infty}.

New Super-quadratic Conditions for Asymptotically Periodic Schrödinger Equations

We study the semilinear Schrödinger equationwherefis a superlinear, subcritical nonlinearity. It focuses on the casewhereV(x) =V0(x)+V1(x),V0∊C(RN),V0(x) is 1-periodic in each of x1 , x2 , . . . , xN, supinf, and. A new super-quadratic condition is obtained that is weaker than some well-known results.

Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\ \end{array}\right. \end{array}$$in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.

Uniqueness of solutions to singular p-Laplacian equations with subcritical nonlinearity

We present a geometric approach to the study of quasilinear elliptic p-Laplacian problems on a ball in ${\mathbb{R}^{n}}$ using techniques from dynamical systems. These techniques include a study of the invariant manifolds that arise from the union of the solutions to the elliptic PDE in phase space, as well as variational computations on two vector fields tangent to the invariant manifolds. We show that for a certain class of nonlinearities f with subcritical growth relative to the Sobolev critical exponent ${p^{*}}$, there can be at most one such solution satisfying ${\Delta_{p}u+f(u)=0}$ on a ball with Dirichlet boundary conditions.

Positive solutions for a class of quasilinear problems with critical growth in ℝN

In this paper, we study the existence, multiplicity and concentration of positive solutions for a class of quasilinear problemswhere —Δp is the p-Laplacian operator for is a small parameter, f(u) is a superlinear and subcritical nonlinearity that is continuous in u. Using a variational method, we first prove that for sufficiently small ε > 0 the system has a positive ground state solution uε with some concentration phenomena as ε → 0. Then, by the minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials. Finally, we obtain some sufficient conditions for the non-existence of ground state solutions.