For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$ converging to zero and initial data $f\in H^s(\mathbb {R}^N)$, $N\geq 2$, we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$, which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$ belongs to Lorentz space ${\ell }^{r,\infty }(\mathbb {N})$, then the a.e. convergence results hold for $s>\min \{\frac {r}{\frac {N+1}{N}r+1},\,\frac {N}{2(N+1)}\}$. Inspired by the work of Lucà-Rogers, we construct a counterexample to show that our a.e. convergence results are sharp (up to endpoints). Our results imply that when $0< r<\frac {N}{N+1}$, there is a gain over the a.e. convergence result from Du-Guth-Li and Du-Zhang, but not when $r\geq \frac {N}{N+1}$, even though we are in the discrete case. Our approach can also be applied to get the a.e. convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $\Omega.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicity and radial solutions, and finally we prove some stability results and discuss some related questions concerning certain approximate solutions in the 2-Dimensional cases.

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

We give a comprehensive study of the 3D Navier–Stokes–Brinkman–Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors

We establish asymptotic formulas for all the eigenvalues of the linearization problem of the Neumann problem for the scalar field equation in a finite interval

\[ \begin{cases} \varepsilon^2u_{xx}-u+u^3=0, & 0< x<1,\\ u_x(0)=u_x(1)=0. \end{cases} \]

$\varepsilon ^2u_{xx}+u-u^3=0$

$-3$

$0$

$-3$

$0$

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.

First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{**}$.

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara–Kadomtsev–Petviashvili equation are locally and globally exponentially stable. The main novelty of this work is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

We consider the system of partial differential equations

\[ \begin{cases} \eta_t - \alpha u_{xxx} - \beta \eta_{xx} = 0 \\ u_t + \eta_x + \beta u_{xx} = 0 \end{cases} \]

\[ \varkappa=\alpha-\beta^2. \]

$\varkappa >0$

$\varkappa \leq 0$

For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.

Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or nonexistence properties, power- and exponential-type Sobolev regularity results, and the boundary behaviour of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$. We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$, $s \in \mathbb {R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$.

In this paper, we give necessary conditions for an $N$-expansive homeomorphism of a compact metric space to be nonchaotic in the Li–Yorke sense. As application we give a partial answer to a conjecture in [2].