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.

Let $\Omega \subset \mathbb {R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$. Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, where $\mu \in {\mathbb {R}}$. We study boundary value problems ($P_\pm$) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$, where $p>1$, $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$. Different critical exponents for the existence of a solution to ($P_\pm$) appear according to concentration of $\nu$. The solvability for problem ($P_+$) was proved in [3, 29] in subcritical ranges for $p$, namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$, we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ($P_+$) in supercritical ranges for $p$, i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ($P_-$) under a smallness assumption on $\nu$.

Gramsch and Lay [8] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$,

\[ \widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), \]

$\widetilde {\sigma }_i$

$f$

$f$

$\widetilde \sigma$

In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.

Let $X,\,Y$ be Banach spaces and fix a linear operator $T \in \mathcal {L}(X,\,Y)$ and ideals $\mathcal {I},\, \mathcal {J}$ on the nonnegative integers. We obtain Silverman–Toeplitz type theorems on matrices $A=(A_{n,k}: n,\,k \in \omega )$ of linear operators in $\mathcal {L}(X,\,Y)$, so that

\[ \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\mathcal{I}\text{-}\lim \boldsymbol{x}) \]

$X$

$\boldsymbol {x}=(x_0,\,x_1,\,\ldots )$

$\mathcal {I}$

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

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

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

$-3$

$0$

$-3$

$0$

We consider two questions on the geometry of Lipschitz-free $p$-spaces $\mathcal {F}_p$, where $0< p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in (0,\,1)$ and $0< p\leq 1$. An upper bound on the Banach–Mazur distance between the spaces $\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and $\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of $p$, $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot |_1)$, where $K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection $\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in $\mathbb {R}^d$ with side length $R>0$, into the Lipschitz-free $p$-space $\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

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 $\Omega$.

In this paper, we derive new differential Harnack estimates of Li–Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form

\[ {\mathscr L}_\phi^{\mathsf a} [w]:= \left[ \frac{\partial}{\partial t} - \mathsf{a}(x,t) - \Delta_\phi \right] w (x,t) = \mathscr G(t, x, w(x,t)), \quad t>0, \]

$({\mathsf k},\, m)$

In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely many small covers.

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$. We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$. Arens regularity of $L_E^1(G)$, the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$), which is not a $\Lambda (1)$-set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard 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 consider Calderón's problem for the connection Laplacian on a real-analytic vector bundle over a manifold with boundary. We prove a uniqueness result for this problem when all geometric data are real-analytic, recovering the topology and geometry of a vector bundle up to a gauge transformation and an isometry of the base manifold.