In this paper, we introduce the anisotropic Herz spaces with two variable exponents and establish their block decomposition. Using this decomposition, we obtain some boundedness on the anisotropic Herz spaces with two variable exponents for a class of sublinear operators.

Linear and nonlinear degenerate abstract parabolic equations with variable coefficients are studied. Here the equations and boundary conditions are degenerated on all boundary and contain some parameters. The linear problem is considered on the moving domain. The separability properties of elliptic and parabolic problems and Strichartz type estimates in mixed $L_{\mathbf{p}} $ spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

We investigate the initial value problem for an inhomogeneous nonlinear Schrödinger equation with a combined power nonlinearity. We prove global well-posedness in the defocusing case. In the focusing case, we prove existence of ground state and nonlinear instability of standing waves.

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^*$. Let $T:X\to X^*$ be demicontinuous, quasimonotone and $\alpha$-expansive, and $C: X\to X^*$ be compact such that either (i) $\langle Tx+Cx, x\rangle \geq -d\|x\|$ for all $x\in X$ or (ii) $\langle Tx+Cx, x\rangle \geq-d\|x\|^2$ for all $x\in X$ and some suitable positive constants $\alpha$ and $d.$ New surjectivity results are given for the operator $T+C.$ The results are new even for $C=\{0\}$, which gives a partial positive answer for Nirenberg's problem for demicontinuous, quasimonotone and $\alpha$-expansive mapping. Existence result on the surjectivity of quasimonotone perturbations of multivalued maximal monotone operator is included. The theory is applied to prove existence of generalized solution in $H^{1}_{0}(\Omega)$ of nonlinear elliptic equation of the type \begin{align*} \begin{split} \left\{\begin{array}{cc} -\sum\limits_{i=1}^{N}{\frac{\partial}{\partial x_i} a_i(x, u(x), \nabla u(x))})+G_{\lambda}(x, u(x))=f(x) &\textrm{in $\Omega$}\\ u(x)=0&\textrm{$x\in\partial \Omega$},\\ \end{array}\right. \end{split} \end{align*} where $f\in L^{2}(\Omega)$, $\Omega$ is a nonempty, bounded and open subset of $\mathbb{R}^{N}$ with smooth boundary, $\lambda>0$, $ G_{\lambda}(x, u)=-div (\beta (\nabla u(x)))+\lambda u(x)+a_0(x, u(x), \nabla u(x))+g(x, u(x))$, $\beta: \mathbb{R}^{N}\to\mathbb{R}^{N}$, $a_i: \Omega\times \mathbb{R}\times \mathbb{R}^{N}\to\mathbb{R}$ ($i=0, 1, 2, ..., N$) and $g:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfy certain conditions.

In [4], Kodama, Top, Washio studied the maximality of a family of elliptic curves, mostly of genus 3, over a finite field. They used the Jacobians of the curves and differential forms to obtain their results. In this note, in order to prove the maximality of the curves under study, we use analytical tools, namely character and Jacobsthal sums, together with an important result which says that if a curve is the image of a maximal curve under a rational map, then it is itself maximal. Character sums are suitable for counting the number of points on a curve over a finite field, and their use makes the proofs natural and rather elementary. The norm and trace curves are utilized to construct rational maps to the hyperelliptic curves. As an application, the maximality of a certain hyperelliptic curve is used to find the explicit value of a character sum.

This paper deals with viscosity solutions of Hamilton-Jacobi equations in which the Hamiltonian $H$ is weakly monotone with respect to the zero order term: this leads to non-uniqueness of solutions, even in the class of periodic or almost periodic (briefly a.p.) functions. The lack of uniqueness of a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution and to study its properties. The classes of asymptotically almost periodic (briefly a.a.p.) and pseudo almost periodic (briefly p.a.p.) functions are also considered.

The Egoroff theorem for measurable ${\mathbb X}$-valued functions and operator-valued measures ${\mathbb m}: \Sigma \to L({\mathbb X}, {\mathbb Y})$ is proved, where $\Sigma$ is a $\sigma$-algebra of subsets of $T \neq \emptyset$ and ${\mathbb X}$, ${\mathbb Y}$ are both locally convex spaces.

We prove in this article new fixed point theorems for positive maps having approximative minorant and majorant at $0$ and $\infty$ in specific classes of operators. Then, the new fixed point theorems are used to obtain existence results for positive solutions to boundary value problems involving a generalized $p(t)$-Laplacian operator.