This paper contains a construction of a scale of almost periodic functions spaces, extending from the space of functions representable as sums of absolutely convergent series of complex exponentials, up to the space of Besicovitch, the largest for which the Parseval equality holds. Instead of using integral norms, that have been used in constructing almost periodic functions (Stepanov, Weyl, Besicovitch), one uses Minkowski's norms in the linear space of trigonometric polynomials, then completes the last space, endowed with Minkowski's norms with various indices between 1 and 2.

We consider a parabolic-elliptic system of drift-diffusion type in the entire two-dimensional Euclidean space, modeling chemotaxis and self-attracting particles. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, we give global existence and decay estimates of nonnegative solutions to the Cauchy problem for this system.

In this paper we consider the von Kármán plate system subject to boundary conditions of memory type. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore generalizes and improves earlier results in the literature.

In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of the cubic nonlinear Schrödinger equation in $H^s$ for every $s>\frac{2}{3}$ on such manifolds.

In this article, we consider the nonlinear Schrödinger equation \begin{equation} -\Delta u + V(x)u=|u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N. \end{equation} Here $V$ is invariant under an orthogonal involution. The basic tool employed here is the concentration--compactness principle. A theorem on existence of a solution which changes sign exactly once is given.

We study quasilinear degenerate singular elliptic equations of the type $ -\Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t} $ in a smooth bounded domain $\Omega$ in ${{\mathbb R}^{N}}={{\mathbb R}^{k}}\times{\mathbb R}^{N-k}$, $x=(y,z)\in{{\mathbb R}^{k}}\times{\mathbb R}^{N-k}$, $2 \leq k<N$ and $N\geq 3$, $1<p<2$, $0\leq s\leq p$, $0\leq t\leq s$ and $p^*(s)=\frac{p(n-s)}{n-p}$. We study existence of solutions for $t<s$, non-existence in a star-shaped domain for $t=s$ and $s<k\big(\frac{p-1}{p}\big)$. We also show that the solution is in $C^{1, {\alpha}}(\Omega)$ for some $0< {\alpha}<1$ provided $t<\frac{k}{N}\big(\frac{p-1}{p}\big)$. The regularity of the solution can be improved to the class $W^{2,p}(\Omega)$ when $t<k(\frac{p-1}{p})$. We also study some properties of the set of degeneracy in a cylindrically symmetric domain using the method of symmetry.

We study the existence of solutions for the equation $u_{tt}-\Delta_{p(x)} u - \Delta u_{t} + g(u) = f(x,t), \; x \in \Omega$ (bounded) $ \subset \mathbb R^n, \; t>0$ in both the isotropic case $(p(x) \equiv p$, a constant) and the anisotropic case $(p(x)$ a measurable function). Furthermore, in the isotropic case we obtain results concerning the asymptotic behavior of solutions. Since uniqueness for this type of problem seems rather difficult, a method implementing generalized semiflows is being used to prove the existence of a global attractor in the phase space $W_0^{1,p}(\Omega)\times L^2(\Omega)$, when $p\geq n$.

We study a nonlinear diffusive predator-prey model with modified Leslie-Gower and Holling-type II functional responses. Making use of global bifurcation theory, we obtain two sufficient conditions for the existence of positive solutions and describe the coexistence region $R$. Moreover, we find that the coexistence region $R$ spreads as $\beta$ increases and narrows for large $\alpha$. At last, we derive the nonlinear effect of large $\beta$ on bifurcation structures in the special case of $\alpha=0$. Some a priori estimates for positive solutions will play an important role in the proof.