We state sufficient conditions for the existence of the positive, almost periodic or ergodic solutions of the following neutral integral equation: \begin{equation*} x(t)=\gamma x(t-\sigma )+(1-\gamma )\displaystyle\int_{t-\sigma }^{t}f(s,x(s))ds, \end{equation*} where $0\leq \gamma < 1$ and $f:\mathbb R\times \mathbb R^{+}\rightarrow \mathbb{R} ^{+}$ is a continuous map. We also treat the asymptotically, weakly and pseudo almost periodic solutions. Our results do not need the monotonicity of $f(t,.)$.

In this paper, we study the existence of multiple solutions for operator equations involving homogeneous potential operators. With the help of the Nehari manifold and the fibering method, we prove that some such equations have at least two nonzero solutions. Furthermore, we apply this result to prove the existence of two positive solutions for some quasilinear elliptic problems involving sign-changing weight functions.

In the present note, we deal with small perturbations of an infinite cylinder in three-dimensional Euclidian space. We find minimal disc-type surfaces embedded in the cylinder and intersecting its boundary perpendicularly. The existence and localization of those minimal discs is a consequence of a non-degeneracy condition for the critical points of a functional related to the oscillations of the cylinder from the flat configuration.

In this paper, we consider a nonlinear Timoshenko system, in a bounded domain, where the memory-type damping is acting on a part of the boundary. 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 consider the problem of existence and uniqueness for a class of abstract integral Volterra equations in a Hilbert space $H$ perturbed by an additive noise of L\'evy type, with nonLipschitz nonlinearities. We find the solution in the space of mean square continuous and predictable processes from ${\mathbb R} _+$ into $H$.

We study a degenerate parabolic equation derived from the kinetic theory using R\'enyi-Tsallis entropy with the critical exponent. In this paper, we show the existence of the threshold mass for a solution blowing up in finite time.

We consider the wave equation in a smooth domain subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of memory-delay type on the remainder of the boundary, where a general Borel measure is involved. Under quite weak assumptions on this measure, using the multiplier method and a standard integral inequality, we show the exponential stability of the system. Some examples of measures satisfying our hypotheses are given, recovering and extending some of the results from the literature.

The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves of finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves tends to infinity, their velocities approach the limiting long-wave speed $c_0$. It is also shown that there can be no solitary waves with velocities much greater than $c_0$. Finally, numerical approximations of some periodic traveling waves are presented. It is found that there is a periodic wave of greatest height $\sim 0.642 h_0$. Periodic traveling waves with increasing wavelengths appear to converge to a solitary wave.

We prove that weak supersolutions to equations similar to the evolutionary $p$-Laplace equation have lower semicontinuous representatives. The proof avoids the use of Harnack's inequality and, in particular, the use of parabolic BMO. Moreover, the result gives a new point of view to approaching the continuity of the solutions to a second-order partial differential equation in divergence form.

We are interested in the global well posedness of the axisymmetric Navier-Stokes system with initial data belonging to the critical Besov spaces ${B}_{p, 1}^{1+\frac{3}{p}}$. We obtain uniform estimates of the viscous solutions $(v_\nu)$ with respect to the viscosity in the spirit of the work [2] concerning the axisymmetric Euler equations. We provide also a strong convergence result in the $L^p$ norm of the viscous solutions $(v_\nu)$ to the Eulerian one $v$.

In this work, we obtain a result of global existence for weak solutions of the three-dimensional incompressible Vlasov-Navier-Stokes equations, the coupling being done through a drag force which is linear with respect to the relative velocity of the fluid and particles.