The author is concerned with the asymptotic stability of traveling wave solutions of integral differential equations arising from synaptically coupled neuronal networks. By using complex analytic functions, he proves that there is no nonzero spectrum of some linear operator $\mathcal L$ in the region Re $\lambda \geq 0$, and $\lambda =0$ is a simple eigenvalue. By applying linearized stability criterion, he shows that the traveling wave solutions are asymptotically stable. Additionally, some explicit analytic functions are found for a scalar integral differential equation.

In this paper it is shown that the Cauchy problem for a fifth order modification of the Camassa-Holm equation is locally well-posed for initial data of arbitrary size in the Sobolev space $H^s(\mathbb{R})$, $s>1/4$, and globally well-posed in $H^1(\mathbb{R})$. The proof is based on appropriate bilinear estimates obtained using Fourier analysis techniques.

We consider the Navier--Stokes equations in~$\mathbb R^3$, in an axisymmetric setting: the data and the solutions only depend on the radial and on the vertical variable. In [7], a unique solution is constructed in a scale invariant function space~$L^2_0$, equivalent to~$L^2$ at finite distance from the vertical axis. We prove here a weak--strong uniqueness result for such solutions associated with data in~$L^2 \cap L^2_0$.

It is well known that the coupled Klein-Gordon-Schrödinger system possesses a compact global attractor into a suitable energy space. We prove the asymptotical smoothing effect for this system, i.e., we prove that the attractor is in fact embedded into a smaller energy space.

This paper deals with a model of solitary waves, in three space dimensions, which are characterized by a topological invariant called charge; these waves behave as relativistic particles. We study the interaction with an electromagnetic field. The Lagrangian density of the system is the sum of three terms: the first is that of the free soliton, the second is the classical Lagrangian density of an electromagnetic field, the third, which is due to the interaction, is chosen so that the electric charge coincides with the topological charge. We prove the existence of a static solution for every fixed value of the charge. The energy functional is strongly unbounded from above, as from below; after a reduction argument, the critical points are found by means of the Principle of Symmetric Criticality.

In this paper we study the existence of periodic solutions of the second order differential equation $$ x''+f(x)x'+n^2x+\varphi(x)=p(t), \quad n\in {{\bf N}} . $$ We assume that the limits $$ \lim_{x\to\pm\infty}\varphi(x)=\varphi(\pm\infty),\quad \lim_{x\to\pm\infty}F(x)=F(\pm\infty)\quad \Big( F(x) =\int_0^xf(u)du \Big) $$ exist and are finite. We prove that the given equation has at least one $2\pi$-periodic solution provided that (for $A=\int_0^{2\pi}p(t)\sin nt dt, B=\int_0^{2\pi}p(t)\cos nt dt$) one of the following conditions is satisfied: $$ 2(\varphi(+\infty)-\varphi(-\infty))>\sqrt{A^2+B^2} $$ $$ 2n(F(+\infty)-F(-\infty))>\sqrt{A^2+B^2} $$ $$ 2(\varphi(+\infty)-\varphi(-\infty))=\sqrt{A^2+B^2}, \quad F(+\infty)\not=F(- \infty) $$ $$ 2n(F(+\infty)-F(-\infty))=\sqrt{A^2+B^2}, \quad \varphi(+\infty)\not=\varphi(- \infty). $$ On the other hand, we prove the non-existence of $2\pi$-periodic solutions provided that the inequality $$ 2(\varphi(+\infty)-\varphi(-\infty))+2n(F(+\infty)-F(-\infty))\leq\sqrt {A^2+B^2} $$ and other conditions hold. We also deal with the existence of $2\pi$-periodic solutions of the equation when $\varphi$ satisfies a one-sided sublinear condition and $F$ is bounded.

In this paper we study a linear elliptic equation having mixed boundary conditions, defined in a connected open set $\Omega $ of $\mathbb{R}^{n}$. We prove a comparison result with a suitable ``symmetrized'' Dirichlet problem which cannot be uniformly elliptic depending on the regularity of $ \partial \Omega $. Regularity results for non-uniformly elliptic equations are also given.