In this work we study existence and uniqueness of solutions to the abstract framework related to the Kirchhoff model for small deformations of a membrane \[ u'' +M ( \| u \| ^{2} ) Au+ F(u)+Au'=0, \] when $M$ is degenerate and $A,$ $F$ are operators. Furthermore, we obtain the asymptotic behavior, as $t \rightarrow\infty,$ of solutions in a concrete case.

We discuss the solvability of the following strongly nonlinear non-autonomous boundary-value problem: \[ (P) \quad \left \{ \begin{array}{ll} (a(x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in {\mathbb R} & \\ x(-\infty)=\nu^- ,\ \ x(+\infty)= \nu^+ & \end{array} \right . \] with $\nu^-< \nu^+$, where $\Phi:{\mathbb R} \to {\mathbb R}$ is a general increasing homeomorphism, with $\Phi(0)=0$, $a$ is a positive, continuous function and $f$ is a Caratheódory nonlinear function. We provide some sufficient conditions for the solvability of $(P)$ which turn out to be optimal for a large class of problems. In particular, we highlight the role played by the behavior of $f(t,x,\cdot)$ and $\Phi(\cdot)$ as $y\to 0$ related to that of $f(\cdot,x,y)$ as $|t|\to +\infty$. We also show that the dependence on $x$, both of the differential operator and of the right-hand side, does not influence in any way the existence or non-existence of solutions.

Let $(M,g(t))$, $0\le t\le T$, be an n-dimensional complete noncompact manifold, $n\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\frac{{\partial} u}{{\partial} t}=\Delta u-au\log u-qu,$ where $a\in\mathbb R$ is a constant and $q$ is a smooth function on $M\times [0,T]$.

We consider nonlinear Schrödinger-type equations on $S^1$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of their solutions. The key is to derive an iteration bound based on a frequency decomposition of the solution, which is different than the iteration bound first used by Bourgain in [4]. We first look at the NLS equation with nonlinearity of degree $\geq 5$. For $q=5$, Bourgain in [9] derives stronger bounds using different techniques. However, our approach works for higher nonlinearities, where the techniques from [9] don't seem to apply. Furthermore, we study non-integrable modifications of the cubic NLS, among which is the Hartree equation, with sufficiently regular convolution potential. For most of the equations obtained this way, we obtain better bounds than for the other equations, due to the fact that we can use higher modified energies, as in the work of the I-Team [18, 20].

In this paper we are concerned with Boussinesq equations which model heat transport by natural convection inside a viscous incompressible fluid in $\mathbb{R} ^{n} $. We prove the well posedness of mild solutions and existence of self-similar ones in the framework of Morrey spaces. Our results allow us to consider singular and unbounded gravitational fields. We also study the long-time behavior of solutions and obtain existence of a basin of attraction for each self-similar solution. Moreover, conditions on initial data for solutions to vanish at infinity are given.

In this paper the substantiation of the method of partial averaging for fuzzy impulsive differential inclusions is considered. Such variants of the averaging method also lead to the simplification of the initial inclusion and happen to be useful when the average of some mappings does not exist or their presence in the system does not complicate its research. These results generalize the results of [24] for impulsive differential inclusions with Hukuhara derivative and of [18] for fuzzy impulsive differential equations.

We prove that the solution-map $ u_0 \mapsto u $ associated with the KdV equation cannot be continuously extended in $ H^s(\mathbb R) $ for $ s<-1 $. The main ingredients are the well-known Kato smoothing effect for the mKdV equation as well as the Miura transform.

We develop an abstract averaging method for a periodic problem $$\left\{ \begin{array}{ll} \ddot u(t) + \beta \dot u(t) + A u + F(t,u(t)) = 0 , \ \ \ t\in [0,T] \\ u(0)=u(T), \dot u(0) = \dot u(T) , \end{array}\right. $$ where $A:D(A)\to X$ is a self-adjoint operator on a Hilbert space $X$ with compact resolvent and $\alpha>0$ such that $A+\alpha$ is strongly $m$-accretive, $\beta>0$ and $F:\mathbb R\times X^{1/2}\to X$, with $X^{1/2}$ being the fractional space of $A+\alpha$, is compact and $T$-periodic in the first variable. The so-called resonant case is considered, i.e., when ${\mathrm{Ker}}\, A\neq \{0\}$ and $F$ is bounded. By use of the topological degree applied to the translation along trajectories operator, effective conditions guaranteeing the existence of $T$-periodic solutions are given. The general result is applied to a damped hyperbolic partial differential equation and the Landesman-Lazer type criterion is derived.

A model for the simultaneous growth and division of a cell population structured by size is examined. The case considered here is that of asymmetrical cell division when cells are dividing into $\beta_1$ and $\beta_2$ daughter cells at a constant rate and the parameters for growth and mortality are constants. The model has a steady-size distribution solution which satisfies a nonlocal differential equation. The solution is in the form of a Dirichlet series which is shown to be the unique probability density function for the steady-size distribution.