The paper concerns the monotonicity with respect to $\lambda$ of the $L^2$-norm of the branch of positive solutions of the nonlinear eigenvalue problem $$ u''(x)+ g(x, u(x)^2) \ u(x) + \lambda u(x) = 0, \ x\in {\bf R}, \ \lim\limits_{|x|\rightarrow \infty} \ u(x) = 0.$$ For the particular case $ g(x) = p(x) + s^{\sigma},$ with $\sigma > 0$ and $p(x)$ an even function, decreasing for $x > 0$ and with $p(\infty) = 0$, the main theorem implies that the $L^2$-norm decreases as we increase $\lambda$ if $\sigma \leq 2$. It is also shown that this is no longer true if $\sigma > 2$. The result has implications for the orbital stability of standing waves of the nonlinear Schrödinger equation.

Kadomtsev-Petviashvili (KP) equations model weakly nonlinear dispersive waves, which are essentially unidimensional, when weak transverse effects are taken into account. These equations can be formally obtained by asymptotic methods from the Euler equations, but there is no rigorous justification yet. In this paper we consider an intermediate equation (BL) derived first by Benney and Luke. (BL) reduces formally to (KP) if we seek waves propagating essentially in one direction, with a weak variations in time and in the transverse direction measured by a small parameter ${\varepsilon}$. We show rigorously that the $L^2(\mathbb R^2)$-norm of the difference between the amplitude of the wave given by (KP) and the one given by (BL) is of order $\mathcal{O}({\varepsilon}^{3/4})$ during a growing with ${\varepsilon}$ time.

The aim of this paper is to investigate the well-posedness of the Cauchy problem $$ \begin{cases} \frac {\partial u}{\partial t} = \Delta u +V(x)u & \qquad \text {in }\mathbb R^N\times (0,T),\quad N\geq 3,\\ u(x,0) = u_0 (x), & \qquad \text {on }\mathbb R^N \end{cases} $$ where the potential $V$ is defined by $V=V(x):=\lambda/|x|^2$, $0\leq \lambda < (N-2)^2/4$. Roughly speaking, we prove that a sufficient condition for existence and uniqueness of the solution is to restrict the growths of the solution $u$ and of the initial datum $u_0$ as $ \vert x \vert \rightarrow \infty $ (at most like $e^{c|x|^2}$, with $c\in \mathbb R_+$) and near the origin (at most like $k|x|^\alpha$, with $k\in \mathbb R_+$, while $\alpha$ is a parameter depending on $\lambda$). For $\lambda >0$, the solution shall present a lack of regularity in the origin which is due only to the presence of the singular potential.

We consider some nonlinear heat equations with nonlocal reaction term in space, in smoothly bounded domains. We prove that all global solutions are uniformly bounded, in other words that blow-up in infinite time cannot occur.

This paper deals with the study of the nonlinear boundary-value problem with initial data, modelling incompressible water infiltration into a homogeneous, isotropic, unsaturated soil, for nonhomogeneous Dirichlet boundary conditions. For some well-known hydraulic models, Richards' equation that describes the evolution of the volumetric water content has the particularity that the diffusion coefficient blows up at a certain value of the soil moisture. In this paper, for a problem of this type, a result of existence and uniqueness of the solution is proved for the unsaturated flow, and its properties are deduced. Conditions under which saturation occurrence is possible are finally discussed.

We study the initial value problem (IVP) associated to a higher-order nonlinear Schrödinger equation with variable coefficients. Under some regularity on its coefficients we establish local well-posedness for the IVP for data in $H^s(\mathbb R)$, $s\ge1/4$, improving previous results [22]. The main ingredient in our proof is an estimate of the maximal function associated to the linear solution similar to the sharp one obtained for linear solutions of the Schrödinger and Korteweg-de Vries equations.

A regularized long-wave--Kadomtsev-Petviashvili equation of the form $$(u_{t}-u_{xxt}+u_{x}+u^{p}u_{x})_{x}-u_{yy}=0, \tag*{(*)} $$ is considered. It is shown that if $p\geq4$, certain initial data can lead to a solution that blows up in finite time. More precisely, under the above condition the solution cannot remain in the Sobolev class $H^{2}(\mathbb R)$ for all time. Also demonstrated here is the solitary-wave solutions $ u(x, y, t) = \phi_c (x-ct, y) $, which exist if and only if $1\leq p <4 $ and $ c > 1 $, when considered as solutions of the initial-value problem for (*), are nonlinearly unstable to perturbations of the initial data, if ${4\over3} <p <4$ and $ 1 < c < {4p\over 4+p}$.