We discuss the life span of the Cauchy problem for the one-dimensional Schrödinger equation with a single power nonlinearity $\lambda |u|^{p-1}u$ ($\lambda\in\mathbb{C}$, $2\le p < 3$) and initial data of the form $\varepsilon\varphi$ prescribed. Here, $\varepsilon$ stands for the size of the data. It is not difficult to see that the life span $T(\varepsilon)$ is estimated by $C_0 \varepsilon^{-2(p-1)/(3-p)}$ from below, provided $\varepsilon$ is sufficiently small. In this paper, we consider a more precise estimate for $T(\varepsilon)$ and we prove that $\liminf_{\varepsilon\to 0}\varepsilon^{2(p-1)/(3-p)}T(\varepsilon)$ is larger than some positive constant expressed only by $p$, $\mathrm{Im}\lambda$ and $\varphi$.

We are interested in studying semilinear Cauchy problems in which the closed linear operator is not Hille-Yosida and its domain is not densely defined. Using integrated semigroup theory, we study the positivity of solutions to the semilinear problem, the Lipschitz perturbation of the problem, differentiability of the solutions with respect to the state variable, time differentiability of the solutions, and the stability of equilibria. The obtained results can be used to study several types of differential equations, including delay differential equations, age-structure models in population dynamics, and evolution equations with nonlinear boundary conditions.

We study the asymptotic behavior of nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a supercritical nonlinearity. It is known that there are initial data such that the corresponding solution decays to zero with an algebraic rate. Furthermore, any algebraic rate which is slower than the self-similar rate occurs as decay rate for some solution. In this paper we prove that the convergence to zero can take place with an "arbitrarily" slow rate, if the initial data are chosen properly.

We study the Dirichlet problem for fully nonlinear elliptic operators: $G(D^2u, \nabla u, u, x) = f(x)$ in $\Omega$, where $\Omega$ is a bounded regular domain, and $f$ is continuous. We prove the existence, the nonexistence and the multiplicity of solutions for some particular right-hand side $f$ when $G$ has its two principal eigenvalues of different sign.

In this paper we extend some existence results concerning generalized eigenvalues for fully nonlinear operators, singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the $p$-Laplace operator by Del Pino and Manasevich.

We consider radial and positive solutions to a parabolic-elliptic system in $\mathbf{R}^2$. This system was introduced as a simplified version of the Keller-Segel model. The system has the critical value of the total mass. If the total mass of a solution is more than the critical value, the solution blows up in finite time. If the total mass of a solution is less than the critical value, the solution exists globally in time. Recently, some properties of solutions whose total mass is equal to the critical value have been investigated. In this paper, we construct a grow-up solution whose total mass is equal to the critical value. Furthermore, we show that the grow-up rate of the solution is equal to $O((\log t)^2)$.

A problem of dynamical reconstruction of boundary controls in a nonlinear parabolic equation is considered. In the case when a control is concentrated in the Neumann boundary conditions, a solving algorithm which is stable with respect to informational noises and computational errors is described. The algorithm is based on the ideas of the theory of feedback control.