We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u, & x\in\Omega,\,t > 0, \\ \nabla u\cdot\nu(x)=u^p,\qquad &x\in\partial\Omega,\,\,t > 0, \\ u(x,0)=\varphi(x)\ge 0, & x\in\Omega, \end{array} \right. $$ where $N\ge 1$, $p > 1$, $\Omega$ is a smooth domain in ${\bf R}^N$ and $\varphi(x)=O(e^{\lambda d(x)^2})$ as $d(x)\to\infty$ for some $\lambda\ge 0$. Here, $d(x)=\mbox{dist}\,(x,\partial\Omega)$. Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary $\partial\Omega$.

We reconsider the asymptotic behavior as $t \to +\infty$ of the $L^{2}$-norm of solutions to strongly damped wave equations in ${\bf R}^{n}$ with weighted initial data in the $n=1,2$ dimensional case. As an application, we shall apply those results to the analysis of the linearized compressible Navier-Stokes equations in ${\bf R}^{3}$.

We prove local well-posedness for the initial-boundary value problem associated to some quadratic nonlinear Schrödinger equations on the half-line. The results are obtained in the low regularity setting by introducing an analytic family of boundary forcing operators, following the ideas developed in [14].

Using the bounded mountain pass lemma and the Ekeland variational principle, we prove a bounded version of the Pucci-Serrin three critical points result in the intersection of a ball with a wedge in a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems.

In this paper, we classify all regular sign changing~solutions~of $$ \Delta ^2 u=u_+^{p} \,\,\,\mbox {in}\, \, \mathbb R^n\ \ \,\,u_+^{p}\in L^1(\mathbb R^n), $$ where $\Delta ^2$ denotes the biharmonic operator in $\mathbb R^n$, $1 < p\leq \frac{n}{n-4}$ and $n\geq 5$. We prove by using the procedure of moving parallel planes that such solutions are radially symmetric about some point in $\mathbb R^n$. We also present a sup+inf type inequality for regular solutions of the following equation: $$ (-\Delta )^m u=u_+^{p}\,\,\,\mbox{in}\,\,\, \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^n$, $m\geq1$, $n\geq 2m+1$ and $p\in (1,(n+2m)/(n-2m) )$.

We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.

We give a recipe to generate ``nonlocal'' constants of motion for ODE Lagrangian systems and we apply the method to find useful constants of motion which permit to prove global existence and estimates of solutions to dissipative mechanical systems, and to the Lane-Emden equation.