We exhibit a class of convex and nonsymmetric domains $\Omega$ in $\mathbb R^N,$ $N\ge4,$ such that the slightly subcritical problem $$ \begin{cases} -\Delta u=u^{{N+2\over N-2}-{\varepsilon}} & \text{ in $\Omega,$ } \\ u>0 & \text{ in $\Omega,$ } \\ u=0 & \text{ on $\partial\Omega $} \end{cases} $$ does not have any solutions blowing up at more than one point in $\Omega$ as ${\varepsilon}$ goes to zero. Moreover if $\Omega$ is a small perturbation of a convex and symmetric domain, we prove that such a problem has a unique solution provided ${\varepsilon}$ is small enough.

We study the limit at infinity of sonic travelling waves of finite energy in the Gross-Pitaevskii equation in dimension $N \geq 2$ and prove the nonexistence of nonconstant sonic travelling waves of finite energy in dimension two.

We show that the periodic Cauchy problem for a modified Camassa-Holm equation with analytic initial data is analytic in the space variable $x$ for time near zero. By differentiating the equation and the initial condition with respect to $x$ we obtain a sequence of initial-value problems of KdV-type equations. These, written in the form of integral equations, define a mapping on a Banach space whose elements are sequences of functions equipped with a norm expressing the Cauchy estimates in terms of the KdV norms of the components introduced in the works of Bourgain, Kenig, Ponce, Vega, and others. By proving appropriate bilinear estimates we show that this mapping is a contraction, and therefore we obtain a solution whose derivatives in the space variable satisfy the Cauchy estimates.

We study positive $C^1(\bar{\Omega})$ solutions to classes of boundary-value problems of the form \begin{eqnarray*} -\Delta_p u & = & \lambda f(u)\mbox{ in } \Omega \\ u & = & 0 \mbox{ on } \partial \Omega , \end{eqnarray*} where $ \Delta_{p} $ denotes the p-Laplacian operator defined by $$ \Delta_p z:= \mbox{div}(|\nabla z|^{p-2}\nabla z);\, p > 1,\, \lambda > 0 $$ is a parameter and $ \Omega $ is a bounded domain in $ R^{N} $; $ N \geq 2 $ with $\partial \Omega$ of class $ C^{2}$ and connected. (If $N=1$, we assume that $\Omega$ is a bounded open interval.) In particular, we establish existence of three positive solutions for classes of nondecreasing, p-sublinear functions $f$ belonging to $C^1([0,\infty))$. Our proofs are based on sub-supersolution techniques.

We study the dependence of the finite blow-up time with respect to the initial data for solutions of the equation $u_t= \Delta u^m + u^{p}$. We obtain Lipschitz continuity for a certain class of initial data and Hölder regularity for wider classes.

We prove the global existence of continuous solutions of the semilinear parabolic problem $\Delta u- {\partial \over {\partial t}}u+ V u^p =0$ in $\mathbb{R}^n_+\times (0,\infty)$, where ${\mathbb{R}^n_+}$ is a half-space in ${\mathbb{R}^n},\, n\geq 3$ . The potential $V$ is in some functional class ${\mathcal K}^{\infty}$. Our approach uses the Shauder fixed-point theorem.

We study the Schrödinger equation: $iu_t+u_{xx}=0,$ $ x\in {\bf R}_+,$ $ t>0$ with a nonlinear boundary condition $-u_x(0,t)=\vert u(0,t)\vert ^{p-1} u(0,t),$ $ t>0$. We show that if $1 <p <3,$ every solution is global in $H^1({\bf R}_+)$, while if $p\ge 3$, then nonglobal solutions exist.

In this paper, we present some conditions under which asymptotic properties of an evolution family ${{\mathcal{U}}}:={(U(t,s))_{t\geq s\geq 0}}$ persist under perturbations by a family $\mathcal B:=(B(t),D(B(t))_{t\in{{\mathbb{R_+}}}}$ of linear operators on a Banach space $X$ satisfying suitable conditions. Our results concern asymptotic properties like boundedness, periodicity, and asymptotic almost periodicity (even in the sense of Eberlein). An application of the abstract results to nonautonomous partial differential equations with delay is given.

We establish local and global existence results for Dirac-Klein-Gordon equations in one space dimension, employing a null-form estimate and a fixed-point argument.

We make precise a class of functions for which we have the strong convergence of the directional derivative of the rearrangement mapping. We apply this result to give a new approximation of the plasma physics equations in a Stellerator.

By introducing the term $-{\varepsilon} ^2 \Delta p$ into the equation of continuity and an additional Neumann boundary condition for the pressure $p$, a strongly elliptic system is obtained which is a singular perturbation of the Stokes system. We use parameter-dependent Sobolev norms to derive asymptotically precise estimates for solutions to the perturbed problem as ${\varepsilon} \searrow 0$. This results in optimal estimates for the difference between solutions to both problems; such estimates are not available by the usually applied energy methods. Under additional regularity assumptions for the data, for the energy estimates, the order of convergence with respect to ${\varepsilon}$ is improved, and convergence in $H^{s+1}$ and $H^s$ norms is obtained for the velocity and pressure with $s\in [0,3/2)$. We verify the asymptotic precision of the estimates by constructing the boundary layers.

We study the energy decay of a network of vibrating elastic strings where the strings are coupled at a common end in a star-shaped configuration. We prove that the solutions are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data. These estimates depend on the diophantine approximations properties of the lengths of the strings of the network.

We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schrödinger equations bounded in the energy space. The result applies for these equations set in any domain of $\mathbb R^N,$ including the whole space. This also holds for a large class of nonlinearities, thereby extending the results obtained by Hayashi and Ozawa in [9] and by the author in [2].

In this work we study the existence and stability for some neutral partial functional differential equations. We suppose that the linear part is not necessarily densely defined and satisfies the Hille-Yosida condition on a Banach space $X$. The nonlinear term takes its values in a space larger than $X$, namely the extrapolated Favard class corresponding to the extrapolated semigroup corresponding to the linear part. Our approach is based on the theory of the extrapolation spaces.