The structure of the blow-up in finite time of a solution of the Generalized Korteweg-de Vries equation arising from a perturbed unstable solitary wave is studied numerically. The computed solution is observed to blow-up in the $L^\infty$-norm in finite time by forming a spike of infinite height at $x=x^*$ and at $t=t^*$. Scaled coordinates are introduced to examine the detailed structure of the solution in the immediate neighborhood of the blow-up. The appropriately rescaled solution is observed to converge in these coordinates as $t\to{t^*}^-$, indicating self-similar behavior. A best-fit solution $w(\xi)$ of the nonlinear ODE satisfied by self-similar profiles is computed for the statistical data compiled from this convergence. The asymptotics at $\pm\infty$ of this solution of the ODE are studied, and found to coincide with those of solutions $w_\pm(\xi)$ of the linearized ODE as $\pm\xi\to\infty$. The self-similar part of the solution is also matched (numerically) to the part of the solution more removed from the blow-up point, showing how rapidly decaying initial data can give rise to self-similar blow-up. Heuristic explanations of how nonlinearity and dispersion cooperate to yield existence of a solution $w(\xi)$ of the ODE with the desired asymptotics as $\pm\xi\to\infty$ are discussed.

We study the uniqueness of positive radial solutions to the Dirichlet boundary value problem for the semilinear elliptic equation $\Delta u+f(|x|,u)=0$ in a finite ball or annulus in $R^n$, $n\ge 3$. Applying our main results to the cases when $f$ is independent of $t$, or $f$ is of the form $K(t)u^p$, we can establish some earlier known results and obtain some new results in an easier and unified approach.

For an eigenvalue problem with exponential nonlinearities we give necessary and sufficient conditions for the existence of a periodic solution. Multiplicity is also discussed.

We consider the dynamical von K\'arm\'an system describing the nonlinear vibrations of a thin plate. We take into account thermal effects as well as a rotational inertia term in the system. Our main result states that the total energy of the system, $E (t)$, satisfies the following estimate: there exist $C>0$ and $\omega > 0$ such that $$ E (t) \le C e^{-\frac{\omega}{1+R^2}\; t} E(0)\qquad \hbox{ as } \qquad t \to +\infty $$ provided $E (0) \le R$ and this for any $R>0$. The result is proved by constructing a Lyapunov function which is a suitable perturbation of the energy of the system that satisfies a differential inequality leading to this decay estimate.

We establish convergence results for sequences of multivalued linear operators $A_n$. When each $A_n$ generates a differentiable semigroup, we state conditions for convergence of the corresponding semigroups and of the solutions of linear differential inclusions governed by $A_n$. Through a method already considered in the literature, this enables us to prove new convergence properties for a sequence of degenerate evolution equations. Applications to partial differential problems are also given.