In looking for particular solutions to the Kuramoto-Sivashinsky equation the following ordinary differential equation arises: $$ \epsilon w^{\prime\prime\prime} +w^\prime = 1- w^2. $$ In some applications $\epsilon$ is small and a singular perturbation problem arises. In this paper we first show that for each $\epsilon>0$, there exists a unique monotonic solution which satisfies $w(0)=0 $ and $w(\infty)=1$. We also show that this solution is not odd by demonstrating that $w^{\prime\prime}(0) \ne 0$. Finally, we prove that $w^{\prime\prime}0)$ is asymptotically small beyond all orders of $\epsilon$ and we derive the asymptotic formula $w^{\prime\prime}(0) \sim -A \epsilon^{-2} \hbox{exp}\left(\frac{-\pi}{2 \sqrt \epsilon } \right) \rm{as}\, \epsilon \to 0$, where $A$ is some positive constant.

We prove that Problem $(1)$ below possesses infinitely many continua of radial solutions branching off at $\lambda=0$ from the trivial solution, each continuum being characterized by nodal properties. The nonlinearities $h$ and $g$ are neither assumed to be odd, nor required to satisfy any growth restriction. For some classes of problems we also study the global behaviour of the continua.

We consider bifurcation problems for classes of nonlinear ordinary differential equations which arise via the study of radial solutions of quasilinear elliptic partial differential equations similar to equations involving the p-Laplacian.

We study optimal control problems governed by differential systems involving measures as controls and where the state vector is a function of bounded variation. Such problems are defined as extensions to $BV$-spaces of classical problems which do not admit solutions in the class of absolutely continuous functions. We give a definition of generalized solutions for differential systems with measures, for which we prove a stability result for the weak-star topology of measures. We next prove existence of $BV$-solutions for control problems. A relaxation theorem is given: the classical control problem defined for $AC$-functions and the control problem extended to $BV$-spaces have the same value and the solutions of extended problem are cluster points of minimizing sequences for the initial problem. We finally characterize $BV$-solutions of control problems by means of Lipschitz solutions of an auxiliary control problem.

The Liouville-Green (WKB) approximation is generalized to a matrix differential equation with an analytic, symmetric matrix coefficient. Some applications to oscillation problems are given.

We consider the Dirichlet problem for the Schrödinger operator in a half-space with boundary data having an arbitrary growth at infinity. A solution is constructed as the generalized Poisson integral. Uniqueness of the solution is investigated also.

A complete characterization of the long-time behaviour of nonnegative solutions to the problem $$ u_t -\Delta u + u + \sum b_j u^{r_j} - \sum a_i u^{p_i} = 0 , \ u=u(x,t) , \ x \in \rn , \ t>0 $$ $$ 1 <r_j < p_i \leq {N\over N-2}, \ a_i, \ b_j > 0 $$ is given. For any nonnegative compactly supported datum $ \bar u \not\equiv 0 $, there is a constant $ \alpha_c >0 $ such that the solution converges to a ground state for $ u(0)= \alpha_c \bar u $, blows up at a finite time if $ u(0) = \alpha \bar u ,$ $ \alpha > \alpha_c $ and tends to zero provided $ u(0) = \alpha \bar u $ with $ 0 \leq \alpha < \alpha_c $. The proof is based on a combination of the concentrated compactness, comparison theorems and some recent results on uniqueness for the corresponding stationary equation.