We construct self-similar focusing solutions for the nonlinear parabolic equation $$ u_t=\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u), $$ usually called the (evolution) $p$-Laplacian equation. We take the parameter $p>2$, so that finite propagation holds and free boundaries occur. We consider the problem posed in the whole space $R^n$ and work with nonnegative solutions. A self-similar solution is a solution $ u(x,t)$ that preserves its shape in time up to scaling. By focusing we mean that the solution vanishes for, say, $t<0$, in a ball of radius $s(t)$ centered at the origin and as $t\to 0$ we get $s(t)\to 0$, so that the hole disappears at $t=0$. We have $s(t)=c\,(-t)^\nu$ and the main point is the calculation of the anomalous exponent $\nu=\nu(n,p)$. The behaviour of the solution near $(0,0)$ is important in the regularity theory because when focusing occurs in dimension greater than $1$ a singularity appears. Hence, focusing solutions supply concrete bounds for the regularity of general solutions of the equation, which are presumably optimal. A characteristic of the focusing problem for the PLE is the fact that $\nu$ is not a monotone function of $p$ so that minimal regularity at the focusing is obtained for an intermediate $p$ in the interval $(2,\infty)$.

Using a continuation theorem dealing with nonlinear equations in absence of a priori bounds, we prove the existence of infinitely many radially symmetric solutions, with prescribed nodal properties, for a Dirichlet problem having superlinear growth and involving a non homogeneous $p$-Laplacian-like operator.

We prove the existence of a bounded positive solution of a semilinear elliptic equation involving a supercritical exponent. Using the Lusternik-Schnirelman theory of critical points we also obtain the existence of multiple solutions.

The existence of the global attractor for a weakly damped, forced Korteweg-de Vries equation is derived in higher-order Sobolev spaces. The main result is the regularity of the global attractor, which is proved to be as regular as the forcing term. This result is attained through a nonobvious decomposition of the solutions used to overcome the lack of a suitable regularization of the linear part of the equation with respect to either the initial condition or the nonhomogeneous term.

By a classical Sturm-Liouville problem, we mean a two-point boundary value problem of the form \begin{equation} -y''+q(x)y = \lambda y,\quad 0\leq x\leq\pi, \tag{0.1} \end{equation} \begin{equation} y'(0) - hy(0) = 0, \tag{0.2} \end{equation} \begin{equation} y'(\pi)+Hy(\pi) = 0, \tag{0.3} \end{equation} where $q(x)$ is a real continuous function, $h$ and $H$ are real numbers, and $\lambda$ is an eigenvalue. The isospectrality problem is that of describing all problems of the form (0.1)--(0.2)--(0.3) that have the same spectrum. This problem has been studied in detail by Trubowitz et al. in three papers ([1], [2], [3]). Some of the main results of our paper coincide with results in their papers, but the methods are completely different. Our main tools are the Gelfand-Levitan integral equation and transmutation (transformation) operators. The results in the papers [1]--[3] are obtained under the assumption that the potential $q(x)$ belongs to $L^2(0,\pi).$ Instead of this we suppose that the first derivative of $q(x)$ belongs to $L^2(0,\pi).$ This assumption facilitates the proof of the differentiability of the function $\mathcal{F}(x,y)$ (see below) but we omit the proof of differentiability. We hope that this paper will also be useful in studying the cited papers of Trubowitz et al\. and the book [4].