In this paper we obtain a good geometric understanding of the Fučík spectrum of several fourth-order problems. This problem was raised by S. Fučík in [5], where it is asked to give a description of this set. The situation resembles the known cases of second-order equations, where analytic branches bifurcate from the eigenvalues in the diagonal.

A semilinear elliptic equation \[ \Delta u + \{1 + {\varepsilon} k(|x|) \} u^p=0, \quad x \in {\bf R}^n, \] is studied, where $n>2$ and ${\varepsilon}$ is a small parameter. It is known that for $p=(n+2)/(n-2)$ fixed, the structure of radial solutions drastically changes under the perturbation ${\varepsilon} k(|x|)$. In this paper it is shown that such a structural change can be understood in a natural way if the exponent $p$ also is taken as a parameter. The Pohozaev identity plays an important role in the perturbation analysis.

In this paper we will prove the global existence and uniqueness of solutions of the Ricci flow equation on $R^2$ $u_t=\Delta \text{ log }u$, $u>0$, in $R^2\times (0,\infty)$, $u(x,0) =u_0(x)$ for $x\in R^2$, satisfying the inequality $u_t\le u/t$ in $R^2\times (0,\infty)$ and the condition $\liminf_{r\to\infty}$ log $u(x,t)/\text{log }r\ge -2$ uniformly on any compact subset of $(0,\infty)$ as $r=|x|\to\infty$ for any $u_0\not\in L^1(R^2)$, $u_0\ge 0$, satisfying $u_0\in L_{loc}^p(R^2)$ for some $p>1$.

We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in $L^1$. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as $t \to \infty$. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts.

We prove that the Benney-Luke solitary waves, as well as their derivatives, are analytic functions and decay at infinity with an optimal algebraic rate.