Let $\Omega\subset\mathbb R^2$ be a bounded domain, $\gamma\in C^{3,\alpha}(\partial\Omega;\mathbb R^3 )$ $(0 <\alpha <1)$ and $H>0$. We consider the asymptotic behavior of large solutions of an $H$-system $$\Delta u=2Hu_{x_1}\wedge u_{x_2}\quad\text{in $\Omega$}, \qquad u=\gamma\quad \text{on $\partial\Omega$}$$ as $H\to 0$ or as $\gamma\to 0$. We show that large solutions blow up at exactly one point in $\Omega$. The exact blow-up rate and location of blow-up point are studied.

This paper is concerned with first-order time-dependent Hamilton-Jacobi equations. Exploiting some ideas of Barron and Jensen [9], we derive lower-bound estimates for the gradient of a locally Lipschitz-continuous viscosity solution $u$ of equations with a convex Hamiltonian. Using these estimates in the context of the level-set approach to front propagation, we investigate the regularity properties of the propagating front of $u$, namely $\Gamma_t = \{ x \in \mathbb R^n : u(x,t)=0 \}$ for $t \geq 0$. We show that, contrary to the smooth case, such estimates do not guarantee, in general, any expected regularity for $\Gamma_t$ even if $u$ is semiconcave.

As a first step, we consider an evolution linear problem, the symbol of which is a real polynomial of degree three with time-dependent coefficients. We get for this problem smoothing effects known when these coefficients are constant. In particular, by using the theory of Calderón-Zygmund operators and the David and Journé T1 Theorem, we establish a local smoothing effect on the solution of the linear problem. In a second step, we study a nonlinear dispersive equation the linear part of which is the one studied above. We use the previous smoothing properties and a regularization method to establish that the Cauchy problem is locally well-posed in the Sobolev spaces $H^s(\mathbb R)$ for $s>3/4$.

The notion of semigroups of Lipschitz operators is introduced and a characterization of continuous infinitesimal generators of such semigroups is obtained. An application of the abstract theory obtained here is given to the Cauchy problem for quasi-linear wave equation with damping.