Gonda and Gomi (T.Gonda, H.Gomi, Ann. Glaciology, 6 (1985), 222--224) have grown large elongated ice crystals from supersaturated vapor. Theoretically this problem may be recast in a framework similar to that used by Seeger (A.Seeger, Philos. Mag., ser. 7, 44, no 348, (1953) 1--13) for studies of planar crystals. The resulting set of equations is of Stefan type. We also include the Gibbs-Thomson relation on the crystal surface. In order to make this system tractable mathematically we assume that the Wulff crystal is a fixed cylinder. Subsequently we study a weak form of our system. We show local in time existence of solutions assuming that the initial shape is an arbitrary cylinder. We comment on properties of weak solutions.

It is proved that the Hopf--Lax formula provides the unique viscosity solution of the Cauchy problem \begin{align*} u'_t(t,x)+H(u'_x(t,x)) & =0, \qquad(t,x)\in(0,T]\times {\bf R}^n,\\ \lim_{t\downarrow0} u(t,x) & =\varphi(x)\qquad\text{for all $x\in {\bf R}^n$.} \end{align*}

We study the equation $$ (\star)\qquad \dot{x}(t)=- h(x(t-1)) + f(x(t)) \ \text{ for } \ t\ge 0, \ x_{|_{[-1,0]}}=x_0, $$ where $h$ is an odd function defined by $h(y)$ is equal to $a$ if $0 < y <c$, equal to $b$ if $y \ge c$, $a>b>0$ and $c>0$ and $f$ is an odd ${{\mathcal C}}^1$ function such that $\sup |f(x)| <b$. We first consider the equation $\dot{x}(t)=- h(x(t-1))$, corresponding to $f\equiv 0$. We find the admissible shapes of rapidly-oscillating symmetric periodic solutions and we show that these periodic solutions are all unstable. We then extend these results to our general equation $(\star)$.

In this paper, we consider stochastic differential equations with heredity in a Hilbert space. In [14], the existence and uniqueness of a solution of such a class of systems was studied using linear growth and some less-restrictive conditions (Ousgood or Holder type) than the Lipschitz condition on the nonlinear terms. Our objective here is to study the existence problem by dropping even the linear growth condition and instead replacing it by some less restrictive conditions. We prove our existence result by the method of successive approximations and a comparison principle.

In this paper we study the null controllability of the semilinear heat equation studying the properties of the minimum energy that one needs to steer an initial state $x$ in $0$. We prove this is locally Lipschitz, and consequently we obtain the expected optimal feedback law. We also characterize the value function as the unique positive viscosity solution (of the corresponding Hamilton--Jacobi equation with singular final data) which tends to $0$ on admissible trajectories, or as the minimal positive viscosity supersolution.