The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation

We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the precise final asymptotic profile, under the form $$u_y(x,y,T) \sim d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{as $(x,y)\to (0,0)$.}$$ Interestingly, this result displays a new phenomenon of strong anisotropy of the profile, quite different to what is observed in other blowup problems for nonlinear parabolic equations, with the exponents $1/(p-1)$ in the normal direction $y$ and $2/(p-2)$ in the tangential direction $x$. Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation, whereas the normal profile remains self-similar.Comment: Int. Math. Res. Not. IMRN, to appea

Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with pp-Laplacian diffusion

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta_p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion ($p=2$). The analysis in the case $p>2$ is considerably more delicate

Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile, which had long remained open, was answered negatively by Guo and Souplet for the case of interior points, and we then showed that touchdown can actually be ruled out in subregions of the domain where the permittivity is positive but suitably small. The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio between f and its maximum, below which no touchdown occurs outside of the bumps. The ratio is rigorously obtained as the solution of a suitable finite-dimensional optimization problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio are not "small" but actually up to the order 0.3, which could hence be quite appropriate for robust use in practical MEMS design. The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a multi-parameter family of refined, nonlinear auxiliary functions with cut-off

Excluding blowup at zero points of the potential by means of Liouville-type theorems

We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation $u_t= \Delta u+V(x)f(u)$, we rule out the possibility of blowup at zero points of the potential $V$ for monotone in time solutions when $f(u)\sim u^p$ for large $u$, both in the Sobolev subcritical case and in the radial case. This solves a problem left open in previous work on the subject. Suitable Liouville-type theorems play a crucial role in the proofs