We characterize the limiting behavior of scalar phase-field equations with infinitely many potential wells as the density of potential wells tends to infinity. An example of such a family of equations is \[ \mathcal{u}^e_t = \Delta\mathcal{u}^e - \frac1{\epsilon^{1+\alpha}}W'(\frac{\mathcal{u}^e}{\epsilon^{1-\alpha}}), \] where $W$ is a periodic function. We prove that solutions of the above equation converge to solutions of the Mean Curvature partial differential equation for a range of positive values of the parameter $\alpha$, and we also determine the limiting equation when $\alpha = 0$. We show that our techniques can be modified to apply to fully nonlinear equations and to other classes of infinite-well equations. We discuss some applications to questions of interaction between wave fronts in dynamic phase transitions.

We study the asymptotic behaviour near a finite blow-up time $t=T$ of the solutions to the initial-boundary value problem for the quasilinear equation $$ u_t = \nabla \cdot (|\nabla u|^\sigma \nabla u) + e^u \mbox{ in } \{ | x| < R\} \times (0,T), \ \ \sigma >0, $$ with zero Dirichlet boundary condition and a radial symmetric initial function $u_0 (|x|) >0$ in $\{ |x| <R\}, \ u'_0 (r)<0$ in $(0,R)$. We prove single point blow-up and different sharp lower and upper estimates of the solution as $t \rightarrow T^-$ and of the final-time profile. For the one-dimensional problem the asymptotic behaviour is proved to be described by nonconstant self-similar solutions of the form $u_* (x,t) =-\log (T-t) + \theta (\xi), \ \xi = x/(T-t)^{1/(\sigma+2)}$, where $\theta (\xi) \sim - (\sigma+2) \log \xi$ as $\xi \rightarrow\infty$.

The existence, uniqueness, and global exponential stability of traveling wave solutions of a class of nonlinear and nonlocal evolution equations are established. It is assumed that there are two stable equilibria so that a traveling wave is a solution that connects them. A basic assumption is the comparison principle: a smaller initial value produces a smaller solution. When applied to di↵erential equations or integro-di↵erential equations, the result recovers and/or complements a number of existing ones.