Asymptotic Analysis - Volume 56, issue 2

Authors: Schimperna, Giulio | Segatti, Antonio

Article Type: Research Article

Abstract: We address a parabolic equation of the form α(ut )−Δu+W′(u)=f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions. The “double nonlinearity” is due to the simultaneous presence of the maximal monotone function α and of the derivative W′ of a smooth, but possibly nonconvex, potential W; f is a source term. After recalling an existence result for weak solutions, we show that, among all the weak solutions, at least one for each admissible choice of the initial datum “regularizes” for t>0. Moreover, the class of “regularizing” solutions constitutes a semiflow 𝒮 for which we prove unique continuation …for strictly positive times. Finally, we address the long time behavior of 𝒮. In particular, we can prove existence of both global and exponential attractors and investigate the structure of ω-limits of single trajectories. Show more

Citation: Asymptotic Analysis, vol. 56, no. 2, pp. 61-86, 2008

Authors: Ohtsuka, Takeshi

Article Type: Research Article

Abstract: We consider the singular limit of an Allen–Cahn type equation with a periodic nonlinear term. In this case we can find a multi-step internal transition layer when the interface thickness parameter tends to zero. We give a rigorous proof of the convergence of internal transition layers to interfaces which move under a mean curvature flow with a driving force, even if no traveling wave solutions connecting two nonadjacent stable equilibria exist.

Keywords: Allen–Cahn equation, multiple-well potential, mean curvature flow, viscosity solution

Citation: Asymptotic Analysis, vol. 56, no. 2, pp. 87-123, 2008