We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ${(-A)}^{\beta }$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already...

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period $\epsilon$ of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the...

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates...

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations.

This paper is devoted to the study of the linear parabolic problem $\epsilon {\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x/\epsilon ,t/{\epsilon }^3)\nabla u_{\epsilon }(x,t)\right)=f(x,t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\epsilon$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems....

We study the homogenization of parabolic or hyperbolic equations like$${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}\Omega \times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in $Ø$) take possibly high values on a $\epsilon$-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

We study the homogenization of parabolic or hyperbolic equations like $${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}Ø\times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$ when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^2(0,T;H_0^1\left(\Omega \right))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ${\epsilon }^p{\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x{\epsilon }^{-1},x{\epsilon }^{-2},t{\epsilon }^{-q},t{\epsilon }^{-r})\nabla u_{\epsilon }(x,t)\right)=f(x,t)$, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic...

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type...