We investigate the approximation of the evolution of compact hypersurfaces of ${ℝ}^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f:{ℝ}^s\to ℝ$ can be approximated with arbitrary accuracy by an infinite sum ${\sum }_{r=1}^{\infty }{H}_r(x₁,...,x_s)\in C^{\infty }\left({ℝ}^s\right)$ of analytic functions $H_r$, each solving the same system of universal partial differential equations, namely $P(x_{\sigma };H_r,\partial H_r/\partial x_{\sigma },...,\partial ⁵H_r/\partial x{⁵}_{\sigma }⁵)=0$ (σ = 1,..., s).

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper...

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...

We consider the linear convection-diffusion equation associated to higher order elliptic operators⎧  ut + Ltu = a∇u   on Rnx(0,∞)⎩  u(0) = u0 ∈ L1(Rn),where a is a constant vector in Rn, m ∈ N*, n ≥ 1 and L0 belongs to a class of higher order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on Rn. The aim of this paper is to study the asymptotic behavior, in Lp (1 ≤ p ≤ ∞), of the derivatives Dγu(t) of the solution of the convection-diffusion equation...

We consider the second order parabolic partial differential equation    ${\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i(x,t)u_{x_i}+c(x,t)u-u_t=0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^{\alpha }\left[u^{\alpha }\right]+{\sum }_{\beta =1}^N{c}^{\alpha \beta }(x,t)u^{\beta }=f^{\alpha }(x,t)$, where    $L^{\alpha }\left[u\right]\equiv {\sum }_{i,j=1}^n{a}_{ij}^{\alpha }(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i^{\alpha }(x,t)u_{x_i}-u_t$, must decay as t → ∞.