Gaseous detonation is a phenomenon with very complicated dynamics which has been studied extensively by physicists, mathematicians and engineers for many years. Despite many efforts the problem is far from a complete resolution. Recently the theory of subsonic detonation that occurs in highly resistant porous media was proposed in [4]. This theory provides a model which is realistic, rich and suitable for a mathematical study. In particular, the model is capable of describing the transition from...

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset {ℝ}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha \left(t\right)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and...

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....

In this paper we review the concept of regional boundary observability, developed in (Michelitti, 1976), by means of sensor structures. This leads to the so-called boundary strategic sensors. A characterization of such sensors which guarantees regional boundary observability is given. The results obtained are applied to a two-dimensional system, and various cases of sensors are considered. We also describe an approach which leads to the estimation of the initial boundary state, which is illustrated...

On démontre des résultats de régularité $L^{\infty }$ et höldérienne pour la solution d’une inéquation parabolique, formulation faible du problème suivant :$$\left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}-\sum _{i=1}^N\frac{\partial }{\partial x_i}{B}_i(x,t,u,\nabla u)+B_0(x,t,u,\nabla u)=0\text{dans}\Omega \times ]0,T[;}\hfill \\ {\displaystyle u\ge 0,\frac{\partial u}{\partial {\nu }_B}\ge 0,\frac{\partial u}{\partial {\nu }_B}=0\text{dans}\partial \Omega \times ]0,T[;u(x,0)=u_0\left(x\right)\text{dans}\Omega .}\hfill \end{array}\right.$$

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

Using as a main tool the time-regularizing convolution operator introduced by R. Landes, we obtain regularity results for entropy solutions of a class of parabolic equations with irregular data. The results are obtained in a very general setting and include known previous results.

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut - div ap (x, ∇u) = f in ] 0,T [xΩ with initial datum in L1(Ω) and assuming Dirichlet's boundary condition, where ap(.,.) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1 (]0,T[xΩ) and Ω is a domain in RN. We find spaces of type Lr(0,T;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian...