Strong regularizing effect of a gradient term in the heat equation with the Hardy potential

We deal with the following parabolic problem,

     

Decay solution for the renewal equation with diffusion

In this paper we consider age structured equation with diffusion under nonlocal boundary condition and nonnegative initial data. We prove existence, uniqueness and the positivity of the solution to the above problem. Our main result is to get an exponential decay of the solution for large times toward such a study state. To this end we prove a weighted Poincaré–Wirtinger’s type inequality in unbounded domain.     

Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary

In this paper we consider the problem

(P)     

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

We deal with the following parabolic problem, $$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$ where is a bounded regular domain or ${\Omega = \mathbb{R}^N}$ , ${1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}$ are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all ${f \in L^1(\Omega_T ), f \geq 0}$ , problem (P) has a positive solution. Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.     

Some remarks on quasilinear parabolic problems with singular potential and a reaction term

In this paper we deal with the following quasilinear parabolic problem $$\left\{\begin{array}{l@{\quad}l} (u^\theta)_t - \Delta_p {u} = \lambda \frac{u^{p - 1}}{|x|^{p}} + u^q + f,\,\, u \geq 0 \quad {\rm in} \;\;\Omega \times (0, T),\\ u(x, t) = 0 \quad\qquad\qquad\qquad\qquad\qquad\qquad {\rm on}\; \partial \Omega \times(0, T),\\ u(x, 0) = u_0(x), \,\,\, \qquad\qquad\qquad\qquad\qquad x \in\; \Omega,\end{array}\right.$$ where θ is either 1 or (p ? 1), \({N \geq 3, \,\Omega \subset \mathcal{IR}^N}\) is either a bounded regular domain containing the origin or \({\Omega \equiv \mathcal{IR}^N}\) , 1 < p < N, q > 0 and u ₀ ≥  0, f ≥  0 with suitable hypotheses. The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, we analyze blow-up phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle. We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power.     

Periodization of random media and representative volume element size for linear composites

Several existing numerical studies show that the effective linear properties of random composites can be accurately estimated using small volumes subjected to periodic boundary conditions – more suitable than homogeneous strain or stress boundary conditions – providing that a sufficient number of realizations are considered. Introducing the concept of periodization of random media, this Note gives a new definition of representative volume element which leads to estimates of its minimum size in agreement with existing theoretical results. A qualitative convergence criterion for the numerical simulations is proposed and illustrated with finite element computations. To cite this article: K. Sab, B. Nedjar, C. R. Mecanique 333 (2005).     

Modeling long-term creep rupture by debonding in unidirectional fibre-reinforced composites

Delayed fracture due to debonding can be observed in many unidirectional fibre-reinforced composites when the fibre/matrix interface experiences creep. The aim of this work is to describe such a phenomenon within the recently proposed modeling framework of transverse isotropy that allows for a neat decomposition of the mechanical behavior into fibre-directional, transverse, and pure shear parts. Specifically, debonding is here chosen to be governed by the tension transverse to the fibres. One can then speak of a mode-I debonding if use is made of the terminology adopted in fracture mechanics. On another hand, the time-dependent response is attributed to the matrix constituent. As the role of this latter is to deform and support stresses primarily in shear, a viscoelastic behavior is introduced that affects solely the pure shear part of the behavior. We show that both characteristics can be easily embedded into the aforementioned formulation. Among others, the occurrence of tertiary creep is made possible to predict. It is otherwise found that the predicted debonding path always propagates along the direction of the fibres in agreement with many experimental observations found in the literature. On the numerical side, the algorithmic treatment of debonding is independent of the one for viscoelasticity. This renders the implementation within the context of the finite element method very easy.