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Global classical solutions to the viscous Hamilton-Jacobi equation ut − ∆u = a |∇u| p in (0, ∞) × Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W 1,∞ (Ω) at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to zero is also obtained in one space dimension but the rate depends on a and… (More)

- BELGACEM REBIAI, SAÏD BENACHOUR
- 2010

The aim of this study is to construct the invariant regions in which we can establish the global existence of classical solutions for reaction-diffusion systems with a general full matrix of diffusion coefficients. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of… (More)

- BELGACEM REBIAI, SAÏD BENACHOUR
- 2009

The aim of this study is to prove global existence of classical solutions for problems of the form ∂u ∂t − a∆u = −f (u, v), ∂v ∂t − b∆v = g(u, v) in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in R n , a > 0, b > 0, a = b and f , g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(ξ, η) ≤… (More)

- SAÏD BENACHOUR
- 2008

Sharp temporal decay estimates are established for the gradient and time derivative of solutions to the Hamilton-Jacobi equation ∂tvε + H(|∇xvε|) = ε ∆vε in R N × (0, ∞), the parameter ε being either positive or zero. Special care is given to the dependence of the estimates on ε. As a by-product, we obtain convergence of the sequence (vε) as ε → 0 to a… (More)

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