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The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider the following Darboux problem for the functional differential equation ∂ ² u / ∂ x ∂ y ( x , y ) = f ( x , y , u ( x , y ) , ∂ u / ∂ x ( x , y ) , ∂ u / ∂ y ( x , y ) ) a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] 0 , a ] × ( 0 , b ] , where the function u ( x , y ) : [ - a ₀ , 0 ] × [ - b ₀ , 0 ] → ℝ k is defined by u ( x , y ) ( s , t ) = u ( s + x , t + y ) for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

The geometrical quantity in damped wave equations on a square

Pascal Hébrard, Emmanuel Humbert (2006)

ESAIM: Control, Optimisation and Calculus of Variations

The energy in a square membrane Ω subject to constant viscous damping on a subset ω ⊂ Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate τ ( ω ) of this decay satisfies τ ( ω ) = 2 min ( - μ ( ω ) , g ( ω ) ) (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here μ ( ω ) denotes the spectral abscissa of the damped wave equation operator and  g ( ω ) is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion...

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