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Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Barbara Szomolay (2003)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction u ¨ = - γ u ˙ + m ( ∥ ∇ u ∥ 2 ) Δ u - δ | u | α u + f , which is known as degenerate if m ( · ) ≥ 0 , and non-degenerate if m ( · ) ≥ m 0 > 0 . We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of α . Our aim is to extend the validity of previous results in [5] to α ≥ 0 both to the degenerate and non-degenerate cases of m . We extend our results to equations with...

Decay rates of Volterra equations on ℝN

Monica Conti, Stefania Gatti, Vittorino Pata (2007)

Open Mathematics

This note is concerned with the linear Volterra equation of hyperbolic type ∂ t t u ( t ) - α Δ u ( t ) + ∫ 0 t μ ( s ) Δ u ( t - s ) d s = 0 on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

Decomposition and Moser's lemma.

David E. Edmunds, Miroslav Krbec (2002)

Revista Matemática Complutense

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

Degenerate Eikonal equations with discontinuous refraction index

Pierpaolo Soravia (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

Kaouther Ammar (2010)

Open Mathematics

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v 0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A 1, A 2,] with A 1 ≤ 0 ≤ A 2 so that the problem is of parabolic-hyperbolic type.

Currently displaying 21 – 40 of 115