Energy decay of solutions of a degenerate Kirchhoff equation with a weak nonlinear dissipation

In this paper we study decay properties of the solutions to the degenerate Kirchhoff equation with a weak nonlinear dissipative term.     

Traveling wave solutions of the porous medium equation with degenerate interfaces

We prove the existence of a one-parameter family of solutions of the porous medium equation in which the interface is a half-plane which advances at constant speed.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the problem of decay rate for the solutions of the initial-boundary value problem to the wave equation, governed by localized nonlinear dissipation and without any assumption on the dynamics (i.e., the control geometric condition is not satisfied). We treat separately the autonomous and the non-autonomous cases. Providing regular initial data, without any assumption on an observation subdomain, we prove that the energy decays at last, as fast as the logarithm of time. Our result is a generalization of Lebeau (in: A. Boutet de Monvel, V. Marchenko (Eds.), Algebraic and Geometric Methods in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996, pp. 73) result in the autonomous case and Nakao (Adv. Math. Sci. Appl. 7 (1) (1997) 317) work in the non-autonomous case. In order to prove that result we use a new method based on the Fourier-Bross-Iaglintzer (FBI) transform.     

Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries

Ricerche di Matematica - In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic...     

Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity

Calculus of Variations and Partial Differential Equations -     

Blow up of positive initial energy solutions for a wave equation with fractional boundary dissipation

In this paper, we consider a strongly damped wave equation with fractional damping on part of its boundary and also with an internal source. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result with positive initial energy is established.     

Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

We consider the Cauchy problem for the damped wave equation with absorption

     

Asymptotic behaviour of solutions for the wave equation with an effective dissipation around the boundary

We consider the initial-boundary value problem for the wave equation with a dissipation a(t,x)u_t in an exterior domain, whose boundary meets no geometrical condition. We assume that the dissipation a(t,x)u_t is effective around the boundary and a(t,x) decays as |x|→∞. We shall prove that the total energy does not in general decay, and the solution is asymptotically free as the time goes to infinity. Further, we shall show that the local energy decays like O(t⁻¹) (t→∞).