Nonlinear dissipative wave equations

Nonlinear Dissipative Wave Equations with Potential

We study the long time behavior of solutions of the wave equation with absorbtion |u|p−1u and variable damping V (x)ut, where V (x) ∼ V0|x|−α for large |x|, V0 > 0. For α ∈ [0, 1) and 1 < p < (n + 2)/(n − 2) we establish decay estimates of the energy, L2 and Lp+1 norm of solutions. We find three different regimes of decay of solutions depending on the exponent of the absorbtion term. We show th...

Global Attractors for Nonlinear Wave Equations with Nonlinear Dissipative Terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x, ut) = a(x)|ut|rut.

Wave equations with non-dissipative damping

Decay Rates for Dissipative Wave equations

We derive decay rates for the energy of solutions of dissipative wave equations. The metod of proof combines multiplier techniques and the construction of suitable Lyapunov functionals. Without imposing any growth condition at the origin on the nonlinearity we show that this Lyapunov functional, which is equivalent to the energy of the system, is bounded above by the solution of a differential ...

Entropy-dissipative discretizations of nonlinear diffusive equations

We present numerical discretizations which preserve the entropy structure of the analyzed nonlinear diffusive equations. More precisely, we develop numerical schemes for which the discrete entropy is stable or even dissipating. The key idea is to ”translate” entropydissipation methods to the discrete case. We consider two situations. First, an implicit Euler finite-volume approximation of porou...