Asymptotic parabolicity for strongly damped wave equations
Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli · Proceedings of symposia in pure mathematics · 2013
For S S a positive selfadjoint operator on a Hilbert space, \[ d 2 u d t ( t ) + 2 F ( S ) d u d t ( t ) + S 2 u ( t ) = 0 \frac {d^2u}{dt}(t) + 2 F(S)\frac {du}{dt}(t) + S^2u(t)=0 \] describes a class of wave equations with strong friction or damping if F F is a positive Borel function. Under suitable hypotheses, it is shown that \[ u ( t ) = v ( t ) + w ( t ) u(t)=v(t)+ w(t) \] where v v satisfies \[ 2 F ( S ) d v d t ( t ) + S 2 v ( t ) = 0 2F(S)\frac {dv}{dt}(t)+ S^2v(t)=0 \] and \[ w ( t ) ‖ v ( t ) ‖