Linear hyperbolic equations equations with time-dependent propagation speed
and strong damping
(Unabridged printout of the lectures of a PhD course) (Freiberg, Mai 2017)
Massimo Gobbino
Contents
Lecture 1. Preliminaries and functional spaces: Hilbert spaces, multiplication opera- tors, Sobolev spaces, distributions, Gevrey spaces and Gevrey ultradistributions.
Continuity moduli. . . 4 Lecture 2. Approximated energy estimates for ODEs. Kovaleskian energy and hyper-
bolic energy. Classical applications: integrable coefficients, Lipschitz coefficients, omega-continuous coefficients. . . 9 Lecture 3. Classical well-posedness theory for equations with time-dependent propa-
gation speed. Higher space-regularity of data compensates lower time-regularity of the propagation speed. . . 14 Lecture 4. Equations with constant propagation speed and strong damping: well-
posedness results (with a full description of possible phase spaces) and higher reg- ularity for positive times. . . 20 Lecture 5. Equations with time-dependent propagation speed and strong damping:
well-posedness in the supercritical case. Heuristics and proof through an adapted Kovaleskian energy. . . 24 Lecture 6. Equations with time-dependent propagation speed and strong damping:
well-posedness in the subcritical case. Heuristics and proof through an adapted hyperbolic energy. . . 30 Lecture 7. ODEs with time-dependent propagation speed (with or without dissipation):
examples of solutions whose energy grows exponentially. . . 34 Lecture 8. Equations with time-dependent propagation speed: how to build counterex-
amples showing the optimality of the well-posedness results. . . 39
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