Anzeige
Erschienen in:
2019 | OriginalPaper | Buchkapitel
Semilinear Damped Klein-Gordon Models with Time-Dependent Coefficients
verfasst von : Giovanni Girardi
Erschienen in: New Tools for Nonlinear PDEs and Application
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)u
t and mass term m(t)2
u, and a power nonlinearity |u|p: We discuss how the interplay between an effective time-dependent damping term and a time-dependent mass term influences the decay rate of the solution to the corresponding linear Cauchy problem, in the case in which the damping term is dominated by the mass term, i.e. liminft→∞ (m(t)∕b(t)) > 1∕4.
$$\displaystyle \begin {cases} u_{tt}-\varDelta u+b(t)u_t+m^2(t)u=|u|{ }^p, & t\geq 0, \ x\in \mathbb R^n,\\ u(0,x)=f(x), \quad u_t(0,x)=g(x). \end {cases} $$
Then we use the obtained estimates of solutions to linear Cauchy problems to prove that a unique global in-time energy solution to the Cauchy problem with power nonlinearity |u|p at the right-hand side of the equation exists for any p > 1, assuming small data in the energy space (f, g) ∈ H
1 × L
2.