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A decay estimate for a class of hyperbolic pseudo-differential equations

Sandra LucenteGuido Ziliotti — 1999

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the equation u t i Λ u = 0 , where Λ = λ D x is a first order pseudo-differential operator with real symbol λ ξ . Under a suitable convexity assumption on λ we find the decay properties for u t , x . These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem u t i Λ u = f u , u 0 , x = g x . If f u is a smooth function which satisfies f u u p near u = 0 , and g is small in suitably Sobolev norm, we prove global existence theorems provided p is greater than a critical exponent.

Nonlinear Wave Equation with Vanishing Potential

Lucente, Sandra — 1999

Serdica Mathematical Journal

We study the Cauchy problem for utt − ∆u + V (x)u^5 = 0 in 3–dimensional case. The function V (x) is positive and regular, in particular we are interested in the case V (x) = 0 in some points. We look for the global classical solution of this equation under a suitable hypothesis on the initial energy.

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