Linear and Nonlinear Hyperbolic Equations

Modulus of Continuity and Decay at infinity in Evolution Equations with Real Characteristics

speaker: Massimo Cicognani (Università di Bologna)

abstract: In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces and the modulus of continuity of the coefficients are deeply connected. This holds true in the more general framework of evolution equations with real characteristics \[D_t^2u-\sum_{k=0}^{2p}a_k(t,x)D_x^ku=0\] (\(p=1\) hyperbolic equations, \(p=2\) vibrating beam models,...) where a sharp scale of Hoelder continuity, with respect to the time variable \(t\), for the \(a_k\)'s has been established.

We show that, for \(p\geq2\), a lack of regularity in \(t\) can be compensated by a decay as the space variable \(x\to\infty\). This is not true in the hyperbolic case \(p=1\) because of the finite speed of propagation.

timetable:
Thu 4 Jul, 15:00 - 15:50, Aula Dini
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