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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