In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces is strictly related to the modulus of continuity of the coefficients. This holds true for p-evolution equations with real characteristics (p=1 hyperbolic equations, p=2 vibrating plate and Schrödinger type models, …). We show that, for p≥2 a lack of regularity in t can be balanced by a damping of the too fast oscillations as the space variable x→∞. This cannot happen in the hyperbolic case p=1 because of the finite speed of propagation.
A well-posed Cauchy problem for an evolution equation with coefficients of low regularity
CICOGNANI, MASSIMO;
2013
Abstract
In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces is strictly related to the modulus of continuity of the coefficients. This holds true for p-evolution equations with real characteristics (p=1 hyperbolic equations, p=2 vibrating plate and Schrödinger type models, …). We show that, for p≥2 a lack of regularity in t can be balanced by a damping of the too fast oscillations as the space variable x→∞. This cannot happen in the hyperbolic case p=1 because of the finite speed of propagation.File in questo prodotto:
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