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.
M. Cicognani, F. Colombini (2013). A well-posed Cauchy problem for an evolution equation with coefficients of low regularity. JOURNAL OF DIFFERENTIAL EQUATIONS, 254, 3573-3595 [10.1016/j.jde.2013.01.033].
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
