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

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 ${D^2_t}u - \sum^{2p}_{k=0}a_k(t,x){D^k_x}u=0.$(p = 1 hyperbolic equations, p = 2 vibrating beam models,...)where a sharp scale of Hölder continuity, with respect to the time variable t, for the a_k’s has been established.

We show that, for $p\geq 2$, a lack of regularity in t can be compensated by a decay as the space variable x $x \rightarrow \infty$This is not true in the hyperbolic case p = 1 because of the finite speed of propagation.

Mathematics Subject Classification. 35G10; 35L15.