Hyperbolic equations and inequalities in octants with the lateral Cauchy data at coordinate planes are consdered. Lipschitz stability estimate is established in the case when both the inhomogeneous right hand side and (unknown) initial conditions at {t = 0} have a finite support. This is the first stability estimate for such a Cauchy problem in an infinite domain. Refocusing of time reversal fi...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

Given an injective closed linear operator A defined in a Banach space X, and writing CFDtα the Caputo–Fabrizio fractional derivative of order α∈(0,1), we show that unique solution abstract Cauchy problem (∗)CFDtαu(t)=Au(t)+f(t),t≥0, where f is continuously differentiable, given by first u′(t)=Bαu(t)+Fα(t),t≥0;u(0)=−A−1f(0), family bounded operators Bα constitutes Yosida approximation Fα(t)→f(t)...