Functional calculus via Laplace transform and equations with infinitely many derivatives

Abstract

We study nonlocal linear equations of the form f(∂t)ϕ=J(t), t≥0, in which f is an entire function. We develop an appropriate functional calculus via Laplace transform, we solve the aforementioned equation completely in the space of exponentially bounded functions, and we analyze the delicate issue of the formulation of initial value problems for nonlocal equations.

The initial value problem for ordinary differential equations with infinitely many derivatives

“…We note that with this definition, the operator f (∂ t ) is a linear operator on D, a fact on which we did not insist in [16]. This definition is reasonable, in the sense that it is directly related to (7) and it generalizes the standard case of polynomial functions f .…”

“…Proof. Condition (14) implies that the series d j−1 /s j converges absolutely for |s| > R. It follows from Lemma 2.1 in [16] that the series r is in fact an entire function, and then Proposition 2.1 of [16] allow us to conclude that (φ, r) ∈ D f .…”

Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces

“…Besides their intrinsic interest, we are interested in Sobolev spaces in this general context because we wish to consider some nonlinear equations appearing in physical theories [4,10,11,28,30] in settings beyond Riemannian manifolds, and also because we wish to use them as a tool for a better understanding of pseudodifferential operators defined on locally compact abelian groups, motivated by the papers [22] and [14], and also by the recent treatise [23] in which the authors study in detail pseudo-differential operators on compact Lie groups. As a first application of our Sobolev-type theorems, we investigate the existence of continuous solutions to the generalized euclidean bosonic string equation…”

Global Existence of Weak Solutions for a Logarithmic Wave Equation Arising From Q-Ball Dynamics

“…The problem (1.2) is a relativistic version of logarithmic quantum mechanics introduced in [3,4] and can also be obtained by taking the limit p → 1 for the p-adic string equation [15,16,21,22]. It is easy to see that the problem (1.2) can be see as one dimensional case of (1.1) without the first order derivative term and cubic term.…”