First-order linear evolution equations with càdlàg-in-time solutions

Abstract

In this work we study first-order linear parabolic evolution PDEs over ${\mathbb{R}}^d\times \mathbb{R}$ and ${\mathbb{R}}^d\times {\mathbb{R}}^{+}$ comprising a spatial operator defined through a symbol function and a source term such that its spatial Fourier transform is a slow-growing measure over ${\mathbb{R}}^d\times \mathbb{R}$. When the source term is required to has its support on ${\mathbb{R}}^d\times {\mathbb{R}}^{+}$, it is shown that there exists a unique solution such that its spatial Fourier transform is a slow-growing measure with support in ${\mathbb{R}}^d\times {\mathbb{R}}^{+}$, which in addition has a càdlàg-in-time behaviour. This allows to well-pose and analyse an initial value problem associated to this class of equations and to consider cases where the spatial operator can be a pseudo-differential operator. We also look at for solutions to the cases where the source term is such that its spatial and spatio-temporal Fourier transforms are slow-growing measures over ${\mathbb{R}}^d\times \mathbb{R}$. In such a case, it is shown that when the real part of the symbol function of the spatial operator is inferiorly bounded by a strictly positive constant, there exists a unique solution whose both spatial and spatio-temporal Fourier transforms are slow-growing measures over ${\mathbb{R}}^d\times \mathbb{R}$, and which also has a càdlàg-in-time behaviour. In addition, it is proven that the solution to an associated Cauchy problem converges spatio-temporally asymptotically to this unique solution as the time flows long enough.