Existence Results to a Class of Nonlinear Parabolic Systems Involving Potential and Gradient Terms

Abstract

In this paper, we investigate the existence of solutions to a nonlinear parabolic system, which couples a non-homogeneous reaction-diffusion-type equation and a non-homogeneous viscous Hamilton-Jacobi one. The initial data and right-hand sides satisfy suitable integrability conditions and non-negative. In order to simplify the presentation of our results, we will consider separately two simplified models : first, vanishing initial data, and then, vanishing right-hand sides.

On some nonlocal parabolic systems with gradient source terms

“…By solution, we essentially mean solution in the sense of distributions, see Definition 3.1. This work extends both the results of [2] for the local (s = 1) version of System (S), and those of [6] pertaining to the single nonlocal KPZ equation (1.2). Those results will either be directly reused or serve as an inspiration in this paper.…”

“…, we are able to prove existence of local solutions to the three following relevant models of System (S), for 2 3 < s < 1:…”

“…They also studied the case where the right-hand sides mix potential and gradient terms, namely, ∂ t u − ∆u = v q + f , ∂ t v − ∆v = |∇u| p + g. Let us mention that the elliptic version of the latter system, namely, ∆u = v q + λf , ∆v = |∇u| p + µg was studied by the same authors in [3]. ⋄ Second, other existence results for local parabolic systems with gradient term are found in [2,36,30,43,47,58] and the references therein. Let us also mention that the case where the gradient term acts as absorption was considered in [54], see also [17,30] and the references therein.…”

“…Let us review the known results about such systems. ⋄ First, in [2] the fourth author and his coworkers investigated the local version (s = 1) of System (S). They also studied the case where the right-hand sides mix potential and gradient terms, namely, ∂ t u − ∆u = v q + f , ∂ t v − ∆v = |∇u| p + g. Let us mention that the elliptic version of the latter system, namely, ∆u = v q + λf , ∆v = |∇u| p + µg was studied by the same authors in [3].…”

“…Although it has been well known for quite a few years, this work also shows significant difference between the local and nonlocal cases. For example, when s = 1, the existence of nonnegative solutions to System (S) is proved for a large class of (p, q) provided that f and g satisfy some suitable integrability assumptions; see [2,Theorems 4.1 and 4.2].…”