2018/09/07 by Irene Benedetti, Benedetti, Irene, Luisa Malaguti +3
Computer Science · Engineering · Mathematics · #34B10 #35K20 (Primary) #47H11 #93D30 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1809.02353
openalex publication_date 2018/09/07 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
The paper deals with second order parabolic equations on bounded domains with\nDirichlet conditions in arbitrary Euclidean spaces. Their interest comes from\nbeing models for describing reaction-diffusion processes in several frameworks.\nA linear diffusion term in divergence form is included which generates a\nstrongly elliptic differential operator. A further linear part, of integral\ntype, is present which accounts of nonlocal diffusion behaviours. The main\nresult provides a unifying method for studying the existence and localization\nof solutions satisfying nonlocal associated boundary conditions. The Cauchy\nmultipoint and the mean value conditions are included in this investigation.\nThe problem is transformed into its abstract setting and the proofs are based\non the homotopic invariance of the Leray-Schauder topological degree. A\nbounding function (i.e. Lyapunov-like function) theory is developed, which is\nnew in this infinite dimensional context. It allows that the associated vector\nfields have no fixed points on the boundary of their domains and then it makes\npossible the use of a degree argument.\n