Coupling local and nonlocal evolution equations
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Calculus of Variations
Coupling local and nonlocal evolution equations Alejandro Gárriz1 · Fernando Quirós1 · Julio D. Rossi2 Received: 30 April 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the decay rates of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way. Mathematics Subject Classification 35K55 · 35B40 · 35A05
1 Introduction and main results The best known linear diffusion equation is for sure the classical heat equation u t = u,
(1.1)
which is naturally associated with the energy |∇u|2 E(u) = , 2
(1.2)
in the sense that (1.1) is the gradient flow in L 2 associated to E(u); see [4,7,9,27].
Communicated by M. Del Pino.
B
Fernando Quirós [email protected] Alejandro Gárriz [email protected] Julio D. Rossi [email protected]
1
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
2
Departamento de Matemáticas, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pab. 1 (1428), Buenos Aires, Argentina 0123456789().: V,-vol
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If you go one step further and consider nonlocal diffusion problems, one popular choice u t (x, t) = J (x − y)(u(y, t) − u(x, t)) dy, (1.3) RN
where J : → R is a nonnegative, radial function with R N J = 1. Notice that the diffusion of the density u at a point x and time t depends on the values of u at all points in the set x + supp J , which is what makes the diffusion operator nonlocal. Evolution equations of this form and variations of it have been recently widely used to model diffusion processes; see for instance [5,6,15,18,20,23,28,29,35–37]. As stated in [28], if u(x, t) is thought of as the density of a single population at the point x at time t, and J (x − y) is regarded as the probability distribution of jumping from location y to location x, then the rate at which individuals are arriving to position x from all other places is given by R N J (y − x)u(y, t) dy, while the rate at which they are leaving location x to travel to all other sites is given by − R N J (y − x)u(x, t) dy = −u(x, t). Therefore, in the absence of external or internal sources, the density u satisfies equation (1.3). In this case there is also an energy that governs the evolution problem, namely 1 J (x − y)(u(y) − u(x))2 dxdy. (1.4) E(u) = 4 RN RN RN
In the present paper we consider an energy which is local in certain subdomain and
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