Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

Abstract

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reactiondiffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.2010 Mathematics Subject Classification. 35J92, 35A15, 35B41,35K65.

Transmission problems with nonlocal boundary conditions and rough dynamic interfaces

“…[7] for other examples and for more details on this subject). Moreover Ω has the W 1,p -extension property and the restriction of H d to ∂Ω is an upper d-Ahlfors measure (see [6,24,29]).…”

“…Following [21, theorem 2.3] (see also [22,24]), (4.6) is a consequence of the same recursive inequality for ( ) E t mk from (3.34). Arguing in a similar fashion as in our recent work [24], (3.34) allows us to deduce the following stronger inequality…”

“…Remark 2.2. We mention that if Ω has the W 1,p -extension property, then W 1,p (Ω) = W 1,p (Ω) and hence, Ω has the W Moreover Ω has the W 1,p -extension property and the restriction of H d to ∂Ω is an upper d-Ahlfors measure (see [6,24,29]).…”

Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions

“…This section consists of two main parts. At first we will establish the existence and uniqueness of a (local) strong solution on a finite time interval using the theory of monotone operators exploited in [15] and [16]. Then exploiting a Moser iteration argument we show that the local solution is actually a global solution.…”

“…We first state a Poincaré-type inequality associated with the quadratic form a δ Ω as defined by (2.24). As in [15,16] it is crucial in the proof of the existence of strong solutions to semilinear parabolic equations with fractional diffusion and dynamic boundary conditions.…”