Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interface or the corresponding prefractal interface is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed. 1. Introduction In this paper we study the parabolic semilinear second-order transmission problem which we formally state as where is the bounded open set , and is a “cylindrical” layer dividing the set into two subsets and (see Figure 2). When is the Koch-type surface , where is the snowflake and (see Section 2), is the energy functional introduced in (12); when is the prefractal surface , is the energy functional introduced in (24). is a nonlinear function from a subset of into . denotes the restriction of to , denotes the jump of across , denotes the Laplace operator defined on the layer (see (12) in Section 3), and denotes the jump of the normal derivatives across , to be intended in a suitable sense. More precisely, we assume that is a nonlinear mapping from to for any fixed , locally Lipschitz; that is, Lipschitz on bounded sets in with Lipschitz constant when restricted to , satisfying suitable growth conditions (see conditions (i) and (ii) in Section 4). Examples of this type of nonlinearity include, for example, which occur in combustion theory (see [1]) and in the Navier-Stokes system (see [2]). In the recent years there has been an increasing interest in the study of linear transmission problems across irregular layers of fractal type and the corresponding prefractal layers [3–7]. Problems of this type are also known in the literature as problems with Ventcel’s boundary conditions [8] or second-order transmission conditions. Fractal layers can provide new interesting settings in those model problems, in which the surface absorption of tension, electric conduction, or flow is the relevant effect. The literature on semilinear equations on smooth domains is extensive (see e.g., [9–13] and the recent review in [14]); the fractal case is more awkward (see e.g., [15–19]). In our case one has to take into account that the diffusion phenomenon takes place both across the smooth domain and the cylindrical layer ; this fact has a counterpart in the structure of the energy functional and hence on problem . In [18] the authors proved local existence and uniqueness results of the “mild” solution of an abstract evolution transmission problem across a prefractal or fractal interface (see (36) and (37)). In this paper we