Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains

Abstract

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of such problems towards the solution of the fractional-in-time Venttsel’ problem in the corresponding fractal domain. This is achieved via the convergence (in the Mosco–Kuwae–Shioya sense) of the approximating energy forms in varying Hilbert spaces.

Keywords

• fractional Caputo time derivative
• Venttsel’ problems
• fractal domains
• asymptotic behavior
• varying Hilbert spaces
• resolvent families