Abstract
A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interface S or the corresponding prefractal interface Sh 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.
Highlights
In this paper we study the parabolic semilinear second-order transmission problem which we formally state as (P){{{{{{{{{{{{{{{{{{{{uu−uut1c((0(tt(Δt,t,PPLPP))u))==(−=t00,ΔuPu2) (t, =(t, P) = J (u (t, [ ∂u (t, ∂n P) ] P)){u (0, P) = φ in [0, T] × Qi, on [0, T] × L, on [0, T] × ∂Q, on [0, T] × L, on [0, T] × ∂L on Q, (1)
In this paper we give a strong interpretation of the abstract problem studied in [18],;namely, we prove that the solution of the abstract problem solves problem (P) in a suitable sense
The proof of the convergence of the solution of the prefractal problems to the one of the fractal problem relies on the convergence, in the Mosco’s sense, of the energy forms which, in turn, implies the convergence of semigroups in the strong operator topology of L2(Q)
Summary
In this paper we study the parabolic semilinear second-order transmission problem which we formally state as (P). It turns out that the restriction uhi of the solution uh to Qhi belongs to suitable weighted Sobolev spaces (see the proof of Theorem 22) This regularity result is important in itself and in the numerical approximation procedure; to this regard, see [20]. The proof of the convergence of the solution of the prefractal problems to the one of the (limit) fractal problem relies on the convergence, in the Mosco’s sense, of the energy forms which, in turn, implies the convergence of semigroups in the strong operator topology of L2(Q) (see Theorem 16). In Appendices A and B, for the reader convenience, we introduce the functional spaces and traces involved
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