Semilinear Venttsel ’ Problems in Fractal Domains

We study a semilinear parabolic problem with a semilinear dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution, are established via a semigroup approach. A sufficient condition on the initial datum for global existence is given.


Introduction
In this paper we study a semilinear problem in a fractal domain with semilinear dynamical boundary conditions.
The model problem, we consider can be formally stated as follows: ( ) where Ω is the (open) snowflake domain and F = ∂Ω is the union of three Koch curves (see Section 2).J is a non linear function from a subset of ( ) ; m is the sum of the 2-dimensional Lebesgue measure and of the Hausdorff measure of F (see Section 2.1).F ∆ denotes the Laplace operator defined on F (see (3.4)  Examples of this type of non linearity include e.g. ( ) 1 , 1 p J u u u p − = > which occurrs in combustion theory (see [1]) and in the Navier Stokes system (see [2]).
Problem ( ) P presents a non linear dynamical boundary condition (known also as Venttsel' boundary condition [3]).Problem ( ) P models a fluid diffusion within a semipermeable membrane and heat flow subject to non linear cooling on the boundary (see [4] [5]).The literature on boundary value problems with dynamical conditions is huge, we refer to [6] for a derivation of such boundary conditions and to [7] and the references listed in.All these papers deal with smooth domains.The case of irregular domains is studied in [8]- [12].
In the present case we consider the case in which the non linearity appears both in bulk and on the boundary.We study the problem by a semigroup approach.More precisely we consider the corresponding abstract Cauchy problem: where ( ) ( ) ( ) is the generator associated to the energy form E introduced in (3.8), T is a fixed positive real number, φ is a given function in ( ) A is the generator of the analytic contraction positivity preserving semigroup ( ) , , L m Ω associated to E .We study problem ( ) In order to prove the existence of the solutions to (1.3) the usual way is to use a contraction argument in suitable Banach spaces see e.g.[13].Usually the functional setting is that of an interpolation space between the domain of the generator A and ( ) or the domain of a fractional power of A − , we refer the reader to [13]- [17].In our fractal case we do not know the domain of .
A We stress the fact that it is not neither known a characterization of the domain of the fractal Laplacian .
F ∆ To overcome this difficulty we adapt the abstract approach in [18] to prove local existence and uniqueness results for the mild solution.The key tool in [18] is an assumption on the estimate of the semigroup ( ) T t as a bounded operator from ( ) L Ω (see (2.1)  in [18]).In the present case we take into account that our problem has a probabilistic interpretation [19]; this, in turn, allows us to deduce an analogue estimate of ( ) 15).We then deal with the strong formulation of the B.V.P. satisfied by the mild solution, which is of course of great interest in the applications, actually we prove that the solution of problem ( ) P solves in a suitable sense Problem ( ) P see Theorems 5.1 and 5.2.The layout of the paper is the following in Section 2 we recall the preliminaries on the geometry and the functional spaces.In Section 3 we consider the energy forms and the associated semigroups.In Section 4 we consider the abstract Cauchy problem ( ) P and we prove local and global existence results.Finally in Section 5 we prove that the solution of the abstract Cauchy problem ( ) P solves problem ( ) P in a suitable sense.

Geometry
In the paper we denote by ( ) The measure µ has the property that there exist two positive constants 1 c , 2 c such that , , , where log 4 log 3 and where ( ) , B P r denotes the Euclidean ball in 2   .As µ is supported on F , it is not ambiguous to write in (2.2) ( ) ( ) In the terminology of the following section we say that F is a d-set with f d d = according to [21].Remark 2.1.The Koch snowflake can be also regarded as a fractal manifold (see [22]).We denote by Ω the (open) snowflake domain.

Functional Spaces
By ( ) 2 L ⋅ we denote the Lebesgue space with respect to the Lebesgue measure 2  on subsets of 2  , which will be left to the context whenever that does not create ambiguity.By ( ) 2 L F we denote the Hilbert space of square summable functions on F with respect to the invariant measure .
µ Let  be a closed set of 2 at every point P ∈  where the limit exists.It is known that the limit (2.3) exists at quasi every P ∈  with respect to the ( ) measure µ  with suppµ =   such that for some constants ( ) ( ) The measure µ is a d-measure.
Throughout the paper c will denote possibly different constants.
We now come to the definition of the Besov spaces.
Actually there are many equivalent definitions of these spaces see for instance [21] and [26].We recall here the one which best fits our aims and we will restrict ourselves to the case 0 1 ; the general setting being much more involved see [18].By ( ) H Ω in the following sense: For the proof we refer to Theorem 1 of Chapter VII in [21], see also [26].From now on we denote 0 u γ by F u .

The Energy Form E
In Definition 4.5 of [22] a Lagrangian measure F  on F and the corresponding energy form F E as with domain ( ) , which is a Hilbert space with norm has been characterized in terms of the domains of the energy forms on i K (see [22] Theorem 4.6).
In the following we will omit the subscript F , the Lagrangian measure will be simply denoted by ( ) , an analogous notation will be adopted for the energies.In the following we shall also use the form ( ) It can be proved as in Proposition 3.1 of [22], that: Proposition 3.1.In the previous notations and assumptions the form F E with domain ( ) is a Hilbert space under the intrinsic norm (3.2).For the definition and properties of regular Dirichlet forms we refer to [27].We now define the Laplace operator on F .As ( ) ( ) L F , there exists (see Chap. 6, Theorem 2.1 in [28]) a unique self-adjoint, non positive operator F ∆ on ( ) . We now introduce the Laplace operator on the fractal F as a variational operator from and for all ( ), is the duality pairing between ( ) ( ) . We use the same symbol F ∆ to define the Laplace operator both as a self-adjoint operator in (3.4) and as a variational operator in (3.5).It will be clear from the context to which case we refer.
In the following we denote by where b denotes a strictly positive continuous function in .

L F Consider now the space of functions :
u The space ( ) defined on the domain ( ) F Ω is a Hilbert space equipped with the scalar product , , , , , We denote by Resolvents and Semigroups Associated to Energy Forms As ( ) ( ) , there exists (see chap. 6 Theorem 2.1 in [28]) a unique self-adjoint non positive operator A on ( ) Moreover in Theorem 13.1 of [27] it is proved that to each closed symmetric form E a family of linear operators { } , 0 G α α > can be associated with the property and this family is a strongly continuous resolvent with generator A, which also generates a strongly continuous semigroup With similar arguments it can be proved that there exists a nonnegative self-adjoint operator F A with domain ( ) ( )  Proof.The contraction property follows from Lumer Phillips Theorem on dissipative operators (Chapter 1 Theorem 4.3 in [16]).In order to prove the analyticity it will be enough to prove that there exists a positive α + ≥ (see Proposition 3 Section 6 in Chapter XVII in [29]).Moreover since the semigroup is Markovian it is positive preserving.□ Remark 3.4.It is well known that the symmetric and contraction analytic semigroup ( )

T t uniquely determines analytic semigroups on the space
,1 p L p ≤ < ∞ see (Theorem 1.4.1 [30]) which we still denote by ( )
Proof.The result follows by using the equivalence between (3.14) and Nash inequality.Actually it holds that for any ( ) (see [34]).□ From Theorem 2.11 in [19] the following estimate on the decay of the heat semigroup holds.Proposition 3.6.There exists a positive constant M such that ( ) ( We will consider the case 2 n = and 1 s d = .We remark that this property is called supercontractivity ( see e.g.[30]).From now on we set We recall that for every 1 q > ( ) : q q T t L L → , and ( ) From interpolation result theory (see e.g.[35]), it can be proved that for every 1 p r < < ( ) In particular we will often use that ( ) Taking into account 2.6 and ( ) , for every 0,1 , for every 1, .

The Abstract Cauchy Problem: Local and Global Existence
We study the solvability of the Cauchy problem: is the generator associated to the energy form E introduced in (3.8), T is a fixed positive real number, φ is a given function in ( ) . We assume that J is a mapping from , , , We also assume that ( ) This assumption is not necessary in all that follows but it simplifies the calculations (see [18]).In order to prove the local existence theorem we make the following assumption on the growth of ( ) Following the approach in Theorem 2 in [18] and adapting the proof of Theorem 5.1 in [8] we have: There is a 0 T > and a unique [ ] ( ) ( ) ( ] ( ) ( ) 0, , , 0, , , with the integral being both an 2 L -valued and 2 p L -valued Bochner integral.The claim of the Theorem is proved by a contraction mapping argument on suitable spaces of continuous functions with values in Banach spaces.We adapt the proof of Theorem 5.1 in [8] to the new functional setting and for the reader's convenience we recall it.
Proof.Let Y be the complete metric space defined as follows equipped with the metric Since condition (g) holds we choose N such that ( ) . By using arguments similar to those used in the proof of Lemma 2.1 of [36] we can prove that and of course ( ) . We now prove that 2 for all 0, .lim sup Taking into account (4.3) there exists 0 . .
, 0 In order to prove that it is a contraction it's enough to choose K such that ( )  [18], the following regularity result holds (see also Theorem 5.3 in [8]).
Theorem 4.3.Under the assumptions of Theorem 4.1 we have.
a) The solution ( ) u t can be continuously extended to a maximal interval ( ) 0,T φ as a solution of (4.4), until Proof.As to the proof of condition a), we follow Theorem 4.2 in [18].From the proof of Theorem 4.1 it turns out that the minimum existence time for the solution to the integral equation is as long as (see also Corollary 2.1.in [18]).
To prove that the mild solution is classical we use the classical regularity results for linear equations (see e.g.Theorem 4.3.4. in [13]) by proving that ( ) J u is Hölder continuous on ( ] , hence Hölder continuous with any exponent L and from Theorem 11.3 and 12.1 in [37] there exists a constant c such that We now give a sufficient condition on the initial datum in order to obtain a global solution adapting Theorem 3 (b) in [38] see also Theorem 5.4 in [8].φ Ω is sufficiently small, then there exists a nonnegative which is a global solution of (4.4).
Proof.Since 2 q p < , from (3.15) it follows that ( ) T t is a bounded operator from q L into 2 p L with

.
From Theorem 4.3 (a) to show that ( ) u t is a global solution it is enough to show that use the notations of the proof in Theorem 4.1.

Ω
must remain bounded.□ In the following we denote by , E u v , we will denote the corresponding bilinear form