Classical solutions to quasilinear parabolic problems with dynamic boundary conditions

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.

(1.1) With the term "classical solutions", we mean solutions possessing all the derivatives appearing in (1.1) in pointwise sense: so we look for conditions guaranteeing the existence of solutions u(t, x) with Dtu, D α x u (|α| ≤ 2) continuous in [0, τ ] × Ω, for some τ > 0. If the functions bj and h are suitably regular, these conditions, together with the second equation in (1.1) (the boundary condition) imply that the map t → Dtu(t, ·) should be continuous with values in C 1 (∂Ω). However, it is well known that, in order to get neat results for parabolic problems, it is often advisable to replace continuous with Hölder continuous functions. So a natural class of solutions could be the set of functions in C 1+β/2,2+β ((0, τ ) × Ω) (see (1.7)), such that the restriction of Dtu to [0, τ ] × ∂Ω is bounded (as a function of t) with values in C 1+β (∂Ω).
In order to frame our results, I begin by recalling some previous literature, concerning nonlinear parabolic problems with dynamic boundary conditions. The first paper I quote is [9]: here the author considers the system Dx j [aj(u, ·, ∇xu)Dx j u] + a0(u, ·) = f (t, ·) in R + × Ω, Dtu + n j,k=1 a jk (u, ·)νjDx k u + b0(u, ·)u = g1(u) on R + × Γ1, n j,k=1 a jk (u, ·)νj Dx k u + b1(u, ·) = 0 on R + × Γ2, where I have indicated with ν(x ′ ) the unit normal vector to ∂Ω in x ′ ∈ ∂Ω, pointing outside Ω. Employing monotonicity assumptions and Rothe's method, imposing a polynomial growth of the coefficients, he constructs a generalized solution. The initial datum u0 is taken in a suitable space W 1,p (Ω), with p connected with the growth conditions of the functions aj. Some results of regularity are proved (for example, the solution in continuos with values in H 2 (Ω ′ ) for every Ω ′ with compact closure in Ω).
Replacing in the boundary condition the time derivative with the second term of the parabolic equation, one is formally reduced to a stationary boundary condition of second order, which is usually called "generalized Wentsell boundary condition". So, in [5] the authors consider the operator Au = φ(x, u ′ (x))u ′′ (x) + ψ(x, u(x), u ′ (x)), with suitable assumptions on φ and ψ, in the interval [0, 1]. General boundary conditions in the form B(u)(j) := αj (Au)(j) + βj u ′ (j) ∈ γj(u(j)) (j ∈ {0, 1}), with γj maximal monotone are imposed. Then it is proved that A, equipped with such conditions, is m−dissipative and generates a nonlinear contraction semigroup in C([0, 1]).
The same author considers in [11] the p−Laplacian Apu = div(a(x)|∇u| p−2 )∇u), with a(x) > 0 in Ω and the nonlinear Wentzell boundary condition on ∂Ω 0 ∈ Apu + b|∇u| p−2 ∂u ∂ν + β(·, u), Here β(x, ·) is the sub differential of the functional B(x, ·). He proves that it generates a nonlinear sub-Markovian C0−semigroup on suitable L 2 −spaces and, in case b(x) ≥ b0 > 0, he obtains the existence of nonlinear, non expansive semigroups in L q spaces, for every q ∈ [1, ∞). A related situation with dynamic boundary conditions is treated in [6]. Finally, the asymptotic behavior of semilinear parabolic system with dynamic boundary conditions is studied in [1]. The nonlinearity is only in the boundary condition. A generalization (with a first order nonlinear term in the parabolic equation) is given in [2].
In this paper, differently from these papers, we want to show the existence and uniqueness of local solutions to (1.1) which are regular up to t = 0. Observe that we consider the case that the coefficients of the elliptic operator in the parabolic equation depend also on ∇u. Moreover, we consider systems which are not necessarily in divergence form and no particular connection between the elliptic operator and the first order operator n j=1 bj(t, x ′ , u)Dx j is required: the only structural condition that we impose is that n j=1 bj(t, x ′ , u)νj(x ′ ) > 0 for every x ′ ∈ ∂Ω. Finally, we do not impose any kind of growth condition on the coefficients.
The organization of this paper is the following: in this first section, we introduce some notations, together with some known facts, which we are going to employ in the sequel. The second section contains a careful study of linear non autonomous parabolic systems, which may have some interest in itself, in particular Theorem 2.6. This study is preliminary to the final third section, containing the main result, Theorem 3.5. Such theorem states the existence and uniqueness of a local solution u to (1.1) such that u ∈ C 1+β/2,2+β ((0, τ ) × Ω), with Dtu |(0,τ )×∂Ω bounded with values in C 1+β (∂Ω) (β ∈ (0, 1)). Apart some regularity of the coefficients, we require only the strong ellipticity of the operator |α|=2 aα(t, x, u, p)D α x and the condition n j=1 bj(t, x ′ , u)νj(x ′ ) > 0 if x ′ ∈ ∂Ω. Concerning the initial datum u0, it should belong to C 2+β (Ω) and a certain (necessary) compatibility condition ((3.4)) should hold.
The main tool of the proof is a certain maximal regularity result which was proved in [7], and it is stated in Theorem 2.1.
We pass to the aforementioned notations and known facts. C(α, β, . . . ) will indicate a positive real number depending on α, β, . . . and may be different from time to time. The symbol ∇x,ub will indicate the gradient of b with respect to the vector (x, u) (x ∈ R n , u ∈ R). On the other hand, D 2 x j u b stands for ∂ 2 b ∂x j ∂u . Let Ω be an open subset of R n . We shall indicate with C(Ω) the class of complex valued continuous functions and with C(Ω) the subspace of uniformly continuous and bounded functions. If f ∈ C(Ω), it is continuously extensible to its topological closure Ω. We shall identify f with this extension. If m ∈ N, we indicate with C m (Ω)(C m (Ω)) the class of functions f in C(Ω)(C(Ω)), whose derivatives D α f , with order |α| ≤ m, belong to C(Ω)(C(Ω)). C m (Ω) admits the natural norm and m ∈ N0, we set and, of course, which means that there exists C > 0, such that, ∀f ∈ C β 1 (Ω) We pass to consider vector valued functions. Let X be a Banach space. If A is a set, we shall indicate with B(A; X) the Banach space of bounded functions from A to X. If m ∈ N0, β ≥ 0 and Ω is an open subset of R n , the definitions of C m (Ω; X), C m (Ω; X), C β (Ω; X) and of the norms · C β (Ω;X) can be obtained by obvious modifications of the corresponding, in the case X = C. (1.6) can be generalized to vector valued functions.
We shall need also spaces C α,β (I × V ), with V suitably regular submanifold of R n : we shall consider, in particular, the case V = ∂Ω, with Ω open, bounded subset of R n . Of course, in this case C β (V ; C(I)) can be defined by local charts.

Lemma 2.3. We consider a system in the form
with the following conditions: ) and, for certain δ and M in R + , ∀t ∈ [0, T ], ∀u ∈ C 2+β (Ω)), Consider the system (2.5). Then: Let T0 ∈ R + . Assume that (b) and (c) are satisfied and (I) holds, replacing T with T0. Take 0 < T ≤ T0. Then: (II) there exists C(T0, A, B, δ, M ) in R + , such that Proof We prove the result in several steps.
Step 1. We consider the case u0 = 0 and prove an a priori estimate if T is sufficiently small.
We show the existence. We suppose that T does not satisfy one of the majorities in (2.11) and replace it with τ ∈ (0, T ), in such a way that these majorities are satisfied by τ . So we can apply Step 3 and get a unique solution u1 with domain [0, τ ] × Ω. We observe that (2.15) and consider the system u2(0, ξ) = u1(τ, ξ), ξ ∈ Ω, By (2.15) Step 3 is applicable and it is easily seen that, if we set ; C 1+β (∂Ω)) and solves (2.5) if we replace T with (2τ ) ∧ T . In case 2τ < T , we can iterate the procedure and in a finite number of steps we construct a solution in (0, T ) × Ω. II) If T is so small that (2.11) holds and u0 = 0, we get the conclusion from the a priori estimate (2.12). The case u0 = 0, can be deduced applying the foregoing to the solution v to (2.14). If T does not satisfy (2.11), we fix T1 in (0, T ), satisfying it and obtain (II) applying the estimates in an interval of length, less or equal than T1 [T /T1] + 1 times.
(III)-(IV) can be proved with the same arguments employed for the analogous estimates in Lemma 2.2.

So, by Lemma 2.3, Remark 2.4 and Lemma 2.3, there exists
So we can begin by constructing a solution v in (0, τ0 ∧ T ) × Ω. In case τ0 < T , we can consider (2.25) In a finite number of steps we get (I) and (II). (III) and (IV) can be obtained as in the proof of Lemma 2.2.