Nonlinear Parabolic Equation on Manifolds

In this work we investigate the existence and the uniqueness of solution for a nonlinear differential equation of parabolic type on the lateral boundary Σ of a cylinder Q, cf. (1). An important part of our study is to transform this initial value problem into another one whose differential operator equation is of the type ut + a (∫ Γ udx ) Au − ΔΓu + u2k+1 = f on Σ, cf. (9), where k is a positive integer. The operatorA acts in Sobolev spaces on Γ, boundary of Ω. The initial value problem (9) will be studied in Section 4. Thus, we obtain the existence and the uniqueness of weak solution for (9).


Introduction
We consider Ω a bounded open set of R n (n ≥ 2) with C ∞ boundary Γ.By ν we denote the outward normal unit vector field defined on Γ.For each T > 0, Q = Ω × ]0, T [ denotes a cylindrical domain whose lateral boundary will be represented by Σ = Γ × ]0, T [.Our main objective is to investigate existence and uniqueness of solution for the following problem: where k is a positive integer, the derivatives are in the sense of the theory of distributions, ∂w ∂ν is the normal derivative of w, by Δ Γ we denote the Laplace Beltrami operator on Γ, the Laplace operator Δ acts only on space variables and w = w (x, t), x ∈ Ω, 0 < t < T .This work was motivated by J. L. Lions who has considered, in 1969, the existence and uniqueness of solution for nonlinear problems on manifolds whose unknown function satisfies the Laplace equation in Ω and a nonlinear evolution equation on its lateral boundary Σ.
The nonlinearity of the type a Γ wdΓ was motivated by the study of problems of diffusion of population cf.Chipot (2000, Chapters 1 and 12) and also Menezes (2006).
Our paper is organized as follows: in section 2, we establish the appropriate notation and the functional setting to the treatment of our problem.In section 3 we develop a classical formalism in order to investigate (1) as a differential operator equation whose operator A acts on Sobolev spaces defined on the manifold Γ.In this way the Equation (1) 2 is formulated as a differential operator equation of the type for which we can apply a known methodology for the initial value problem (9).In section 4, we investigate the initial value problem (9) by approximate method and we succeed to prove existence and uniqueness for weak solutions.
In the present paper we need the embedding of H s (Γ) into L 4k+2 (Γ), for s ≥ 2 and k a positive integer.In fact, by Sobolev embedding theorem, cf.Lions (2003) or Lions and Magenes (1968) Finally, we suppose a (s), s ∈ R, real continuous function, with bounded derivative and a (s) ≥ a 0 > 0, for all s ∈ R. (6)

Formulation of the Problem (1) on Σ
In (Antunes, Lopez, Silva, & Araújo 2013), we defined an operator A ∈ L H 1/2 (Γ) , H −1/2 (Γ) which is a composition of the traces γ 0 , γ 1 , these are, roughly speaking, respectively, ∂w ∂ν and w restricted to Γ.To avoid duality pairing in the process of approximation we define, in the present argument, an operator A: H 1 (Ω) −→ L 2 (Γ) and we obtain scalar product instead of duality.
We have the Dirichlet problem: This bounded operator will be the "substitute" of the normal derivative in (1).
Moreover, we have, from ( 7) We formulate now the problem (1) on Σ.In fact, we define Thus, the problem (1) can be rewritten as follows: From now on, our objective will be to prove existence and uniqueness of solutions for the problem (9).

Main Results
In this section we formulate and prove existence and uniqueness of weak solutions for the mixed problem (9) on Σ.
as defined above.Then, there exists u: which is the unique weak solution of the initial value problem (9).
About the operator Δ Γ , we obtain its spectral resolution and we realize it as an operator from H 2 (Γ) in L 2 (Γ).For this argument we call attention to the reader to see (Lions & Magenes, 1968, p. 42).In fact, we deduce that the domain of −Δ Γ is H 2 (Γ) and its range is L 2 (Γ).We have the spectral resolution: where the eigenvectors w j are normalized in L 2 (Γ) and complete in H 2 (Γ).
We consider in H 2 (Γ) the complete orthonormal basis w j j∈N of eigenvectors of −Δ Γ and we define V m = [w 1 , ..., w m ] ⊂ H 2 (Γ), the subspace generated by the m first eigenvectors of −Δ Γ .
For each m ∈ N, we look for a function u m (t) = m j=1 g jm (t) w j in V m , such that u m (t) is solution of the approximate problem: Observe that by section 2, we obtained Observe that ( 10) is a nonlinear system of first order ordinary differential equation in g jm (t), the coordinates of the approximations u m (t).It has local solution defined in 0 ≤ t ≤ t m < T .By mean of estimates we extend these local solutions to the interval [0, T ].
Estimate 1 Setting v = 2u m (t) in (10), we have: We observe that: • From (8), we have (Au k+1) 2(k+1),Γ .Going back to (11), employing the results above and the results (8), about A, and ( 6), about a (s), we obtain: From ( 12), applying Gronwall lemma we conclude the existence of a positive constant C 1 depending only on and T , such that: Then, by the extension theorem for ordinary differential equation, the local solution u m (t) has an extension to the whole interval [0, T ].We represent the extension by the same notation u m (t).Thus, for the extension u m (t) we have ( 12) true for all t in [0, T ].Consequentely, it makes sense to integrate (12) on [0, t) ⊂ [0, T ].
Integrating ( 12) on (0, t) ⊂ [0, T ], by the hypothesis of f , the convergence in (10) 2 and the estimate in (13), we obtain: Estimate 2 Setting v = 2u m (t) in (10), we obtain, after some calculus, that Now we observe that where C 4 is a positive constant depending on the measure of Γ.By ( 16) and ( 13) we obtain From ( 17) and the continuity of the function a (s) it follows that Returning with ( 18) in ( 15) we get As the operator A ∈ L H 1 (Γ) , L 2 (Γ) then the last inequality is transformed into Integrating ( 19) from 0 to t, By the hypothesis of f , the convergence in ( 10) 2 and the definition of the norm in H s (Γ), observing ( 13) and ( 14), we obtain: 10), that makes sense because Δ Γ V m ⊂ V m , we have: Remark 1 For all v ∈ C ∞ (Γ), we have , we obtain that −2 u 2k+1 m , Δ Γ u m is non-negative.Note also the elementary inequality for positive real numbers: 2αβ ≤ α 2 + β 2 .From (22), employing (18) and Remark 1, we conclude that Integrating ( 23) from 0 to t ≤ T , we get Considering the convergence in (10) 2 , the hypothesis on f , the fact that the operator A ∈ L H 1 (Γ) , L 2 (Γ) and the limitation obtained in (21), from (24) we obtain the third estimate: Finally we observe that from ( 21), ( 25) and the norm defined in (4) we get Passage to the limit Thus we proved that the sequence of approximatios (u m ) m∈N is bounded in the spaces: L ∞ 0, T ; H 1 (Γ) , by (21); L 2 0, T ; H 2 (Γ) , by ( 26); L ∞ 0, T ; L 2k+2 (Γ) by ( 21) and u m m∈N bounded in L 2 0, T ; L 2 (Γ) by ( 20).
From the above estimates we deduce that there exists u μ μ∈N , subsequence of (u m ) m∈N , and a function u, such that On the other hand, from ( 18) and as the operator A ∈ L H 1 (Γ) , L 2 (Γ) , we conclude that ≤ C 10 a.e. on (0, T ) and therefore From now on, we will consider some subsequeces of u μ μ∈N that will be still denoted by u μ μ∈N .
Therefore, from ( 36) and (37), by a similar argument as that one employed to obtain (35), we get u 2k+1 μ u 2k+1 weakly in L q (0, T ; L q (Γ)) with q = 2k + 2 2k + 1 .Taking into account the convergences in (27) 3 , (27) 4 , ( 35) and ( 38), we can pass to the limit in the approximate equation to obtain We observe that, from the regularity obtained for u and u , we have that u (0) makes sense and, in fact, we can prove that u (0) = u 0 .

Uniqueness
Let us consider u 1 and u 2 solutions of (9).From Theorem 4.1 we know that From (39), we have, after some calculations, that z for all v ∈ L 2 (Γ).