, Manuel Milla Miranda

Local solutions for a hyperbolic equation

Aldo Trajano Lourêdo

, Manuel Milla Miranda

, Marcondes Rodrigues Clark

and Haroldo Rodrigues Clark

1Universidade Estadual da Paraíba, Departamento de Matemática, Campina Grande, PB, Brazil

2Universidade Federal do Piauí, Departamento de Matemática, Teresina, PI, Brazil

3Universidade Federal Fluminense, Instituto de Matemática e Estatística, Niterói, RJ, Brazil

Received 11 October 2014, appeared 27 May 2015 Communicated by Vilmos Komornik

Abstract. LetΩbe an open bounded set ofRⁿ with its boundaryΓconstituted of two disjoint partsΓ0andΓ1withΓ0∩_Γ1 =∅. This paper deals with the existence of local solutions to the nonlinear hyperbolic problem

u⁰⁰− 4u+|u|^ρ= f inΩ×(0,T0), u=0 onΓ0×(0,T0),

∂u

∂ν +h(·,u⁰) =0 onΓ1×(0,T0),

(∗)

where ρ > 1 is a real number, ν(x) is the exterior unit normal at x ∈ Γ1 and h(x,s) (forx ∈Γ1ands∈R) is a continuous function and strongly monotone ins. We obtain existence results to problem (∗) by applying the Galerkin method with a special basis, Strauss’ approximations of continuous functions and trace theorems for non-smooth functions. As usual, restrictions on ρ are considered in order to have the continuous embedding of Sobolev spaces.

Keywords: Galerkin method, local solutions, hyperbolic equation.

2010 Mathematics Subject Classification: 35L05, 35L70, 35G30.

1 Introduction

Motivated by a nonlinear theory of mesons field introduced by L. I. Schiff [27], K. Jörgens in [5, 6] began a rigorous mathematical investigation, from a mathematical point of view, of equations of the type

∂²u

∂t² − 4u+F⁰(|u|²)u=_{0. (1.1)} Specifically, K. Jörgens [6] proved the existence and uniqueness of solutions for the equation

∂²u

∂t² − 4u+µ²u+η²|u|²u=0 inΩ×(0,∞),

BCorresponding author. Email: aldotl@cct.uepb.edu.br

whereΩis a bounded open set ofRⁿwith boundaryΓ. This equation is of the type (1.1) when F(s) =µ²s+¹

2η²s².

Motivated by the works of K. Jörgens [5,6], the authors J.-L. Lions and W. A. Strauss [28]

initiated and developed a large field of research on nonlinear evolution equations that includes K. Jörgens’ model. See also, F. E. Browder [1], J. A. Goldstein [3, 4], L. A. Medeiros [14], I. E. Segal [26], W. A. Strauss [28] and von Wahl [30].

Medeiros et al. [15] proved the existence and uniqueness of global solutions of the nonlin- ear hyperbolic problem

u⁰⁰− 4u+|u|^ρ= f in Ω×(0,∞),

u=0 on Γ×(0,∞),

u(x, 0) =u⁰(x), u⁰(x, 0) =u¹(x) in Ω,

(1.2)

whereρ >1 is a real number with restrictions given by the continuous embedding of Sobolev spaces and the initial datau⁰ andu¹do not have restrictions on their norms.

Considering the boundary Γ of Ω constituted of two disjoint parts Γ0 and Γ1 such that Γ0∩_Γ1 = _∅ and denoting by ν(x) the unit exterior normal vector at x ∈ _Γ1, Milla Miranda and Medeiros [20] studied the existence and uniqueness of solutions of the problem

u⁰⁰−µ(t)4u=0 inΩ×(0,∞),

u=0 on Γ0×(0,∞),

µ(t)^∂u

∂ν +δ(x)u⁰ =0 on Γ1×(0,∞), u(x, 0) =u0(x), u⁰(x, 0) =u₁(x) inΩ.

(1.3)

When µ > 0 is constant, existence and uniqueness of global strong solutions for (1.3) has been proved by Komornik and Zuazua [7], Quinn and Russell [25] applying semigroup the- ory. This method does not work for (1.3) because the boundary condition (1.3)₃ depends on µ(t). For this reason Milla Miranda and Medeiros [20] constructed a special basis where lie approximations of the initial data, so the Galerkin method works well with this basis. Using this approach they proved the well-posedness for (1.3).

The existence of solutions of problem (1.3) with nonlinear boundary conditions has been obtained, by using the theory of monotone operators by Zuazua [7], Lasiecka and Tataru [18], and applying the Galerkin method by Lourêdo and Milla Miranda [12].

Motivated by (1.2) and (1.3) we consider in this paper the following problem:

u⁰⁰− 4u+|u|^ρ= f in Ω×(0,T₀),

u=0 on Γ0×(0,T₀),

∂u

∂ν

+h(·,u⁰) =0 on Γ1×(0,T₀), u(x, 0) =u⁰(x), u⁰(x, 0) =u¹(x) in Ω.

(1.4)

With restrictions on the real numberρ > 1 due to the continuous embedding of Sobolev spaces, we obtain the existence of local solutions to problem (1.4) in two cases: first,h(x,s) = δ(x)p(s) with p Lipschitzian and strongly monotone. In the second case h(x,s)is only con- tinuous in s and strongly monotone in s, and the initial data belong to a class more regular

than in the first case. In our approach, we apply the Galerkin method with a special ba- sis, the Strauss’ approximations of continuous functions and trace theorems for non-smooth functions.

It is worth noting that the termR

Ω|u|^ρu⁰dxdoes not have a definite sign. This fact brings serious difficulties to obtain global solutions to problem (1.4) without considering restrictions on the norms of the initial data.

Hereafter, this paper is organized in three sections, namely, Section 2 is devoted to the notations and statements of the two main results. In Section 3 we present the proof of Theorem 2.1 in which the case h = δp is considered, where p is a Lipschitz continuous function. In Section 4, Theorem2.2is proved which contains the case wheres7→ h(·,s)is only a continuous function ins.

2 Notations and main results

Let Ωbe an open bounded set ofRⁿ with aC² boundary Γ, which has two disjoint parts Γ0

and Γ1 such that measΓ0 > _{0; measΓ1} > _{0; and Γ0}∩_Γ1 = _{∅. Let} ν(x) be the unit normal vector atx ∈_Γ1.

The scalar product and norm of the space L²(_Ω)will be denoted by(·,·)and| · |, respec- tively. We represent by V the Hilbert spaceV = {v ∈ H¹(_Ω)_; v = _{0 on Γ0}}, equipped with the scalar product and norm

((u,v)) =

∑

Z

Ω

∂u

∂x_i(x)^∂v

∂x_i(x)dx and kuk² = ((u,u)), respectively. All scalar functions considered in this paper are real-valued.

In what follows, we introduce necessary hypotheses on some objects of problem (1.4) in order to state our first result.

Let p: R→_Rbe a function satisfying:

pis Lipschitz-continuous and strongly monotone in the second variable, i.e.

(p(s)−p(r))(s−r)≥b₀(s−r)², ∀s, r∈_{R, (b0} is a positive constant). (2.1) The functionδ: Γ1 →_Ris such that

δ ∈W^1,∞(_Γ1)andδ(x)≥δ0, ∀x∈_Γ1 (δ0is a positive constant). (2.2) The real numberρis chosen according to the spatial dimensionn.

ρ>1 ifn=1, 2 and n+1

n ≤ρ≤ ⁿ

n−2 ifn≥3. (2.3)

Theorem 2.1. Suppose (2.1)–(2.3) hold, f ∈ H¹(0,T;L²(_Ω)), {u⁰,u¹} ∈ (V∩H²(_Ω))×V and satisfies the compatibility condition

∂u⁰

∂ν

+δ(·)p(_u¹) =_{0 onΓ1. (2.4)}

Then there exist a real number T₀with0< T₀≤ T and a unique function u in the class u∈L^∞(0,T0;V∩H²(_Ω)),

u⁰ ∈L^∞(_0,T₀;V)_,

u⁰⁰ ∈L^∞(0,T₀;L²(_Ω))∩L²(0,T₀;L²(_Γ1)),

(2.5)

satisfying the equations

u⁰⁰− 4u+|u|^ρ = f in L^∞(0,T0;L²(_Ω)), (2.6)

∂u

∂ν +δp(u⁰) =0 in L^∞(0,T₀;H^1/2(_Γ1)),

∂u⁰

∂ν +δp⁰(u⁰)u⁰⁰ =0 in L^∞(0,T₀;L²(_Γ1)),

(2.7) and the initial conditions

u(0) =u⁰, u⁰(0) =u¹. (2.8) Moreover, T₀is explicitly given by

T₀ =min 1

2L 1

2|u¹|²+ ¹

2ku⁰k²+2(1−ρ)/2

, T

, (2.9)

where

L= (ρ−1) 2

h

2^1/2kfk_L∞(0,T;L²(_Ω))+2^(ρ+1)/2k^ρ₁i

(2.10) and k₁>0is the constant of the continuous embedding of V in L²(_Ω), defined in inequality(3.2).

To state our second result we make the following considerations: let A= −4be the self- adjoint operator of L²(_Ω)defined by the triplet {V,L²(_Ω)_;((·,·))}. Then the domain of −4 is given by

D(−4) =

u∈V∩H²(_Ω); ∂u

∂ν =0 on Γ1

(2.11) and it is known thatD(−4)is dense inV, see this statement for instance, in Lions [10].

We suppose the functionh: Γ1×_R→_Rsatisfies

r 7→h(·,r)∈C⁰(_R;L^∞(_Γ1)), h(x, 0) =0, for almost allx ∈_Γ1, h is strongly monotone in the second variable, i.e.

[h(x,r)−h(x,s)] (r−s)≥d₀(r−s)², ∀r,s∈R, for almost allx ∈_Γ1, (d₀is a positive constant).

(2.12)

Theorem 2.2. Assume that hypotheses(2.3)–(2.12)are satisfied, f ∈ H¹(0,T;L²(_Ω))and{u⁰,u¹} ∈ D(−4)×H₀¹(_Ω).Then there exist a real number T₀ > 0(the same T₀given in Theorem2.1) and at least one function u in the class

u∈ L^∞(0,T₀;V), 4u ∈L^∞(0,T₀;L²(_Ω)), u⁰ ∈ L^∞(0,T₀;V),

u⁰⁰ ∈ L^∞(0,T₀;L²(_Ω))∩L²(0,T₀;L²(_Γ1)),

(2.13)

satisfying

u⁰⁰− 4u+|u|^ρ = f in L^∞(0,T0;L²(_Ω)), (2.14)

∂u

∂ν+h(·,u⁰) =0 in L¹(0,T₀;L¹(_Γ1)), (2.15) and

u(0) =u⁰, u⁰(0) =u¹. (2.16) Remark 2.3. Note that in Theorem2.2the set of initial data satisfies ^∂u_∂ν⁰ +h(·,u¹) =0.

Remark 2.4. Ash(x,s)is only continuous ins, the uniqueness of solutions of Theorem2.2is an open problem.

3 Case p Lipschitz

We begin by making some considerations. Since p:R→_Ris a Lipschitz continuous function then p(v)∈ H^1/2(_Γ1)forv∈ H^1/2(_Γ1)and the mapping p: H^1/2(_Γ1)→ H^1/2(_Γ1), v 7→ p(v) is continuous, for this result we refer to Marcus and Mizel [13].

Remark 3.1. The regularity of the trace mapping of order zero, γ₀: V → H^1/2(_Γ1) _ensures that the mapping pe= p◦γ₀with ep:V →H^1/2(_Γ1)is continuous.

Remark 3.2. Throughout this section, in order to facilitate the notation, the mapping ep(v)for v∈Vwill be denoted just by p(v).

Remark 3.3. Sinceδ ∈W^1,∞(_Γ1)thenδv∈ H^1/2(_Γ1)for all v∈ H^1/2(_Γ1), and the linear oper- atorδ: H^1/2(_Γ1)→ H^1/2(_Γ1), v 7→δvis continuous. In fact, using the theory of interpolation for Hilbert spaces (see for instance the reference [11]) it can be shown that the linear operators δ: H¹(_Γ1)→ H¹(_Γ1), v7→δvandδ: L²(_Γ1)→ L²(_Γ1), v7→δvare continuous.

Remark 3.4. As a consequence of (2.11), the intersectionV∩H²(_Ω)is dense inV.

Proposition 3.5. In V∩H²(_Ω)the norm of H²(_Ω)and the norm

u7→

"

|4u|²+

∂u

∂ν

2 H^1/2(_Γ1)

#1/2

are equivalent.

Proposition 3.6. Letδ ∈ W^1,∞(_Γ1), p: R→ _Rbe a Lipschitz continuous function with p(0) =0, u⁰∈V∩H²(_Ω), u¹∈ V, and

∂u⁰

∂ν +δ(·)p(u¹) =0 onΓ1. Then, for each ε>0there exist w and z in V∩H²(_Ω)such that

kw−u⁰k_V∩H2(_Ω)<ε, kz−u¹k<ε and ∂w

∂ν +δ(·)p(z) =0 on Γ1.

The proof of the preceding propositions can be found in Milla Miranda and Medeiros [20]

and Milla Miranda and Lourêdo [19].

Under the restrictions (2.3) on ρ, we have (ρ−₁)n ≤ _2ρ ≤ _n²ⁿ₋₂ = q for n ≥ 3, and this implies

V ,→L^q(_Ω),→L^2ρ(_Ω),→L^(ρ−1)n(_Ω), n≥3. (3.1) In (3.1) we mean byX ,→Y that the spacesX,Ysatisfy X ⊂Y and the injection of X inY is continuous. We denote by k₀,k₁andk₂the constants immersion that satisfying

kuk_Lq(_Ω)≤ k0kuk, kuk_L2ρ(_Ω) ≤k₁kuk, kuk_L(ρ−1)n(_Ω)≤k2kuk ∀u∈V. (3.2) We now can proceed to the proof of our first result.

Proof of Theorem2.1. Proposition3.6 provides us sequences(u⁰_l)and(u¹_l)of vectors ofV∩ H²(_Ω)such that

llim→_∞u⁰_l =u⁰ inV∩H²(_Ω),

llim→_∞u¹_l =u¹ inV,

∂u⁰_l

∂ν +δp(u¹_l) =0 on Γ1 for l∈ _N.

(3.3)

We now construct a special basis ofV∩H²(_Ω)in the following way: forl∈_Nwe consider the basis{w^l₁,w^l₂, . . . ,w^l_j, . . .}ofV∩H²(_Ω)satisfyingu⁰_l,u¹_l ∈ [w^l₁,w₂^l], where[w^l₁,w^l₂]denotes the subspace generated by w^l₁, w^l₂. According to this basis we determine approximate solutions u_lm(t)of problem (3.4) with h = δp, that is, u_lm(t) = _∑^m_j=1g_jlm(t)w^l_j, where g_jlm(t) is defined as the solutions of the approximate problem

(u⁰⁰_lm(t),v) + ((u_lm(t),v)) + (|u_lm(t)|^ρ,v) +

Z

Γ1

δp(u⁰_lm(t))_{v dΓ}= (f(t),v), ∀v∈V_m^l, u_lm(0) =u⁰_l, u⁰_lm(0) =u¹_l,

(3.4)

whereV_m^l is the subspace generated byw^l₁,w^l₂, . . . ,w^l_m. The solution u_lm of (_3.4)is defined on [0,t_lm) with 0 < t_lm ≤ T₀. The next estimate enables us to extend u_lm to the whole interval [0,T₀].

First estimate: Settingv=u⁰_lm(t)in (3.4), we obtain 1

2 d

dt|u⁰_lm(t)|²+¹ 2

d

dtku_lm(t)k²+

Z

Γ1

δp(u⁰_lm(t))u⁰_lm(t))_dΓ

= (f(t),u⁰_lm(t))−(|u_lm(t)|^ρ,u⁰_lm(t)).

(3.5)

By usual inequalities and (3.2) we get

|(|u_lm(t)|^ρ,u⁰_lm(t))| ≤ ku_lm(t)k^ρ

L^2ρ(_Ω)|u⁰_lm(t)| ≤k^ρ₁ku_lm(t)k^ρ|u⁰_lm(t)|. (3.6) Taking

ϕ_lm(t) = ¹

2|u⁰_lm(t)|²+ ¹

2ku_lm(t)k²+1 (3.7)

and combining (3.6), (3.7), (3.5), and using hypotheses (2.1) and (2.2) on pandδ, we get d

dtϕ_lm(t) +δ0b0

Z

Γ1

u⁰_lm²(t)_dΓ≤ |f(t)||u⁰_lm(t)|+k^ρ₁ku_lm(t)k^ρ|u⁰_lm(t)|. (3.8) Observing that

ku_lm(t)k^ρ ≤2^ρ2ϕ

ρ 2

lm(t) and |u⁰_lm(t)| ≤2¹²ϕ

1 2

lm(t), and together withϕ_lm(t)≥1, we find

|f(t)||u⁰_lm(t)|+k^ρ₁ku_lm(t)k^ρ|u⁰_lm(t)| ≤^h2¹²kfk_L∞(0,T;L²(_Ω))+2^ρ+21k^ρ₁i ϕ

ρ+1 2

lm (t). Combining this inequality with (3.8), we derive

d

dtϕ_lm(t) +δ0b0

Z

Γ1

u⁰_lm²(t)_dΓ≤ Mϕ

ρ+1 2

lm (t),

where

M=2¹²kfk_L∞(0,T;L²(_Ω))+2^ρ+21k^ρ₁. This means that

ϕ⁻

ρ+1 2

lm (t)^d

dtϕ_lm(t)≤ M, (3.9)

and recalling the identity d dt

2 1−ρϕ

1−ρ 2

lm (t)

= ϕ⁻

ρ+1 2

lm (t)^d

dtϕ_lm(t). We obtain by (3.9) that

d dtϕ

1−ρ 2

lm (t)≥ −^ρ−1 2 M, which implies

ϕ

1−ρ 2

lm (t)≥ ϕ

1−ρ 2

lm (0)−Lt. (3.10)

In the above expressionLwas defined in (2.10). By the convergences in (3.3) we find ϕ

ρ−1 2

lm (₀)<

1

2|u¹|²+¹

2ku⁰k²+₂ ^ρ−₂¹

, ∀l≥ l₀, ∀m, which is equivalent to

ϕ

1−ρ 2

lm (0)>

1

2|u¹|²+¹

2ku⁰k²+2 ¹⁻₂^ρ

, ∀l≥ l₀, ∀m.

Thus, from (3.10) it follows that ϕ

1−ρ 2

lm (t)≥ 1

2|u¹|²+¹

2ku⁰k²+2 ¹⁻₂^ρ

−Lt. (3.11)

By hypothesis (2.9), we obtain Lt≤ ¹

2 1

2|u¹|²+¹

2ku⁰k²+2 ¹⁻₂^ρ

, ∀t∈[0,T₀]. Then (3.11) provides

ϕ

1−ρ 2

lm (t)≥ ¹ 2

1

2|u¹|²+¹

2ku⁰k²+2 ¹⁻₂^ρ

, ∀t∈[0,T₀]. Thus

ϕ_lm(t)≤2^ρ−21 1

2|u¹|²+¹

2ku⁰k²+2

= N, ∀t∈[0,T0], ∀l≥l0, ∀m. (3.12) With this limitation and taking into account (3.7), we find

(u_lm)is bounded inL^∞(0,T₀;V), (u⁰_lm)is bounded inL^∞(0,T0;L²(_Ω)), (u⁰_lm)is bounded inL²(_0,T₀;L²(_Γ1))_,

(3.13)

where these limitations are independent ofl≥ l₀ andm.

Second estimate: Differentiating the approximate equation in (3.4)₁ with respect to t and takingv=u⁰⁰_lm(t), we find

1 2

d

dt[|u⁰⁰_lm(t)|²+ku⁰_lm(t)k²] + (ρ|u_lm(t)|^ρ−2u_lm(t)u⁰_lm(t)_,u⁰⁰_lm(t)) +

Z

Γ1

δp⁰(u⁰_lm(t))[u⁰⁰_lm(t)]²dΓ

= (f⁰(t)_,u⁰⁰_lm(t)) _(3.14)

By Hölder’s inequality with ¹_n+¹_q+ ¹₂ =1, we obtain

|(|u_lm(t)|^ρ−2u⁰_lm(t)_,u⁰⁰_lm(t))| ≤ ku_lm(t)k^ρ−1

L^(ρ−1)n(_Ω)ku⁰_lm(t)k_Lq(_Ω)|u⁰⁰_lm(t)|

From this, notations (3.2) and using the constant N introduced in (3.12) that bounds ϕ_lm(t), we get

ρ|(|u_lm(t)|^ρ−2u⁰_lm(t),u⁰⁰_lm(t))| ≤ρk^ρ₂⁻¹k₀ku⁰_lm(t)k^ρ−1ku⁰_lm(t)k|u⁰⁰_lm(t)|

≤ρk^ρ₂⁻¹k₀2^ρ−21N^ρ−21ku⁰_lm(t)k|u⁰⁰_lm(t))|, that is

ρ|(|u_lm(_t)|^ρ−2u⁰_lm(_t)_,u⁰⁰_lm(_t))| ≤ Rku⁰_lm(_t)k|u⁰⁰_lm(_t))|

where

R=ρk^ρ₂⁻¹k₀2^ρ−21N^ρ−21.

Combining the last inequality with (3.14) and considering hypothesis(_2.1)₃, we obtain 1

2 d

dt[|u⁰⁰_lm(t)|²+ku⁰_lm(t)k²] +δ₀b0

Z

Γ1

[u⁰⁰_lm(t)]²_dΓ

≤ ¹

2|f⁰(t)|²+ ^R

2

2 ku⁰_lm(t)k²+¹

2|u⁰⁰_lm(t)|, ∀t∈[0,T₀]. Then

1 2

h|u⁰⁰_lm(t)|²+ku⁰_lm(t)k²ⁱ+δ₀b₀ Z _t

0

Z

Γ1

[u⁰⁰_lm(t)]²_dΓ

≤ ¹

2|u⁰⁰_lm(0)|²+ ¹

2ku_lm(0)k²+¹ 2

Z _T

0

|f⁰(t)|²dt +

Z _t

0

R²

2 ku⁰_lm(s)k²+¹

2|u⁰⁰_lm(s)|

ds, ∀t∈ [0,T₀].

(3.15)

Remark 3.7. To apply Gronwall’s lemma in inequality (3.15) we need to derive an upper bound for(u⁰⁰_lm(0)). This is the key point of the proof of Theorem 2.1. We get this limitation thanks to the choice of the special basis ofV∩H²(_Ω), previously built in this section.

We make t = 0 in the approximate system (3.4)₁, take v = u⁰⁰_lm(0), and after that apply Gauss’ theorem, to obtain

|u⁰⁰_lm(0)|²+ (−4u⁰_l,u⁰_lm(0)) +

Z

Γ1

"

∂u⁰_l

∂ν +δp(u¹_l)

#

u⁰⁰_lm(0)_dΓ+ (|u⁰_l|^ρ,u⁰⁰_lm(0)) = (f(0),u⁰⁰_lm(0)). Using(3.3)₃, we find

|u⁰⁰_lm(0)|² ≤ |4u⁰_l||u⁰⁰_lm(0)|+ku⁰_lk^ρ

L^2ρ(_Ω)|u⁰⁰_lm(0)|+|f(0)||u⁰⁰_lm(0)|. From this and convergence (3.3)₁ and (3.3)₂, and notations (3.2) we get

|u⁰⁰_lm(0)|²≤|4u⁰_l|+k₁ku⁰k+|f(0)|+1+k^ρ₁

|u⁰⁰_lm(0)|, ∀l≥l₀, ∀m.

Thus,

|u⁰⁰_lm(0)| ≤ |4u⁰_l|+k₁ku⁰k+|f(0)|+1+k^ρ₁ =S, ∀l≥l0, ∀m. (3.16) Inequality (3.15), (3.16), convergences (3.3)₁, (3.3)₂and Gronwall’s lemma provide

1

2|u⁰⁰_lm(t)|+ ¹

2ku⁰_lm(t)k²+δ₀b₀ Z _t

0

ku⁰⁰_lm(s)k²_L2(Γ

1)dΓ≤ P, (3.17)

where the constant Pis independent ofl≥l₀, mandt ∈[0,T₀]. With the above estimate, we obtain

(u⁰_lm)is bounded inL^∞(0,T₀;V), (u⁰⁰_lm)is bounded inL^∞(0,T₀;L²(_Ω)), (u⁰⁰_lm)is bounded inL²(0,T0;L²(_Γ1)).

(3.18)

Passage to the limit in m. The constants N and P in (3.12) and (3.17) are independent of l≥ l₀,mandt ∈ [0,T₀]. Thus, the estimates (3.13) and (3.18), allow us to find a subsequence of (u_lm), which still will be denoted by(u_lm), and a functionu_l such that

u_lm →u_l weak star inL^∞(0,T₀;V), u⁰_lm →u⁰_l weak star inL^∞(0,T₀;V), u⁰⁰_lm →u⁰⁰_l weak star inL^∞(0,T₀;L²(_Ω)), u⁰_lm →u⁰_l weak inL²(0,T0;L²(_Γ1)), u⁰⁰_lm →u⁰⁰_l weak in L²(_0,T0;L²(_Γ1))_.

(3.19)

The convergence (3.19)₁, (3.19)₂and the Aubin–Lions theorem provide the convergenceu_lm → u_l in L²(0,T₀;L²(_Ω)). Therefore,

|u_lm(x,t)|^ρ → |u_l(x,t)|^ρ a.e. inΩ×(0,T₀) =Q₀. (3.20)

We also have _Z

Ω[|u_lm(t)|^ρ]²dx≤k₁ku_lm(t)k^2ρ≤k^2ρ₁ 2^2ρN^ρ, wherek1 andNare introduced in (3.2) and (3.12), respectively. From this

k|u_lm|^ρk_L∞(0,T₀;L²(_Ω))≤C, ∀l≥l₀ and ∀m. (3.21) Applying Lemma 3.1 of Lions [8] and using (3.20) and (3.21),

|u_lm|^ρ → |u_l|^ρ weak star inL^∞(0,T₀;L²(_Ω)). (3.22) On the other hand, the convergence(3.19)₂ implies u⁰_lm → u⁰_l weak in L²(0,T0;H^1/2(_Γ1)). This, convergence (3.19)₅ and the Aubin–Lions theorem provideu⁰_lm →u⁰_l in L²(0,T₀;L²(_Γ1)). As p is Lipschitzian, δp(u⁰_lm) → δp(u⁰_l)in L²(0,T₀;L²(_Γ1)). Note that |δp(u⁰_lm)|_L2(_Γ1) ≤ C for alll≥l0, for allmandt∈ [0,T0]. This and the preceding convergence provide

δp(u⁰_lm)→δp(u⁰_l) _inL^∞(_0,T₀;L²(_Γ1))_{. (3.23)} Now taking θ ∈ L²(0,T₀) andv ∈ V then the convergence (3.19), (3.22), and (3.23) allow us to pass to the limit in the approximate equation (3.4). Moreover, observing thatV∩H²(_Ω) is dense inV, we obtain

Z _T0

0

(u⁰⁰_l(t)_,v)θdt+

Z _T0

0

((u_l(t)_,v))θdt+

Z _T0

0

(|u_l(t)|^ρ_,v)θdt+

Z _T0

0

Z

Γ1

δp(u⁰_l(t))vθdΓdt

=

Z _T0

0

(f(t),v)θdt.

(3.24)

Taking, in particular,v∈ D(_Ω)andθ ∈ D(0,T₀), in the preceding equation yields u⁰⁰_l − 4u_l+|u_l|^ρ= f inD⁰(Q₀), whereQ₀ =_Ω×(0,T₀). The regularities ofu_l and f permit us to write

u⁰⁰_l − 4u_l+|u_l|^ρ= f in L^∞(0,T₀;L²(_Ω)). (3.25) Sinceu_l ∈ L^∞(0,T0;V)and4u_l ∈L^∞(0,T0;L²(_Ω)), then ^∂u_∂ν^l ∈ L^∞(0,T0;H^−1/2(_Γ1)), as shown in Milla Miranda [21]. Multiplying both sides of (3.25) byvθ, with v ∈ V andθ ∈ L²(0,T₀), and integrating onQ₀, we obtain

Z _T0

0

(u⁰⁰_l(t),v)θdt+

Z _T0

0

((u_l(t),v))θdt+

Z _T0

0

(|u_l(t)|^ρ,v)θdt+

Z _T0

0

∂u_l(t)

∂ν ,v

θdt

=

Z _T0

0

(f(t)_,v)θdt,

whereh·,·idenotes the duality pairingH^−1/2(_Γ1)andH^1/2(_Γ1). This equality and (3.24) imply

∂u_l

∂ν

+δp(u⁰_l) =0 in L²(0,T₀;H^−1/2(_Γ1)). Sinceu⁰_l ∈ L^∞(0,T₀;H^1/2(_Γ1)), thenδp(u⁰_l)∈L^∞(0,T₀;H^1/2(_Γ1)), and thus

∂u_l

∂ν

+δp(u⁰_l) =_{0 in} L^∞(_0,T₀;H^1/2(_Γ1))_{. (3.26)} By the facts u_l,4u_l ∈ L^∞(_0,T0;L²(_Ω)) _and ^∂ul

∂ν ∈ L^∞(_0,T0;H¹²(_Γ1)), and by Proposition3.5, we conclude

u_l ∈L^∞(0,T₀;V∩H²(_Ω)). (3.27) Differentiating with respect to t the equality (3.26) and noting thatu⁰⁰_l ∈ L²(0,T₀;L²(_Γ1)), we obtain the regularities in (2.7). As estimates (_3.13) _and(_3.18) are independent of l ≥ l₀, we obtain in a similar way a functionuin class (2.5),usatisfying (2.6)-(2.8).

The verification of the initial data (2.8) follows from estimatee (3.19)₁– (3.19)₃.

Uniqueness of solutions. Letuandvbe two functions in class (2.5) which satisfy equations (2.6), (2.7) and initial conditions (2.8). Considering the differencew=u−v, we have

w⁰⁰− 4w+|u|^ρ− |v|^ρ= inL^∞(0,T₀;L²(_Ω)),

∂w

∂ν +δ

p(u⁰)−p(v⁰)=_{0 in} L^∞(_0,T₀;H¹²(_Γ1))_, w(0) =0, w⁰(0) =0.

(3.28)

Multiplying both sides of(3.28)₁byw⁰, integrating onΩand using Gauss’ theorem, we obtain 1

2 d dt

|w⁰(t)|²+kw(t)k²+

Z

Γ1

δ[p(u⁰(t))−p(v⁰(t))]_dΓ=− |u(t)|^ρ− |v(t)|^ρ,w⁰(t). (3.29) By the mean value theorem,|u(x,t)|^ρ− |v(x,t)|^ρ =ρ|ξ|^ρ−2ξw(x,t), whereξ is betweenu(x,t) and v(x,t)_{, and thus}u(x,t)|^ρ− |v(x,t)|^ρ ≤ |g(x,t)|^ρ−2|w(x,t)|_{, where} g(x,t) = |u(x,t)|+

|v(x,t)|. Therefore

|(|u(t)|^ρ− |v(t)|^ρ)w⁰(t)| ≤ρ Z

Ωg^ρ−1|w(t)||w⁰(t)|dx

≤ρkgk^ρ−1

L^(ρ−1)n(_Ω)kw(t)k_Lq(_Ω)|w⁰(t)|

≤ρ h

ku(t)k_L(ρ−1)n(_Ω)+kv(t)k_L(ρ−1)n(_Ω)

iρ−1

kw(t)k_Lq(_Ω)|w⁰(t)|.

(3.30)

From the embedding (3.1) we find ku(t)k_L(ρ−1)n(_Ω) ≤ k₂ku(t)k ≤ C, ∀t ∈ [0,T₀] and simi- larly, kv(t)k_L(ρ−1)n(_Ω) ≤ k₂ku(t)k ≤ C, ∀t ∈ [0,T₀]. Combining (3.30) with the two preceding inequalities, we get

|(|u(t)|^ρ− |v(t)|^ρ)| ≤C₁kw(t)k|w⁰(t)| ≤ ^C

21

2 kw(t)k²+¹

2|w⁰(t)|². This inequality, (3.29) and properties (2.1) of p, imply

1 2

d dt

|w⁰(t)|²+kw(t)k²+δ₀b₀ Z

Γ1

w⁰(t)²dΓ≤ ¹

2|w⁰(t)|+^C

12

2 kw(t)k².

Then the Gronwall inequality provides w⁰(t) = 0 andw(t) = 0 a.e. in [0,T₀]. This concludes the proof of Theorem2.1.

4 Case h ( x, s ) continuous in s

Initially note that, sincehis a continuous function, the following Strauss’ approximations were shown by Louredo and Milla Miranda [12] .

Proposition 4.1. Assume that h satisfies hypotheses (2.11). Then there exists a sequence (h_l) of functions of C⁰(_R;L^∞(_Γ1))satisfying the following conditions:

(i) h_l(x, 0) =0for almost all x inΓ1;

(ii) [h_l(x,s)−h_l(x,r)] (s−r)≥d₀(s−r)²,∀s,r∈ _Rand for almost all x inΓ1; (iii) there exists a function c_l ∈ L^∞(_Γ1)such that

|h_l(x,s)−h_l(x,r)| ≤c_l(x)|s−r|, ∀s,r∈_R for almost all x inΓ1; (iv) (h_l)converges to h uniformly on bounded sets ofRfor almost all x inΓ1.

Proof of Theorem2.1. We proceed as in Theorem 2.1, changing the function δ(x)p(s) into h_l(x,s). Let(u¹_l)be a sequence of functions ofD(_Ω)such that

u¹_l →u¹ in H₀¹(_Ω)_{. (4.1)}

Note that

∂u⁰

∂ν +h_l(·,u¹_l) =0, ∀l. (4.2)

For fixedl∈ _Nwe construct the special basis {w₁^l,w^l₂, . . .}ofV∩H²(_Ω)such that u⁰_l,u¹_l be- long to[w^l₁,w₂^l]. With this basis we determine the approximate solutionsu_lm(t) =_∑^m_j=1g_jlm(t)w^l_j of (4.3), whereg_jlm(t)is defined by the approximate problem

(u⁰⁰_lm(t),v) + ((u_lm(t),v)) + (|u_lm(t)|^ρ,v) +

Z

Γ1

h_l(·,u⁰_lm(t))_{v dΓ}= (f(t),v), ∀v∈V_m^l, u_lm(0) =u⁰, u⁰_lm(0) =u¹_l.

(4.3) In a similar way as we made to obtain the estimates (3.12), (3.16) and (3.17) of Section 3, we find

1

2|u⁰_lm(t)|²+ ¹

2ku_lm(t)k² ≤N, d₀ Z _t

0

Z

Γ1

[u⁰_lm(x,s)]²dΓds≤D, 1

2|u⁰⁰_lm(t)|²+¹

2ku⁰_lm(t)k²+d₀ Z _t

0

Z

Γ1

[u⁰⁰_lm(x,s)]²_dΓds≤P,

(4.4)

where the constants N,D and P are independent of l ≥ l₀, mand t ∈ [_0,T₀]. The estimates (4.4) provide a subsequence of(u_lm), which still will be denoted by(u_lm), such that

u_lm →u_l weak star inL^∞(0,T₀;V), u⁰_lm →u⁰_l weak star inL^∞(0,T₀;V), u⁰⁰_lm →u⁰⁰_l weak star in L^∞(0,T0;L²(_Ω)), u⁰_lm →u⁰_l weak inL²(_0,T₀;L²(_Γ1))_, u⁰⁰_lm →u⁰⁰_l weak in L²(0,T₀;L²(_Γ1)).

(4.5)

In a similar way as in the convergence(3.22)and(3.23), we get

|u_lm|^ρ → |u_l|^ρ weak star inL^∞(0,T₀;L²(_Ω)),

h_l(·,u⁰_lm)→h_l(·,u⁰_l)weak star inL^∞(0,T₀;L²(_Γ1)). (4.6) Convergence (4.5) and (4.6) allow us to pass to limit in m in the approximate equations of (4.3). Therefore,

Z _T0

0

(u⁰⁰_l(t),v)θdt+

Z _T0

0

((u_l(t),v))θdt+

Z _T0

0

(|u_l(t)|^ρ,v)θdt+

Z _T0

0

Z

Γ1

h_l(·,u⁰_l)vθdΓdt

=

Z _T0

0

(f(t),v)θdt,

forv ∈Vandθ ∈L²(0,T0). By analogous arguments used to obtain (3.25) and (3.26), we find u⁰⁰_l − 4u_l+|u_l|^ρ= f in L^∞(0,T₀;L²(_Ω)), (4.7)

∂u_l

∂ν +h_l(·,u⁰_l) =0 in L^∞(0,T₀;H^1/2(_Γ1)). (4.8) Estimates (4.4) imply in the existence of a subsequence of(u_l), which still will be denoted by(u_l), and a functionusuch that

u_l →uweak star inL^∞(0,T₀;V), u⁰_l →u⁰ weak star inL^∞(0,T₀;V), u⁰⁰_l →u⁰⁰ weak star in L^∞(0,T0;L²(_Ω)), u⁰_l →u⁰ weak in L²(_0,T₀;L²(_Γ1))_, u⁰⁰_l →u⁰⁰ weak in L²(0,T₀;L²(_Γ1)).

(4.9)