2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 35–47. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

A SYSTEM OF SEMILINEAR EVOLUTION EQUATIONS WITH

HOMOGENEOUS BOUNDARY CONDITIONS FOR THIN PLATES COUPLED WITH MEMBRANES

JAIRO HERNÁNDEZ

Abstract. In this work we consider a semilinear initial boundary-value problem modelling an elastic thin plate (in the context of the so-called Kirchhoff-Love theory) coupled with an elastic membrane, regarding homogeneous boundary conditions. By means of the theory of strongly continuous semigroups of linear operators applied to abstract semilinear initial valued problems [16], we obtain existence and uniqueness of a weak solution which is defined in a suitable way.

1. Introduction

In this work we consider a semilinear evolution problem which we pose as follows: Let Ω and Ω_{m} be two open bounded connected subsets of ℝ^{2} with sufficiently smooth boundary ∂Ω and ∂Ω_{m} so that Ω_{m} ⊂⊂ Ω. Let Ω_{p} := Ω\\overline{Ω}_{m} and Γ_{1} := ∂Ω_{m}. We decompose ∂Ω in two connected parts Γ_{2} and Γ_{3} with Γ_{2} ∩ Γ_{3} = ∅, σ_{1}(Γ_{2}) ≠ 0 and σ_{1}(Γ_{3}) ≠ 0, where σ_{1} is the surface measure on ∂Ω, induced by the Lebesgue measure on ℝ (see figure 1). Then we consider the system of partial differential equations

\begin{matrix}ρ_{p}h\frac{∂^{2}u_{p}}{∂t^{2}} (t, x) + \frac{h^{3}}{12} ∑_{α,βγ,θ=1}^{2} \frac{∂^{2}}{∂x_{α}∂x_{β}} (A_{αβγθ}(x)\frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} (t, x)) \\ = f_{p}(t, x, u_{p}(t, x)) in ]0, T] × Ω_{p} \end{matrix}  (1.1)

ρ_{m} \frac{∂^{2}u_{m}}{∂t^{2}} (t, x) − CΔu_{m}(t, x) = f_{m}(t, x, u_{m}(t, x)) in ]0, T] × Ω_{m} ,  (1.2)

\begin{matrix}\frac{h^{3}}{12} ∑_{α,β,γ,θ=1}^{2} ν_{α} \frac{∂}{∂x_{β}} (A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} ) + \frac{h^{3}}{12} \frac{∂}{∂τ⃗} ( ∑_{α,β,γ,θ=1}^{2} ν_{α}τ_{β}A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} ) \\ = 0 on ]0, T] × Γ_{2} , \end{matrix} (1.3)

2000 Mathematics Subject Classification. 74H20, 74H25, 74K15.

Key words and phrases. Plates, membranes, coupled structures, transmission problems,

semilinear evolution equations.

\begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos.

Published May 30, 2005.

\begin{matrix}\frac{h^{3}}{12} ∑_{α,β,γ,θ=1}^{2} ν_{α} \frac{∂}{∂x_{β}} (A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} ) + \frac{h^{3}}{12} \frac{∂}{∂τ⃗} ( ∑_{α,β,γ,θ=1}^{2} ν_{α}τ_{β}A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} ) \\ + C \frac{∂u_{m}}{∂ν⃗} = 0 on ]0, T] × Γ_{1} , \end{matrix} (1.4)

∑_{α,β,γ,θ=1}^{2} ν_{α}ν_{β}A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} = 0 on ]0, T] × (∂Ω_{p} \ Γ_{3}),  (1.5)

u_{p} = \frac{∂u_{p}}{∂ν⃗} = 0 on ]0, T] × Γ_{3},  (1.6)
u_{p} = u_{m} on ]0, T] × Γ_{1} ,  (1.7)

with the initial conditions

u_{p}(0, ·) = g_{p}^{0} in Ω_{p},  (1.8)
u_{m}(0, ·) = g_{m}^{0} in Ω_{m},  (1.9)
\frac{∂u_{p}}{∂t} (0, ·) = g_{p}^{1} in Ω_{p},  (1.10)
\frac{∂u_{m}}{∂t} (0, ·) = g_{m}^{1} in Ω_{m}.  (1.11)

Equations (1.1)-(1.11) describe the vibrations of a structure which consists of a thin elastic anisotropic plate (in the context of the so called Kirchhoff-Love theory) with its middle surface occupying the domain Ω_{p}, coupled with a membrane occupying the domain Ω_{m} (see figure 1).

It is supposed that ρ_{p} and ρ_{m} are positive constants, where ρ_{p} (resp. ρ_{m} ) is the density of the middle surface of the plate (resp. the membrane) and h is the thickness of the plate. The coefficients A_{αβγθ} depend on the elastic modulus of the plate and are assumed as C^{∞} functions on \overline{Ω}_{p}; they satisfy the symmetry assumption

A_{αβγθ} = A_{βαγθ}, A_{αβγθ} = A_{αβθγ}, A_{αβγθ} = A_{γθαβ}  (1.12)

and the coercivity hypothesis

∑_{α,β,γ,θ=1}^{2} A_{αβγθ}(x)ξ_{γθ}ξ_{αβ} ≥ ρ ∑_{α,β=1}^{2} ξ_{αβ}^{2}  (1.13)

for all x ∈ Ω_{p} and for all real matrices (ξ_{αβ})_{2×2} with ξ_{αβ} = ξ_{βα} for α, β ∈ {1, 2}, where ρ > 0 is a constant. Moreover it is supposed that the plate is clamped on Γ_{3} (equation (1.6)) and is free on Γ_{2} (see figure 1).

The vector ν⃗ = (ν_{1}, ν_{2}) is the unitary outward normal to ∂Ω_{p} and τ = (τ_{1}, τ_{2}) = (−ν_{2}, ν_{1}) is the positive oriented unitary tangent vector. C is a positive constant depending on the material forming the membrane. f_{p} (resp. f_{m}) is the pressure supported by the plate (resp. the membrane) and depend on the transverse displacement u_{p} (resp. u_{m}) of the plate (resp. the membrane). The initial conditions g_{p}^{0} and g_{p}^{1} (resp. g_{m}^{0} and g_{m}^{1}) are real functions defined on Ω_{p} (resp. Ω_{m}). The equations (1.4) and (1.7) are the boundary conditions expressing the coupling between the plate and the membrane. We give the definition of weak solution for our semilinear problem (1.1)-(1.11) and with help of the theory of C^{0}-semigroups of linear operators we obtain a result of existence and uniqueness for this type of solution. For other works in the area of

[figure: Ω_{m}; Ω_{p}; Γ_{2}; Γ_{3}; Γ_{1}; ν⃗; τ⃗; τ⃗; ν⃗]

Figure 1. \overline{Ω}_{m} (resp. \overline{Ω}_{p}) is occupied by the membrane (resp. the middle surface of the Plate). The Plate is clamped on Γ_{3}.

transmission problems and networks we refer the reader to [2, 3, 4, 6, 7, 10, 11, 12, 13, 14, 15].

2. Notation and mathematical preliminaries

In this section we shall present the concepts and abstract framework that we need for the treatment of our problem (1.1)-(1.11). We shall consider only real valued functions. Let n a positive integer. For any multi-index α = (α_{1}, …, α_{2}) (i.e. α ∈ ℕ_{0}^{n}, where ℕ_{0} is the set of all nonnegative integers), we write

∂^{α} := \frac{∂^{|α|}}{∂x_{1}^{α_{1}} … ∂x_{n}^{α_{n}}} , where |α| := α_{1} + ⋯ + α_{n}.

Sometimes we write ∂_{i} for \frac{∂}{∂x_{i}}, i = 1, …, n. For the rest of this section, let Ω be an open bounded connected set in ℝ^{n} with sufficiently smooth boundary.

For any nonnegative integer k let C^{k}(Ω) be the vector space consisting of all functions ϕ which, together with all their partial derivatives ∂^{α}ϕ of orders |α| ≤ k, are continuous in Ω. C^{∞}(Ω) is the vector space consisting of all functions ϕ, such that ϕ ∈ C^{k}(Ω) for all nonnegative integer k.

We write C^{k}(\overline{Ω}) for the Banach space consisting of all functions ϕ ∈ C^{k}(Ω) for which ∂^{α}ϕ is bounded and uniformly continuous on Ω for |α| ≤ k, with norm given by

∥ϕ∥_{C^{k}(\overline{Ω})} := max_{|α|≤k} sup_{x∈Ω} |∂^{α}ϕ(x)|.

38  J. HERNÁNDEZ  EJDE/CONF/13

For a nonnegative integer k and 1 ≤ p ≤ ∞ let W^{k,p}(Ω) be the usual Sobolev space defined as

W^{k,p}(Ω) := {u ∈ L^{p}(Ω); ∂^{α}u ∈ L^{p}(Ω)forallα ∈ ℕ_{0}^{n}, |α| ≤ k},  (2.1)

where ∂^{α}u is understood in distributional (or weak) sense, with the usual norm

∥u∥_{k,p,Ω} := { ∑_{|α|≤k} ∫_{Ω} |∂^{α}u(x)|^{p}dx}^{1/p} if1 ≤ p < ∞,  (2.2)
∥u∥_{k,∞,Ω} := max_{|α|≤k} ess sup_{x∈Ω} |∂^{α}u(x)|.  (2.3)

As usual we shall write H^{k}(Ω) := W^{k,2}(Ω). Lemma 2.1. The set D(\overline{Ω}) of restrictions to Ω of functions in C_{c}^{∞}(ℝ^{n}) (i.e. the set of all infinitely differentiable functions on ℝ^{n} with compact support) is dense in W^{k,p}(Ω) for 1 ≤ p < ∞.

For the proof of the above lemma, see Adams [1, theorem 3.18,]. Lemma 2.2. If kp = n, then W^{k,p}(Ω) ↪ L^{q}(Ω) for p ≤ q < ∞.

For the proof of the above lemma, see Adams [1, lemma 5.14]. Lemma 2.3. If kp > n, then W^{k,p}(Ω) ↪ C^{0}(\overline{Ω}).

The proof of the above lemma can be found in Evans [9, sec. 5.6, Theorem 6] and in Adams [1, lemma 5.17]. Lemma 2.4. Let 1 ≤ p < ∞. Then there exists a linear operator

γ_{0} : W^{1,p}(Ω) → L^{p}(∂Ω)  (2.4)

such that (i) γ_{0}u = u|_{∂Ω} if u ∈ W^{1,p}(Ω) ∩ C(\overline{Ω}). (ii) ∥γ_{0}u∥_{L^{p}(∂Ω)} ≤ c(p, Ω)∥u∥_{1,p,Ω} for each u ∈ W^{1,p}(Ω), where c(p, Ω) is a

constant depending only on p and Ω. For the proof of the above lemma, see Evans [9, theorem 5.5.1].

Remark 2.5. We call γ_{0}u the trace of order zero of u on ∂Ω. Definition 2.6. Let j, k ∈ ℕ, k > 1, 1 ≤ j ≤ k − 1 and u ∈ W^{k,p}(Ω). We define the trace of order j of u on ∂Ω by

γ_{j}u := ∑_{|α|=j} \frac{j!}{α_{1}! ⋯ α_{n}!}γ_{0}(∂^{α}u)ν_{1}^{α_{1}} ⋯ ν_{n}^{α_{n}},  (2.5)

where ν⃗ = (ν_{1}, …, ν_{n}) is the unit outward normal along ∂Ω. Remark 2.7. γ_{j} : W^{k,p}(Ω) → L^{p}(∂Ω) is a linear operator with

(i) γ_{j}u = \frac{∂^{j}u}{∂ν^{j}} |_{∂Ω} := ∑_{|α|=j} \frac{j!}{α_{1}!···α_{n}!}∂^{α}u|_{∂Ω}ν_{1}^{α_{1}} ⋯ ν_{n}^{α_{n}} for j = 1, …, k − 1 if

u ∈ W^{k,p}(Ω) ∩ C^{k−1}(\overline{Ω}).

(ii) ∥γ_{j}u∥_{L^{p}(∂Ω)} ≤ c(k, p, Ω)∥u∥_{k,p,Ω} for each u ∈ W^{k,p}(Ω) and for all j =

1, …, k − 1.

Now for j, k ∈ ℕ_{0}, 0 ≤ j ≤ k, and 1 ≤ p < ∞ we define the the functional given by

|u|_{j,p,Ω} := { ∑_{|α|=j} ∫_{Ω} |∂^{α}u(x)|^{p}dx}^{1/p}, u ∈ W^{k,p}(Ω).  (2.6)

Clearly, |u|_{0,p,Ω} = ∥u∥_{0,p,Ω} = ∥u∥_{L^{p} (Ω)}. We have the following statement.

Lemma 2.8. The functional

((u))_{k,p,Ω} = {|u|p_{k,p,Ω} + |u|p_{0,p,Ω}}^{1/p}

is a norm on W^{k,p}(Ω), equivalent to the usual norm ∥ · ∥_{k,p,Ω}.

The proof of the above lemma can be found in Adams [1, corollary 4.16].

We need some crucial results of the theory of semigroups of linear operators in Banach spaces. We refer to Pazy [16] or Dautray-Lions [8], chapter XVII, with respect to this theory.

Let V (resp. H) be a real separable Hilbert space with scalar product (·|·)_{V} (resp. (·|·)_{H}) and norm ∥ · ∥_{V} (resp. ∥ · ∥_{H}). We assume V ↪ H and V dense in H. Let a(·|·) : V × V → ℝ be a continuous bilinear form, V -coercive with respect to H i.e., there exists λ_{0} ∈ ℝ and c_{0} > 0 such that

a(v|v) + λ_{0}∥v∥_{H}^{2} ≥ c_{0}∥v∥_{V}^{2} , ∀v ∈ V.  (2.7)

We put

D(A) := {u ∈ V; V ∋ v ↦ a(u|v)is continuous for the topology ofH}. (2.8)

Theorem 2.9. Let A : D(A) ⊂ H → H be the operator given by (Au|v)_{H} = a(u|v) ∀u ∈ D(A) and ∀v ∈ V. Then −A is the infinitesimal generator of a C^{0}- semigroup {T (t)}_{t≥0} in H which satisfies

∥T (t)∥_{L(H)} ≤ e^{λ_{0}t} ∀t ≥ 0 .

For a proof of the above theorem, see Dautray-Lions [8, theorem XVII.3.3]. Now we assume furthermore that a(·|·) is symmetrical (a(u|v) = a(v|u) ∀u, v ∈

V). Let H := V × H. H equipped with the scalar product defined by (u|v)_{H} := a(u_{1}|v_{1}) + (u_{2}|v_{2})_{H} for u = (u_{1}, u_{2})^{t}, v = (v_{1}, v_{2})^{t} ∈ H ( we write the elements of H as columns ) is a Hilbert space (cf. Dautray-Lions [8], Section VII.3.4., p. 331).

Let D(𝔸) := D(A) × V. We define the operator 𝔸 over D(𝔸) by

𝔸u := (\begin{matrix}0 & −id \\ A & 0 \end{matrix}) (\begin{matrix}u_{1} \\ u_{2} \end{matrix}) = (\begin{matrix}−u_{2} \\ Au_{1} \end{matrix}) , ∀u = (\begin{matrix}u_{1} \\ u_{2} \end{matrix}) ∈ D(𝔸).  (2.9)

It follows that D(𝔸) is dense in H and 𝔸 is a closed operator.

Theorem 2.10. −𝔸 is the infinitesimal generator of a C^{0}-semigroup in H.

For the proof of the above theroem, see Dautray-Lions [8, theorem XVII.3.4].

Theorem 2.11. Let −A be the infinitesimal generator of a C^{0}-semigroup of linear operators on a Banach space X and u_{0} ∈ D(A). If f : [t_{0}, T] × X → X is continuously differentiable with bounded partial derivatives then there exists a unique classical solution u ∈ C^{1}([t_{0}, T]; X) of the initial value problem

\begin{matrix}\frac{du(t)}{dt} + Au(t) = f(t, u(t)) in X, on ]t_{0}, T] \\ u(t_{0}) = u_{0} . \end{matrix}  (2.10)

The proof of this lemma ca be found in Pazy [16, theorem 6.1.5].

3. Function spaces and bilinear forms for the semilinear problem plate-membrane

We define the vector space (with the usual vectorial sum and multiplication by scalars)

V := {(u_{p}, u_{m}) ∈ H^{2}(Ω_{p}) × H^{1}(Ω_{m}); u_{p}|_{Γ_{3}} = γ_{1}u_{p}|_{Γ_{3}} = 0, u_{p}|_{Γ_{1}} = γ_{0}u_{m}|_{Γ_{1}} }

(3.1)

(In this work we only consider real vector spaces). The vector space V, endowed with the inner product

((u_{p}, u_{m})|(v_{p}, v_{m}))_{V} := (u_{p}|v_{p})_{H^{2}(Ωp)} + (u_{m}|v_{m})_{H^{1}(Ωm)},  (3.2)

is a separable Hilbert space. The norm in V is given by

∥(u_{p}, u_{m})∥_{V} := (∥u_{p}∥_{2,2,Ω_{p}}^{2} + ∥u_{m}∥_{1,2,Ω_{m}}^{2})^{1/2}.  (3.3)

We consider also

H := L^{2}(Ω_{p}) × L^{2}(Ω_{m})  (3.4)

with inner product and norm given by

((u_{p}, u_{m})|(v_{p}, v_{m}))_{H} := (u_{p}|v_{p})_{L^{2}(Ωp)} + (u_{m}|v_{m})_{L^{2}(Ωm)}  (3.5)

and

∥(u_{p}, u_{m})∥_{H} := (∥u_{p}∥_{0,2,Ω_{p}}^{2} + ∥u_{m}∥_{0,2,Ω_{m}}^{2})^{1/2} .  (3.6)

Also we consider

Ṽ := \{(ũ_{p}, ũ_{m}) ∈ H^{2}(Ω_{p}) × H^{1}(Ω_{m}); (\frac{1}{\sqrt{ρ_{p}h}}ũ_{p}, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}) ∈ V \},  (3.7)

endowed with the norm

∥(ũ_{p}, ũ_{m})∥Ṽ := (\frac{1}{ρ_{p}h}∥ũ_{p}∥_{2,2,Ω_{p}}^{2} + \frac{1}{ρ_{m}} ∥ũ_{m}∥_{1,2,Ω_{m}}^{2})^{1/2}.  (3.8)

We have the imbedding Ṽ ↪ H with Ṽ dense in H. Identifying H with its dual

H′ we obtain Ṽ ↪^{i} H = H′ ↪^{i′} Ṽ′, where i : Ṽ → H is the identity operator and i′ : H → Ṽ′ is the dual operator of i : V → H. Since i : Ṽ → H is injective and its range is dense in H, the same holds for i′ : H → Ṽ′. Furthermore we identify i′f with f for f ∈ H. Therefore we regard H as subspace of Ṽ′.

We consider the symmetric bilinear form

\begin{matrix}a((u_{p}, u_{m})|(v_{p}, v_{m})) \\ := \frac{h^{3}}{12} ∑_{α,β,γ,θ=1}^{2} ∫_{Ω_{p}} A_{αβγθ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} \frac{∂^{2}v_{p}}{∂x_{α}∂x_{β}} dx + C ∫_{Ω_{m}} ∇u_{m} · ∇v_{m}dx \end{matrix}  (3.9)

for (u_{p}, u_{m}), (v_{p}, v_{m}) ∈ V (The symmetry is a consequence of the assumption (1.12)). For technical reasons it is convenient to consider also

ã ((ũ_{p}, ũ_{m})|(ṽ_{p}, ṽ_{m})) := a((\frac{1}{\sqrt{ρ_{p}h}}ũ_{p}, \frac{1}{\sqrt{ρ_{m}}} ũ_{m})|(\frac{1}{\sqrt{ρ_{p}h}}ṽ_{p}, \frac{1}{\sqrt{ρ_{m}}} ṽ_{m})) (3.10)

for (ũ_{p}, ũ_{m}), (ṽ_{p}, ṽ_{m}) ∈ Ṽ.

Lemma 3.1. Under the assumptions introduced for the coefficients A_{αβγθ}, the bilinear form (3.9) (resp. (3.10)) is continuous and V -coercive (resp. Ṽ −coercive) with respect to H.

Proof. From the Schwarz inequality we have the continuity of the bilinear forms (3.9) and (3.10). Now let u = (u_{p}, u_{m}) ∈ V. From Lemma 2.8 we have that there exists c_{p} > 0 such that

((u_{p}))_{2,2,Ω_{p}} ≥ c_{p}∥u_{p}∥_{2,2,Ω_{p}}.

Then

a(u|u) = \frac{h^{3}}{12} ∑_{α,β,γ,θ=1}^{2} ∫_{Ω_{p}} A_{α,β,γ,θ} \frac{∂^{2}u_{p}}{∂x_{γ}∂x_{θ}} \frac{∂^{2}u_{p}}{∂x_{α}∂x_{β}} dx + C ∫_{Ω_{m}} |∇u_{m}|^{2}dx

≥ \frac{h^{3}}{12} ρ ∑_{α,β=1}^{2} ∫_{Ω_{p}} |\frac{∂^{2}u_{p}}{∂x_{α}∂x_{β}} |^{2}dx + C|u_{m}|_{1,2,Ω_{m}}^{2}

= \frac{h^{3}}{12} ρ|u_{p}|_{2,2,Ω_{p}}^{2} + C|u_{m}|_{1,2,Ω_{m}}^{2}

≥ \frac{h^{3}}{12} ρc_{p}∥u_{p}∥_{2,2,Ω_{p}}^{2} − \frac{h^{3}}{12} ρ|u_{p}|_{0,2,Ω_{p}}^{2} + C∥u_{m}∥_{1,2,Ω_{m}}^{2} − C|u_{m}|_{0,2,Ω_{m}}^{2}.

With λ_{0} := max\{\frac{h^{3}}{12} ρ, C\} and c_{0} := min\{\frac{h^{3}}{12} ρc_{p}, C\} we obtain the V -coerciveness of a(·|·) with respect to H. From this follows immediately the Ṽ -coerciveness of ã(·|·) with respect to H. □

LetD(Ã) := Ã^{−1}(H) and Ã := Ã|_{D(Ã)}, where Ã : Ṽ → Ṽ′ is given by ⟨Ãũ|ṽ⟩ = ã(ũ|ṽ), for all ũ, ṽ ∈ Ṽ. We have that −Ã is the infinitesimal generator of a C^{0}- semigroup in H (see [11, p. 54].

4. Weak solution

For the function

(t, x, u) ↦ f_{p}(t, x, u) : [0, T] × Ω_{p} × ℝ → ℝ  (4.1)

we assume the following:

(i) For all t ∈ [0, T], x ↦ f_{p}(t, x, u(x)) : Ω_{p} → ℝ is measurable, if u : Ω_{p} → ℝ is measurable.

(ii) |f_{p}(t, x, u)| ≤ q_{p}(t, x)+k_{p}|u| for all (t, x, u) ∈ [0, T]×Ω_{p}×ℝ, where q_{p}(t, ·) ∈

L^{2}(Ω_{p}) for all t ∈ [0, T] and k_{p} > 0 is a constant. (iii) \frac{∂f_{p}}{∂t} (t, x, u) exists for all (t, x, u) ∈ [0, T] × Ω_{p} × ℝ. It is bounded and

Lipschitz continuous on [0, T] × Ω_{p} × ℝ. (iv) \frac{∂f_{p}}{∂u} (t, x, u) exists for all (t, x, u) ∈ [0, T] × Ω_{p} × ℝ. It is bounded and

Lipschitz continuous on [0, T] × Ω_{p} × ℝ.

For the function

(t, x, u) ↦ f_{m}(t, x, u) : [0, T] × Ω_{m} × ℝ → ℝ  (4.2)

we assume the following:

(i) For all t ∈ [0, T], x ↦ f_{m}(t, x, u(x)) : Ω_{m} → ℝ is measurable, if u : Ω_{m} → ℝ is measurable.

(ii) |f_{m}(t, x, u)| ≤ q_{m}(t, x) + k_{m}|u|, for all (t, x, u) ∈ [0, T] × Ω_{m} × ℝ, where

q_{m}(t, ·) ∈ L^{2}(Ω_{m}) for all t ∈ [0, T] and k_{m} > 0 a constant. (iii) \frac{∂f_{m}}{∂t} (t, x, u) exists for all (t, x, u) ∈ [0, T] × Ω_{m} × ℝ. It is bounded and

Lipschitz continuous on [0, T] × Ω_{m} × ℝ. (iv) \frac{∂f_{m}}{∂u} (t, x, u) exists for all (t, x, u) ∈ [0, T] × Ω_{m} × ℝ. It is bounded and

Lipschitz continuous on [0, T] × Ω_{m} × ℝ.

Let f_{p} : [0, T] × L^{2}(Ω_{p}) → L^{2}(Ω_{p}) and f_{m} : [0, T] × L^{2}(Ω_{m}) → L^{2}(Ω_{m}) be defined by

[f_{p}(t, u_{p})](x) := f_{p}(t, x, u_{p}(x)) for (t, x) ∈ [0, T] × Ω_{p} u_{p} ∈ L^{2}(Ω_{p}) ,  (4.3)
[f_{m}(t, u_{m})](x) := f_{m}(t, x, u_{m}(x)) for (t, x) ∈ [0, T] × Ω_{m} u_{m} ∈ L^{2}(Ω_{m}). (4.4)

From assumptions on (4.1) and (4.2), we see that f_{p}(t, u_{p}) ∈ L^{2}(Ω_{p}) and f_{m}(t, u_{m}) ∈ L^{2}(Ω_{m}), for u_{p} ∈ L^{2}(Ω_{p}) and u_{m} ∈ L^{2}(Ω_{m}).

For technical reasons we introduce the following functions:

f̃_{p}(t, u_{p}) := \frac{1}{\sqrt{ρ_{p}h}}f_{p}(t, \frac{1}{\sqrt{ρ_{p}h}}u_{p}) for t ∈ [0, T] u_{p} ∈ L^{2}(Ω_{p}) ,  (4.5)
f̃_{m}(t, u_{m}) := \frac{1}{\sqrt{ρ_{m}}} f_{m}(t, \frac{1}{\sqrt{ρ_{m}}} u_{m}) for t ∈ [0, T] u_{m} ∈ L^{2}(Ω_{m}) .  (4.6)

Let us suppose that u_{p} : [0, T] × \overline{Ω}_{p} → ℝ and u_{m} : [0, T] × \overline{Ω}_{m} → ℝ are smooth enough in such a way that the system (1.1) - (1.11) for (u_{p}, u_{m}) holds; i.e., we suppose that (u_{p}, u_{m}) is a classical solution of the semilinear problem (1.1)-(1.11). Furthermore we assume that (ũ_{p}(t, .), ũ_{m}(t, .)) ∈ D(Ã) for t ∈]0, T], where (ũ_{p}, ũ_{m}) := (\sqrt{ρ_{p}h}u_{p}, \sqrt{ρ_{m}}u_{m}). If we multiply (1.1) (resp. (1.2)) with \frac{1}{\sqrt{ρ_{p}h}} ṽ_{p} (resp. \frac{1}{\sqrt{ρ_{m}}} ṽ_{m}),

where (ṽ_{p}, ṽ_{m}) ∈ Ṽ, by use of integration by parts, (1.3)-(1.7) and the fact that Ṽ is dense in H we obtain

(\frac{∂^{2}ũ_{p}}{∂t^{2}} (t, ·), \frac{∂^{2}ũ_{m}}{∂t^{2}} (t, ·))+Ã(ũ_{p}(t, ·), ũ_{m}(t, ·)) = (f̃_{p}(t, ũ_{p}(t, ·)), f̃_{m}(t, ũ_{m}(t, ·))) (4.7)

in H, for t ∈]0, T]. On the other hand we have

ũ_{p}(0, ·) = g̃_{p}^{0}, ũ_{m}(0, ·) = g̃_{m}^{0}, \frac{∂ũ_{p}}{∂t} (0, ·) = g̃_{p}^{1}, \frac{∂ũ_{p}}{∂t} (0, ·) = g̃_{m}^{1},  (4.8)

where g̃_{p}^{0} := \sqrt{ρ_{p}h}g_{p}^{0}, g̃_{m}^{0} := \sqrt{ρ_{m}}g_{m}^{0}, g̃_{p}^{1} := \sqrt{ρ_{p}h}g_{p}^{1} and g̃_{m}^{1} := \sqrt{ρ_{m}}g_{m}^{1}. We suppose

(i) (g_{p}^{0}, g_{m}^{0}) ∈ A^{−1}(H), (ii) (g_{p}^{1}, g_{m}^{1}) ∈ V  (4.9)

where A : V → V′ is given by ⟨Au|v⟩ = a(u|v), for all u, v ∈ V.

Equations (4.7) and (4.8) motivate the following definition: Consider the Hilbert space H := Ṽ × H endowed with the inner product

( (\begin{matrix}(ũ_{p}^{1}, ũ_{m}^{1}) \\ (ũ_{p}^{2}, ũ_{m}^{2}) \end{matrix}) | (\begin{matrix}(ṽ_{p}^{1}, ṽ_{m}^{1}) \\ (ṽ_{p}^{2}, ṽ_{m}^{2}) \end{matrix}) )_{H} := a((ũ_{p}^{1}, ũ_{m}^{1})|(ṽ_{p}^{1}, ṽ_{m}^{1})) + ((ũ_{p}^{2}, ũ_{m}^{2})|(ṽ_{p}^{2}, ṽ_{m}^{2}))_{H}.

(4.10)

Moreover let D(𝔸̃) := D(Ã) × Ṽ and 𝔸̃ := (\begin{matrix}0 & −id \\ Ã & 0 \end{matrix}). It follows from theorem

2.10 that −𝔸̃ is the infinitesimal generator of a C^{0}-semigroup of contractions in H. We put

𝔽̃(t, 𝕌̃) := (\begin{matrix}0 \\ (f̃_{p}(t, ũ_{p}^{1}), f̃_{m}(t, ũ_{m}^{1})) \end{matrix}) for 𝕌̃ := (\begin{matrix}(ũ_{p}^{1}, ũ_{m}^{1}) \\ (ũ_{p}^{2}, ũ_{m}^{2}) \end{matrix}) ∈ H,  (4.11)

𝔾̃ := (\begin{matrix}(g̃_{p}^{0}, g̃_{m}^{0}) \\ (g̃_{p}^{1}, g̃_{m}^{1}) \end{matrix}) .  (4.12)

Next we define weak solution for our semilinear problem.

Definition 4.1. Assume that (1.12), (1.13), (4.1), (4.2) and (4.9) are satisfied. We say that a function (u_{p}, u_{m}) ∈ C^{1}([0, T]; V) ∩ C^{2}([0, T]; H) is a weak solution of the semilinear problem (1.1)-(1.11) if the function

(ũ_{p}, ũ_{m}) := (\sqrt{ρ_{p}h}u_{p}, \sqrt{ρ_{m}}u_{m}) ∈ C^{1}([0, T]; Ṽ) ∩ C^{2}([0, T]; H)

has the following properties:

(i)(\frac{d^{2}ũ_{p}(t)}{dt^{2}}, \frac{d^{2}ũ_{m}(t)}{dt^{2}}) + Ã(ũ_{p}(t), ũ_{m}(t)) = (f̃_{p}(t, ũ_{p}(t)), f̃_{m}(t, ũ_{m}(t)))
in H, on ]0, T]
(ii)(ũ_{p}(0), ũ_{m}(0)) = (g̃_{p}^{0}, g̃_{m}^{0}).
(iii)(\frac{dũ_{p}}{dt} (0), \frac{dũ_{m}}{dt} (0)) = (g̃_{p}^{1}, g̃_{m}^{1}).

(4.13)

Lemma 4.2. Assume (1.12), (1.13), (4.1) and (4.2). Then the function (t, 𝕌) ↦ 𝔽(t, 𝕌) : [0, T] × H → H which is defined by (4.11), is continuously differentiable with bounded partial derivatives.

Proof. 1. The assumptions (4.1)(i),(ii) and (4.2)(i),(ii) lead to

f̃_{p}(t, ũ_{p}^{1}) ∈ L^{2}(Ω_{p}) and f̃_{m}(t, ũ_{m}^{1}) ∈ L^{2}(Ω_{m})

for ũ_{p}^{1} ∈ L^{2}(Ω_{p}) and ũ_{m}^{1} ∈ L^{2}(Ω_{m}) and for all t ∈ [0, T] (cf. [5, theorem 2.1]). Then we have 𝔽̃(t, 𝕌̃) ∈ H for (t, 𝕌̃) ∈ [0, T] × H.

2. It follows from (4.1)(iii) that

\frac{∂f_{p}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(·)) ∈ L^{2}(Ω_{p}) ∀t ∈ [0, T] ∀ũ_{p}^{1} ∈ L^{2}(Ω_{p}).

Let t ∈ [0, T]. For τ ∈ ℝ with −t ≤ τ ≤ T − t we have

∥\frac{f̃_{p}(t + τ, ũ_{p}^{1}) − f̃_{p}(t, ũ_{p}^{1})}{τ} − \frac{1}{\sqrt{ρ_{p}h}} \frac{∂f_{p}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(·))∥2_{L^{2}(Ωp)}

= ∫_{Ω_{p}} \frac{1}{ρ_{p}h}| ∫_{0}^{1} [\frac{∂f_{p}}{∂t} (t + ξτ, x, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(x)) − \frac{∂f_{p}}{∂t} (t, x, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(x))]dξ|^{2}dx

≤ ∫_{Ω_{p}} \frac{1}{ρ_{p}h}[ ∫_{0}^{1} |\frac{∂f_{p}}{∂t} (t + ξτ, x, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(x)) − \frac{∂f_{p}}{∂t} (t, x, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(x))|dξ]^{2}dx

≤ \frac{1}{ρ_{p}h}const.μ_{p}(Ω_{p})τ^{2} ⟶_{τ→0} 0

(4.14)

The above inequality because the Lipschitz continuity of \frac{∂f_{p}}{∂t}.

3. It follows from (4.2)(iii) that

\frac{∂f_{m}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}(·)) ∈ L^{2}(Ω_{m}) ∀t ∈ [0, T] ∀ũ_{m}^{1} ∈ L^{2}(Ω_{m}).

Let t ∈ [0, T]. For τ ∈ ℝ with −t ≤ τ ≤ T − t we have as above

∥\frac{f̃_{m}(t + τ, ũ_{m}^{1}) − f̃_{m}(t, ũ_{m}^{1})}{τ} − \frac{1}{\sqrt{ρ_{m}}} \frac{∂f_{m}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}(·))∥2_{L^{2}(Ωm)}  (4.15)

approaches zero as τ → 0.

4. Let (t, 𝕌̃) ∈ [0, T] × H with 𝕌̃ := (\begin{matrix}(ũ_{p}^{1}, ũ_{m}^{1}) \\ (ũ_{p}^{2}, ũ_{m}^{2}) \end{matrix}). We consider the operator

D_{1}𝔽̃(t, 𝕌̃) ∈ L(ℝ; H) which is defined by

D_{1}𝔽̃(t, 𝕌̃)τ := (\begin{matrix}0 \\ (\frac{1}{\sqrt{ρ_{p}h}} \frac{∂f_{p}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(·))τ, \frac{1}{\sqrt{ρ_{m}}} \frac{∂f_{m}}{∂t} (t, ·, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}(·))τ) \end{matrix}) (4.16)

For (t, 𝕌̃) ∈ [0, T] × H and from (4.14) and (4.15) we have that

\frac{∥𝔽̃(t + τ, 𝕌̃) − 𝔽̃(t, 𝕌̃) − D_{1}𝔽̃(t, 𝕌̃)τ∥_{H}}{|τ|} ⟶_{−t≤τ≤T −t, τ≠0, τ→0}^{0} .  (4.17)

Then there exists the partial derivative of 𝔽̃ with respect to t for all (t, 𝕌̃) ∈ [0, T]×H and it is equal to D_{1}𝔽̃(t, 𝕌̃). By the Lipschitz continuity of \frac{∂f_{p}}{∂t} and \frac{∂f_{m}}{∂t} it can be showed that

∥D_{1}𝔽̃(t_{1}, 𝕌̃_{1}) − D_{1}𝔽̃(t_{2}, 𝕌̃_{2})∥_{L(ℝ;H)} ≤ const.(|t_{1} − t_{2}| + ∥𝕌̃_{1} − 𝕌̃_{2}∥_{H}). (4.18)

Then the maping

(t, 𝕌̃) ↦ D_{1}𝔽̃(t, 𝕌̃) : [0, T] × H → L(ℝ; H)

is continuous. The boundedness of \frac{∂f_{p}}{∂t} and \frac{∂f_{m}}{∂t} implied by the boundedness of D_{1}𝔽̃.

5. From (4.1)(iv) and (4.2)(iv) we have

\frac{∂f_{p}}{∂u} (t, ·, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(·))ṽ_{p}^{1} ∈ L^{2}(Ω_{p})

and

\frac{∂f_{m}}{∂u} (t, ·, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}(·))ṽ_{m}^{1} ∈ L^{2}(Ω_{m})

for all t ∈ [0, T] and all (ũ_{p}^{1}, ũ_{m}^{1}), (ṽ_{p}^{1}, ṽ_{m}^{1}) ∈ H. For t ∈ [0, T], 𝕌̃ := (\begin{matrix}(ũ_{p}^{1}, ũ_{m}^{1}) \\ (ũ_{p}^{2}, ũ_{m}^{2}) \end{matrix}) ∈

H and 𝕍̃ := (\begin{matrix}(ṽ_{p}^{1}, ṽ_{m}^{1}) \\ (ṽ_{p}^{2}, ṽ_{m}^{2}) \end{matrix}) ∈ H we put

D_{2}𝔽̃(t, 𝕌̃)𝕍̃ := (\begin{matrix}0 \\ (\frac{1}{ρ_{p}h} \frac{∂f_{p}}{∂u} (t, ·, \frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}(·))ṽ_{p}^{1}, \frac{1}{ρ_{m}} \frac{∂f_{m}}{∂u} (t, ·, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}(·))ṽ_{m}^{1}) \end{matrix}) (4.19)

Since \frac{∂f_{p}}{∂u} (resp. \frac{∂f_{m}}{∂u}) is bounded on [0, T] × Ω_{p} × ℝ (resp. [0, T] × Ω_{m} × ℝ), we see that D_{2}𝔽̃(t, 𝕌̃) ∈ L(H) for all (t, 𝕌̃) ∈ [0, T] × H.

For (t, 𝕌̃) ∈ [0, T]×H and 𝕍̃ ∈ H with ∥𝕍̃∥_{H} ≠ 0 we have (with “const” denoting different constants)

\frac{∥𝔽̃(t, 𝕌̃ + 𝕍̃) − 𝔽̃(t, 𝕌̃) − D_{2}𝔽̃(t, 𝕌̃)𝕍̃∥2_{H}}{∥𝕍̃∥2_{H}}

≤ \frac{const}{∥𝕍̃∥2_{H}} \{ ∫_{Ω_{p}} [ ∫_{0}^{1} |\frac{∂f_{p}}{∂u} (t, x, \frac{1}{\sqrt{ρ_{p}h}}(ũ_{p}^{1}(x) + ξṽ_{p}^{1}(x)))

− \frac{∂f_{p}}{∂u} (t, x, \frac{ũ_{p}^{1}(x)}{\sqrt{ρ_{p}h}})|dξ]^{2} \frac{|ṽ_{p}^{1}(x)|^{2}}{ρ_{p}h} dx

+ ∫_{Ω_{m}} [ ∫_{0}^{1} |\frac{∂f_{m}}{∂u} (t, x, \frac{1}{\sqrt{ρ_{m}}} (ũ_{m}^{1}(x) + ξṽ_{m}^{1}(x)))

− \frac{∂f_{m}}{∂u} (t, x, \frac{ũ_{m}^{1}(x)}{\sqrt{ρ_{m}}})|dξ]^{2} \frac{|ṽ_{m}^{1}(x)|^{2}}{ρ_{m}} dx\}

≤ \frac{const}{∥𝕍̃∥2_{H}} \{\frac{1}{ρ_{p}^{2}h^{2}} ∫_{Ω_{p}} |ṽ_{p}^{1}(x)|^{4}dx + \frac{1}{ρ_{m}^{2}} ∫_{Ω_{m}} |ṽ_{m}^{1}(x)|^{4}dx\} .

(4.20)

The above holds because of the Lipschitz continuity of \frac{∂f_{p}}{∂u} and \frac{∂f_{m}}{∂u}. Since

ṽ_{p}^{1} ∈ H^{2}(Ω_{p}) ↪ C^{0}(\overline{Ω}_{p}) ↪ L^{4}(Ω_{p}) and ṽ_{m}^{1} ∈ H^{1}(Ω_{m}) ↪ L^{4}(Ω_{m})

(see lemmas 2.2 and 2.3), from (4.20), we have

\frac{∥𝔽̃(t, 𝕌̃ + 𝕍̃) − 𝔽̃(t, 𝕌̃) − D_{2}𝔽̃(t, 𝕌̃)𝕍̃∥2_{H}}{∥𝕍̃∥2_{H}}

≤ \frac{const.}{∥𝕍̃∥2_{H}}(\frac{1}{ρ_{p}^{2}h^{2}} ∥ṽ_{p}^{1}∥4_{H^{2}(Ωp)} + \frac{1}{ρ_{m}^{2}} ∥ṽ_{m}^{1}∥4_{H^{1}(Ωm)})

≤ \frac{const.}{∥𝕍̃∥2_{H}}∥(ṽ_{p}^{1}, ṽ_{m}^{1})∥4_{Ṽ}

≤ \frac{const.}{∥𝕍̃∥2_{H}}∥𝕍̃∥4_{H} = const.∥𝕍̃∥2_{H}.

(4.21)

It follows that the partial derivative of 𝔽̃ with respect to the second variable 𝕌̃ exists and it is equal to D_{2}𝔽̃(t, 𝕌̃) for all (t, 𝕌̃) ∈ [0, T] × H. We can show similarly that the Lipschitz continuity (resp. the boundedness) of \frac{∂f_{p}}{∂u} and \frac{∂f_{m}}{∂u} leads to the continuity (resp. the boundedness) of

(t, 𝕌̃) ↦ D_{2}𝔽̃(t, 𝕌̃) : [0, T] × H → L(H).

So the proof is complete. □

Lemma 4.3. Let 𝔽̃ : [0, T] × H → H (resp. 𝔾̃) be defined by (4.11) (resp. (4.12)). Under assumptions (1.12), (1.13), (4.9), (4.1) and (4.2), there exists a unique function 𝕌̃ : [0, T] → H with the following properties:

\begin{matrix}(i) & 𝕌̃ ∈ C^{1}([0, T]; H). \\ (ii) & \frac{d𝕌̃(t)}{dt} + 𝔸̃𝕌̃(t) = 𝔽̃(t, 𝕌̃(t)) in H on]0, T]. \\ (iii) & 𝕌̃(0) = 𝔾̃. \end{matrix}  (4.22)

Proof. 1. It follows from theorem 2.10 that −𝔸̃ is the infinitesimal generator of a C^{0}-semigroup of linear operators in H.

2. From lemma 4.2 we have that 𝔽̃ : [0, T] × H → H is continuously differentiable with bounded partial derivatives.

3. It can be seen that 𝔾̃ belongs to D(𝔸̃).

4. From theorem 2.11 we have the desired result. □

Theorem 4.4. Under assumptions (1.12), (1.13), (4.9), (4.1) and (4.2), there exists a unique weak solution of the semilinear problem (1.1)-(1.11).

Proof. Let

𝕌̃ := (\begin{matrix}(ũ_{p}^{1}, ũ_{m}^{1}) \\ (ũ_{p}^{2}, ũ_{m}^{2}) \end{matrix}) : [0, T] → H

be the unique function satisfying (4.22) (Lemma 4.3). It can be showed that (ũ_{p}^{1}, ũ_{m}^{1}) belongs to C^{1}([0, T]; Ṽ) ∩ C^{2}([0, T]; H) and that it satisfies (4.13). Then (\frac{1}{\sqrt{ρ_{p}h}}ũ_{p}^{1}, \frac{1}{\sqrt{ρ_{m}}} ũ_{m}^{1}) is the desired weak solution. The uniqueness follows from the

uniqueness of 𝕌̃. □

Remark 4.5. For sufficiently smooth solutions in the sense of definition 4.1 we can obtain as usual a classical pointwise solution of system (1.1)-(1.11). See [12].

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Universidad del Norte, Km 5 via a Puerto Colombia, Barranquilla, Colombia

E-mail address: jahernan@uninorte.edu.co