The Wave Equation with Acoustic Boundary Conditions on Non-Locally Reacting Surfaces
Authors: Mugnolo Delio , Vitillaro Enzo 
Published: November 2024

The aim of the paper is to study the problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t t Baseline minus c squared normal upper Delta u equals 0 2nd Column a m p semicolon in double-struck upper R times normal upper Omega comma 2nd Row 1st Column mu v Subscript t t Baseline minus d i v Subscript Sub Subscript normal upper Gamma Baseline left-parenthesis sigma nabla Subscript normal upper Gamma Baseline v right-parenthesis plus delta v Subscript t Baseline plus kappa v plus rho u Subscript t Baseline equals 0 2nd Column a m p semicolon on double-struck upper R times normal upper Gamma 1 comma 3rd Row 1st Column v Subscript t Baseline equals partial-differential Subscript nu Baseline u 2nd Column a m p semicolon on double-struck upper R times normal upper Gamma 1 comma 4th Row 1st Column partial-differential Subscript nu Baseline u equals 0 2nd Column a m p semicolon on double-struck upper R times normal upper Gamma 0 comma 5th Row 1st Column u left-parenthesis 0 comma x right-parenthesis equals u 0 left-parenthesis x right-parenthesis comma u Subscript t Baseline left-parenthesis 0 comma x right-parenthesis equals u 1 left-parenthesis x right-parenthesis 2nd Column a m p semicolon in normal upper Omega comma 6th Row 1st Column v left-parenthesis 0 comma x right-parenthesis equals v 0 left-parenthesis x right-parenthesis comma v Subscript t Baseline left-parenthesis 0 comma x right-parenthesis equals v 1 left-parenthesis x right-parenthesis 2nd Column a m p semicolon on normal upper Gamma 1 comma EndLayout">

{

u

t
t

−

c
2

Δ
u
=
0

a
m
p
;

in 

R

×
Ω

,

μ

v

t
t

−
d
i

v

Γ

(
σ

∇
Γ

v
)
+
δ

v
t

+
κ
v
+
ρ

u
t

=
0

a
m
p
;

on 

R

×

Γ
1

,

v
t

=

∂
ν

u

a
m
p
;

on 

R

×

Γ
1

,

∂
ν

u
=
0

a
m
p
;

on 

R

×

Γ
0

,

u
(
0
,
x
)
=

u
0

(
x
)
,

u
t

(
0
,
x
)
=

u
1

(
x
)

a
m
p
;

in 

Ω

,

v
(
0
,
x
)
=

v
0

(
x
)
,

v
t

(
0
,
x
)
=

v
1

(
x
)

a
m
p
;

on 

Γ
1

,

\begin{equation*} \begin {cases} u_{tt}-c^2\Delta u=0 \qquad &\text {in $\mathbb {R}\times \Omega $,}\\ \mu v_{tt}- div_{_\Gamma }(\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad &\text {on $\mathbb {R}\times \Gamma _1$,}\\ v_t =\partial _\nu u\qquad &\text {on $\mathbb {R}\times \Gamma _1$,}\\ \partial _\nu u=0 &\text {on $\mathbb {R}\times \Gamma _0$,}\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & \text {in $\Omega $,}\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) & \text {on $\Gamma _1$,} \end{cases} \end{equation*}

where

Ω
\Omega

is a open domain of

R

N

\mathbb {R}^N

with uniformly

C
r

C^r

boundary (

N
≥
2

N\ge 2

,

r
≥
1

r\ge 1

),

Γ
=
∂
Ω

\Gamma =\partial \Omega

,

(

Γ
0

,

Γ
1

)

(\Gamma _0,\Gamma _1)

is a relatively open partition of

Γ
\Gamma

with

Γ
0

\Gamma _0

(but not

Γ
1

\Gamma _1

) possibly empty. Here

d
i

v

Γ

div_{_\Gamma }

and

∇
Γ

\nabla _\Gamma

denote the Riemannian divergence and gradient operators on

Γ
\Gamma

,

ν
\nu

is the outward normal to

Ω
\Omega

, the coefficients

μ
,
σ
,
δ
,
κ
,
ρ

\mu ,\sigma ,\delta , \kappa , \rho

are suitably regular functions on

Γ
1

\Gamma _1

with

ρ
,
σ

\rho ,\sigma

and

μ
\mu

uniformly positive while

c
c

is a positive constant. This problem have been proposed long time ago by Beale and Rosencrans, when

N
=
3

N=3

,

σ
=
0

\sigma =0

,

r
=
∞

r=\infty

,

ρ
\rho

is constant,

κ
,
δ
≥
0

\kappa ,\delta \ge 0

, to model acoustic wave propagation with locally reacting boundary.

In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we give precise qualitative results for solutions when

Ω
\Omega

is bounded and

r
=
2

r=2

,

ρ
\rho

is constant,

κ
,
δ
≥
0

\kappa ,\delta \ge 0

. These results motivate a detailed discussion of the derivation of the problem in Theoretical Acoustics and the consequent proposal of adding to the model the integral condition

∫
Ω

u
t

=

c
2

∫

Γ
1

v

\int _\Omega u_t=c^2\int _{\Gamma _1}v

.