bluuebunny/crossref_metadata_2025_split huggingface dataset:
The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations
Authors: Golovaty Dmitry 
Published: December 2016

The asymptotic behavior of a nonlocal Ginzburg-Landau equation
\[

u
t
ε

=
Δ

u
ε

−

1

ε
2

W
′

(

u
ε

)

+

1
ε

g

(

u
ε

)

λ
ε

u_t^\varepsilon = \Delta {u^\varepsilon } - \frac {1}{{{\varepsilon ^2}}}W’\left ( {{u^\varepsilon }} \right ) + \frac {1}{\varepsilon }g\left ( {{u^\varepsilon }} \right ){\lambda ^\varepsilon }

\]
is studied when the small parameter

ε
\varepsilon

tends to zero. Here a Lagrange multiplier

λ
ε

{\lambda ^\varepsilon }

is introduced into the equation to enforce the conservation of mass. An energy-estimates approach is used to show that a limiting solution can be characterized by moving interfaces. It is further shown that the asymptotic limit of solutions of the nonlocal Ginzburg-Landau equation is a weak solution of the nonlocal, mass-preserving mean curvature flow. The weak solutions are constructed within a framework of the theory of viscosity solutions. In addition, the results describing interactions between the interfaces are obtained.
A regularizing effect of nonlinear transport equations
Authors: Otto Felix 
Published: December 2016

We consider the semigroup on

L
1

(

R

n

)

{L^1}\left ( {\mathbb {R}^n} \right )

defined by the nonlinear transport equation for the scalar

s
s

,
\[

∂
t

s
+
d
i
v

(
f

(
s
)

u
)

=
0

i
n

(
0
,
∞
)

×

R

n

{\partial _t}s + div\left ( f\left ( s \right )u \right ) = 0 \qquad in \left ( 0, \infty \right ) \times {\mathbb {R}^n}

\]
for given velocity field

u
u

. We show that this nonlinear semigroup is Hölder continuous for

t
>
0

t > 0

in the uniform operator topology, provided the graph of

f
f

has no linear segments. This continuity property—which expresses a regularizing effect of the nonlinearity in the transport equation—is robust with respect to the spatial behaviour of the time-independent velocity field

u
u

.
Variational problems on flows of diffeomorphisms for image matching
Authors: Dupuis Paul , Grenander Ulf , Miller Michael I. 
Published: December 2016

This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image

T
T

is to be carried via a smooth change of variable into an image that is intended to provide a good fit to the observed data. The images are all defined on an open bounded set

G
⊂

R
3

G \subset {R^3}

. The changes of variable are determined as solutions of the nonlinear Eulerian transport equation
\[

d
η

(
s
;
x
)

d
s

=
v

(
η

(
s
;
x
)

,
s
)

,

η

(
τ
;
x
)

=
x
,

(
0.1
)

\frac {{d\eta \left ( s; x \right )}}{{ds}} = v\left ( \eta \left ( s; x \right ),s \right ), \qquad \eta \left ( \tau ; x \right ) = x, \qquad \left ( 0.1 \right )

\]
with the location

η

(
0
;
x
)

\eta \left ( 0; x \right )

in the canonical image carried to the location

x
x

in the deformed image. The variational problem then takes the form
\[

arg
⁡

min
v

[

‖
v
‖

2

+

∫
G

|

T
o
η

(

0
;
x

)

−
D

(
x
)

|

2

d
x

]

,

(

0.2

)

\arg \min \limits _v {\kern -0.1pt} \left [ {{{\left \| v \right \|}^2} + \int _G {{{\left | {T o \eta \left ( {0; x} \right ) - D\left ( x \right )} \right |}^2}dx} } \right ], \qquad \left ( {0.2} \right )

\]
where

‖
v
‖

\left \| v \right \|

is an appropriate norm on the velocity field

v
(
⋅
,
⋅
)

v( \cdot , \cdot )

, and the second term attempts to enforce fidelity to the data.
Decay rates of solutions to a von Kármán system for viscoelastic plates with memory
Authors: Muñoz Rivera Jaime E. , Perla Menzala Gustavo 
Published: December 2016

We consider the dynamical von Kármán equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially, When the relaxation

g
g

satisfies
\[

−

c
1

g

1
+

1
p

(
t
)

≤

g
′

(
t
)

≤
−

c
0

g

(
t
)

1
+

1
p

,

0
≤
g

(
t
)

≤

c
2

g

1
+

1
p

(
t
)

,

and

- {c_1}{g^{1 + \frac {1}{p}}}\left ( t \right ) \le g’\left ( t \right ) \le - {c_0}g{\left ( t \right )^{1 + \frac {1}{p}}}, \qquad 0 \le g\left ( t \right ) \le {c_2}{g^{1 + \frac {1}{p}}}\left ( t \right ) , \: \textrm {and}

\]

\[

g
,

g

1
+

1
p

∈

L
1

(

R

)

with

p
>
2
,

g, {g^{1 + \frac {1}{p}}} \in {L^1}\left ( \mathbb {R} \right ) \textrm {with} \: p > 2 ,

\]
then the energy decays as

1

(
1
+
t
)

p

\frac {1}{\left ( 1 +t \right )^{p}}

. A new Liapunov functional is built for this problem.
𝐿²-regularity theory of linear strongly elliptic Dirichlet systems of order 2𝑚 with minimal regularity in the coefficients
Authors: Ebenfeld Stefan 
Published: December 2016

In this article, we consider the following Dirichlet system of order

2
m

2m

:
\[

L

(
x
,
∇
)

u
=
f

(
x
)

i
n
Ω

L\left ( x, \nabla \right )u = f\left ( x \right ) \qquad in \Omega

\]
,
\[

∇
k

u
=
0

o
n
∂
Ω

(
k
=
0
,
.
.
.
,
m
−
1
)

{\nabla ^k}u = 0 \qquad on \partial \Omega \left ( k = 0,...,m - 1 \right )

\]
. Here,

Ω
\Omega

is a smooth bounded domain in

R

n

{\mathbb {R}^n}

and the differential operator

L

(
x
,
∇
)

L\left ( x, \nabla \right )

given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients

A

α
β

(
m
)

,

B

α
β

(

k
m

)

,

C
α

(
k
)

A_{\alpha \beta }^{\left ( m \right )}, B_{\alpha \beta }^{\left ( {km} \right )}, C_\alpha ^{\left ( k \right )}

and for

f
∈

H

−
m
+
s

(
Ω
,

R

N

)

f \in \\ {H^{ - m + s}}\left ( \Omega , {\mathbb {R}^N} \right )

, every weak solution

u
∈

H

0

m

(
Ω
,

R

N

)

u \in H_{0}^{m}\left ( \Omega , {\mathbb {R}^N} \right )

is actually in

H

m
+
s

(
Ω
,

R

N

)

{H^{m + s}}\left ( \Omega , {\mathbb {R}^N} \right )

and satisfies an a priori estimate of the following form:
\[

‖
u
‖

M

m
+
s

(
Ω
,

R

N

)

≤

C
^

‖
f
‖

H

−
m
+
s

(
Ω
,

R

N

)

+

K
^

‖
u
‖

L
2

(
Ω
,

R

N

)

{\left \| u \right \|_{{M^{m + s}}\left ( \Omega , {\mathbb {R}^N} \right )}} \le \hat C{\left \| f \right \|_{{H^{ - m + s}}\left ( \Omega ,{\mathbb {R}^N} \right )}} + \hat K{\left \| u \right \|_{{L^2}\left ( \Omega ,{\mathbb {R}^N} \right )}}

\]
. The latter a priori estimate is of particular interest in applications to nonlinear PDEs (see, e.g., [6] and [10]). There the coefficients of

L

(
x
,
∇
)

L\left ( x, \nabla \right )

result from a linearization procedure and consequently they cannot be chosen as smooth as one likes. Therefore, e.g. in [10] (Kato), the author cannot use the famous results stated in [4] (Agmon-Douglis-Nirenberg) but refers to [14] (Milani) instead.
An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions
Authors: Zayed E. M. E. 
Published: December 2016

This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in

R
3

{R^3}

. The trace of the heat semigroup

θ

(
t
)

=

∑

v
=
1

∞

exp
⁡

(
−
t

μ
v

)

\theta \left ( t \right ) = \sum \nolimits _{v = 1}^\infty {\exp \left ( - t{\mu _v} \right )}

, where

{

μ
v

}

v
=
1

∞

\left \{ {{\mu _v}} \right \}_{v = 1}^\infty

are the eigenvalues of the negative Laplacian

−

∇
2

=
−

∑

β
=
1

3

(

∂

∂

x
β

)

2

- {\nabla ^2} = - {\sum \nolimits _{\beta = 1}^3 {\left ( {\frac {\partial }{{\partial {x^\beta }}}} \right )} ^2}

in the

(

x
1

,

x
2

,

x
3

)

\left ( {x^1}, {x^2}, {x^3} \right )

-space, is studied for a general multiply-connected bounded domain

Ω
\Omega

in

R
3

{R^3}

surrounding by simply connected bounded domains

Ω
j

{\Omega _j}

with smooth bounding surfaces

S
j

(
j
=
1
,
.
.
.
,
n
)

{S_j}\left ( j = 1,...,n \right )

, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components

S
i
∗

(
i
=
1
+

k

j
−
1

,
.
.
.
,

k
j

)

S_i^* \left ( i = 1 + {k_{j - 1}},...,{k_j} \right )

of the bounding surfaces

S
j

{S_j}

is considered, such that

S
j

=

∪

i
=
1
+

k

j
−
1

k
j

S
i
∗

{S_j} = \cup _{i = 1 + {k_{j - 1}}}^{{k_j}} S_i^*

, where

k
0

=
0

{k_0} = 0

. Some applications of

θ

(
t
)

\theta \left ( t \right )

for an ideal gas enclosed in the multiply-connected bounded container

Ω
\Omega

with Robin boundary conditions are given. We show that the asymptotic expansion of

θ

(
t
)

\theta \left ( t \right )

for short-time

t
t

plays an important role in investigating the influence of the finite container

Ω
\Omega

on the thermodynamic quantities of an ideal gas.
Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source
Authors: Chan C. Y. , Tian H. Y. 
Published: December 2016

Let

q
q

be a nonnegative real number, and

T
T

be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem:
\[

x
q

u
t

(
x
,
t
)

−

u

x
x

(
x
,
t
)

=

a
2

δ

(
x
−
b
)

f

(
u

(
x
,
t
)

)

f
o
r
0
>
x
>
1
,
0
>
t
≤
T

{x^q}{u_t}\left ( x, t \right ) - {u_{xx}}\left ( x, t \right ) = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) \qquad for 0 > x > 1, 0 > t \le T

\]
,
\[

u

(
x
,
0
)

=
ψ

(
x
)

f
o
r
0
≤
x
≤
1

u\left ( x, 0 \right ) = \psi \left ( x \right ) \qquad for 0 \le x \le 1

\]
,
\[

u

(
0
,
t
)

=
u

(
1
,
t
)

=
0

f
o
r
0
>
t
≤
T

u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 \qquad for 0 > t \le T

\]
, where

δ

(
x
)

\delta \left ( x \right )

is the Dirac delta function, and

f
f

and

ψ
\psi

are given functions. It is shown that the problem has a unique solution before a blow-up occurs,

u
u

blows up in a finite time, and the blow-up set consists of the single point

b
b

. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.
Front motion in multi-dimensional viscous conservation laws with stiff source terms driven by mean curvature and variation of front thickness
Authors: Fan Haitao , Jin Shi 
Published: December 2016

The bistable reaction-diffusion-convection equation
\[

∂
t

u
+
∇
⋅
f

(
u
)

=

−

1
ϵ

g

(
u
)

+
ϵ
Δ
u

,

x
∈

R

n

,

u
∈

R

(
1
)

{\partial _t}u + \nabla \cdot f\left ( u \right ) = {- \frac {1}{\epsilon }g\left ( u \right ) + \epsilon \Delta u}, \qquad {x \in \mathbb {R}{^n}}, {u \in \mathbb {R}} \qquad \left ( 1 \right )

\]
is considered. Stationary traveling waves of the above equation are proved to exist when

f

(
u
)

f\left ( u \right )

is symmetric and

g

(
u
)

g\left ( u \right )

is antisymmetric about

u
=
0

u = 0

. Solutions of initial value problems tend to almost piecewise constant functions within

O

(
1
)

ϵ

O\left ( 1 \right )\epsilon

time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of

f
=
b

u
2

f = b{u^2}

and

g

(
u
)

=
a
u

(
1
−

u
2

)

g\left ( u \right ) = au\left ( 1 - {u^2} \right )

, where

b
∈

R

n

b \in {\mathbb {R}^{n}}

and

0
>
a
∈

R

0 > a \in \mathbb {R}

are constants, the front motion equation takes a more explicit form, showing that the front’s speed is
\[

ϵ

(

k
+

∇
μ

μ

⋅
T

)

\epsilon \left ( {k + \frac {{\nabla \mu }}{\mu } \cdot T} \right )

\]
, where

κ
\kappa

is the mean curvature of the front,

μ
\mu

is the width of the planar traveling of (1) in the normal direction n of the front, and T is a vector tangential to the front. Both

κ
\kappa

and

∇
μ

/

μ
⋅
T

\nabla \mu /\mu \cdot T

T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in

R

2

{\mathbb {R}^{2}}

is found to preserve its shape while shrinking.
Mean field variation in random media
Authors: Beran Mark J. , McCoy John J. 
Published: December 2016

We consider here the basic equation
\[

∂

∂

x
i

[

ϵ

(

X

)

∂
ϕ

∂

x
i

(

X

)

]

=
ρ

(

X

)

\frac {\partial }{{\partial {x_i}}}\left [ { \epsilon \left ( \textrm {X} \right )\frac {{\partial \phi }}{{\partial {x_i}}}\left ( \textrm {X} \right )} \right ] = \rho \left ( \textrm {X} \right )

\]
, where

∈

(

X

)

\in \left ( \textrm {X} \right )

is a random function of position and

ρ

(

X

)

\rho \left ( \textrm {X} \right )

is a prescribed source term. A formal equation is derived that governs

{

ϕ

(

X

)

}

\left \{ {\phi \left ( \textrm {X} \right )} \right \}

, where the braces indicate an ensemble average. The equation, which depends on the boundary conditions of the stochastic problem, is presented in terms of an infinite sequence of correlation functions associated with

∈

(

X

)

\in \left ( \textrm {X} \right )

. The equation is investigated first for the case of an infinite dielectric where isotropy may be assumed. An impulse response function is obtained and an explicit form of this response function is presented for the limit of small perturbations. Further, it is shown that the equation governing

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

is greatly simplified for the case in which all characteristic lengths associated with

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

are large compared to all correlation lengths

l
i

{l_i}

associated with the

∈

(

X

)

\in \left ( \textrm {X} \right )

field. The question of boundary conditions is next considered and as an example a spherical boundary (radius

R
R

) is studied. It is demonstrated, in this case, that if

R
≫

l
i

R \gg {l_i}

the effects of the boundary conditions on the equation governing

{

ϕ

(
x
)

}

\left \{ {\phi \left ( x \right )} \right \}

are negligible except at points within a thin layer near the boundary. The relationship between the ensemble average and the local volume average is also discussed.
Sobolev-type lower bounds on ∥∇𝜓∥² for arbitrary regions in two-dimensional Euclidean space
Authors: Rosen Gerald 
Published: December 2016

This note reports the derivation of lower bounds of the Sobolev type on

‖

∇
ψ

‖

2

≡
∫

R

(
∂
ψ

/

∂

x
1

)
2

+

(
∂
ψ

/

∂

x
2

)
2

)
d

x
1

d

x
2

{\left \| {\nabla \psi } \right \|^2} \equiv \smallint {}_R{(\partial \psi /\partial {x_1})^2} + {(\partial \psi /\partial {x_2})^2})d{x_1}d{x_2}

for generic real scalar

ψ
=
ψ
(

x
1

,

x
2

)

\psi = \psi ({x_1},{x_2})

of function class

C
0

{C^0}

piecewise

C
2

{C^2}

which vanish over the boundary of the (bounded or unbounded) region

R
R

in Euclidean 2-space.
Resonant frequencies in an electromagnetic eccentric spherical cavity
Authors: Kanellopoulos John D. , Fikioris John G. 
Published: December 2016

The interior boundary-value electromagnetic (vector) problem in the region between two perfectly conducting spheres of radii

R
1

{R_1}

,

R
2

{R_2}

and distance

d
d

between their centers is considered. Surface singular integral equations are used to formulate the problem. Use of spherical vector wave functions and related addition theorems reduces the solution of the integral equations to the problem of solving an infinite set of linear equations. Their determinant is evaluated in powers of

k
d
=
2
π
d

/

λ

kd = 2\pi d/\lambda

to a few terms. It is then specialized to the axially symmetric case and set equal to zero. This yields closed-form expressions for the coefficients

g

n
s

{g_{ns}}

in the resulting relations

ω

n
s

(

k
d

)

=

ω

n
s

(
0
)

[

1
+

g

n
s

(

k
d

)

2

+
⋅
⋅
⋅

]

{\omega _{ns}}\left ( {kd} \right ) = {\omega _{ns}}\left ( 0 \right )\left [ {1 + {g_{ns}}{{\left ( {kd} \right )}^2} \\ + \cdot \cdot \cdot } \right ]

for the natural frequencies of the cavity. Numerical results, comparisons and possible generalizations are also included.
Plane harmonic functions in the presence of a surface layer of arbitrary stiffness
Authors: Zahalak George Ireneus 
Published: December 2016

Complex variable techniques are employed to characterize two-dimensional solutions

u

(

x
,
y

)

u\left ( {x, y} \right )

of Laplace’s equation which satisfy the boundary condition

[

β

(

∂
2

u

/

∂

y
2

)

+

(

∂
u

/

∂
x

)

]

x
=
0

=
0

{\left [ {\beta \left ( {{\partial ^2}u/ \\ \partial {y^2}} \right ) + \left ( {\partial u/\partial x} \right )} \right ]_{x = 0}} = 0

, where

β
\beta

is referred to as the surface-stiffness parameter. Simple closed-form singular solutions are derived which satisfy this boundary condition and represent source and dislocation singularities. The former is used to synthesize the field generated by a small inclusion of arbitrary shape on which

u
=
1

u = 1

, in the presence of a boundary at

y
=
0

y = 0

on which

u
=
0

u = 0

. At points not near the inclusion the field has the form of a function of position and surface stiffness multiplied by a strength factor which depends on the size and shape of the inclusion and the surface stiffness. Detailed calculations are presented for two extreme shapes of inclusions—a shallow, wide inclusion on the surface and a deep, narrow inclusion penetrating below the surface—which exhibit the relation between the field near the inclusion and the distant field, and show explicitly the dependence of the strength factor on surface stiffness and inclusion size and shape. The nature and strength of the singularities at the tips of the inclusions are also examined and it is found that a tip singularity at the surface changes character as the surface stiffness varies.
Translational addition theorems for spheroidal scalar and vector wave functions
Authors: Sinha Bateshwar P. , Macphie Robert H. 
Published: December 2016

The translational addition theorems for spheroidal scalar wave functions

R

m
n

(
i
)

(

h
,
ξ

)

S

m
n

(

h
,
η

)

exp
⁡

(

j
m
ϕ

)

;
i
=
1
,
3
,
4

R_{mn}^{\left ( i \right )}\left ( {h, \xi } \right ){S_{mn}}\left ( {h, \eta } \right )\exp \left ( {jm\phi } \right ); i = 1, 3, 4

and spheroidal vector wave functions

M

m
n

x
,
y
,
z

(
i
)

(

h
;
ξ
,

η
,
ϕ

)

,

N

m
n

x
,
y
,
z

(
i
)

(

h
;
ξ
,
η
,
ϕ

)

;
i
=
1
,
3
,
4

M_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \\ \eta , \phi } \right ), N_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \eta , \phi } \right ); i = 1, 3, 4

, with reference to the spheroidal coordinate system at the origin

O
O

, have been obtained in terms of spheroidal scalar and vector wave functions with reference to the translated spheroidal coordinate system at the origin

O
′

O’

, where

O
′

O’

has the spherical coordinates (

r
0

,

θ
0

,

ϕ
0

{r_0},{\theta _0},{\phi _0}

) with respect to

O
O

. These addition theorems are useful in acoustics and electromagnetics in those cases involving spheroidal radiators and scatterers.
On a uniform asymptotic expansion of a Fourier-type integral
Authors: Wong R. 
Published: December 2016

An alternative derivation is given for a uniform asymptotic expansion of the integral
\[

I

(
h
)

=

∫
q
z

g

(

y
2

−

q
2

)

sin
⁡
y
h
d
y

(

h
→
+
∞

)

I\left ( h \right ) = \int _q^z {g\left ( {\sqrt {{y^2} - {q^2}} } \right )\sin yh dy \qquad \left ( {h \to + \infty } \right )}

\]
, which has been obtained recently by Schmidt. Here

q
q

varies in the interval

[

0
,

q
0

]

\left [ {0,{q_0}} \right ]

,

q
0

{q_0}

is some fixed number less than

z
z

, and

z
z

is finite. A similar result is obtained for integrals having an infinite range of integration. Realistic bounds are also provided for the error terms associated with the expansions. Our approach is based on a summability method introduced by Olver.
On some epidemic models
Authors: Gripenberg Gustaf 
Published: December 2016

The qualitative behavior of the solution

x
x

of the equation
\[

x

(
t
)

=
k

(

p

(
t
)

−

∫
0
t

A

(

t
−
s

)

x

(
s
)

d
s

)

(

f

(
t
)

+

∫
0
t

a
(
t
−
s
)
x
(
s
)
d
s

)

,
t
≥
0

x\left ( t \right ) = k\left ( {p\left ( t \right ) - \smallint _0^tA\left ( {t - s} \right )x\left ( s \right )ds} \right )\left ( {f\left ( t \right ) + \smallint _0^ta(t - s)x(s)ds} \right ),t \ge 0

\]
is studied. This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity.
New third-order bounds on the effective moduli of 𝑁-phase composites
Authors: Phan-Thien N. , Milton G. W. 
Published: December 2016

We develop some new bounds on the effective moduli of

N
N

-phase composites. These new bounds are accurate up to and including terms of third order in

O

(

|

K
i

−

K
j

|

,

|

μ
i

−

μ
j

|

)

O\left ( {\left | {{K_i} - {K_j}} \right |,\left | {{\mu _i} - {\mu _j}} \right |} \right )

, where

K
i

{K_i}

and

μ
i

{\mu _i}

are the bulk and shear modulus, respectively, of phase

i
i

. These bounds use the same statistical information as McCoy’s and Beran-Molyneux’s bounds but are tighter than, or at worst coincident with, the latter bounds. We also present in the appendix a new perturbation solution for the effective moduli which only requires that

|

δ
μ

|

=
O

(

|

μ
i

−

μ
j

|

)

\left | {\delta \mu } \right | = O\left ( {\left | {{\mu _i} - {\mu _j}} \right |} \right )

be small.
Nonoscillatory differential equations with retarded and advanced arguments
Authors: Gopalsamy K. 
Published: December 2016

Sufficient conditions are derived for a vector-matrix system of the form
\[

d
n

X

(
t
)

d

t
n

+

(

−
1

)

n
−
1

[

P

(
t
)

X

(

t
−

τ
1

(
t
)

)

+
Q

(
t
)

X

(

t
+

τ
2

(
t
)

)

]

=
0

\frac {{{d^n}X\left ( t \right )}}{{d{t^n}}} + {\left ( { - 1} \right )^{n - 1}}\left [ {P\left ( t \right )X\left ( {t - {\tau _1}\left ( t \right )} \right ) + Q\left ( t \right )X\left ( {t + {\tau _2}\left ( t \right )} \right )} \right ] = 0

\]
to be nonoscillatory.
Finite element approximation of a reaction-diffusion equation. I. Application of topological techniques to the analysis of asymptotic behavior of the semidiscrete approximations
Authors: Khalsa Sat Nam S. 
Published: December 2016

The initial-boundary value problem for a reaction-diffusion equation
\[

u
t

=

u

x
x

+
f

(
u
)

,

f

(
u
)

=
−
u

(

u
−
b

)

(

u
−
1

)

,

0
>
b
>
1

/

2
,

(
∗
)

{u_t} = {u_{xx}} + f\left ( u \right ), \qquad f\left ( u \right ) = - u\left ( {u - b} \right )\left ( {u - 1} \right ), \qquad 0 > b > 1/2, \qquad \left ( * \right )

\]
was analyzed in [4, 17] by using the Conley index. In this paper we study the asymptotic behavior of the semidiscrete finite element approximations, with interpolation of the coefficients in the nonlinear terms. We show that for small

h
h

the spectrum of the linearized discrete steady-state problem is a “good” approximation for the spectrum of the linearized continuous steady-state problem. Using the interpretation of the Conley index as the dimension of an unstable manifold of a steady-state solution, we establish that the properties of the semidiscrete approximations are completely analogous to those of the solutions of

(
∗
)

\left ( * \right )

. The asymptotic, as

t
→
∞

t \to \infty

, optimal order convergence of the approximate solutions is proved.
Oscillations in a delay-logistic equation
Authors: Gopalsamy K. 
Published: December 2016

Sufficient conditions are derived for all nonconstant nonnegative solutions of the equations of the form
\[

d
x

(
t
)

d
t

=
x

(
t
)

{

a
−

∑

j
=
1

n

b
j

x

(

t
−

τ
j

)

}

\frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - \sum \limits _{j = 1}^n {{b_j}x\left ( {t - {\tau _j}} \right )} } \right \}

\]
and
\[

d
x

(
t
)

d
t

=
x

(
t
)

{

a
−
b

∫

−
∞

t

k

(

t
−
s

)

x

(
s
)

d
s

}

\frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - b\int _{ - \infty }^t {k\left ( {t - s} \right )x\left ( s \right )ds} } \right \}

\]
to be oscillatory about their respective positive steady states. The results are complementary to those in [15].