• Nie Znaleziono Wyników

On the asymptotic behavior of solutions of second order parabolic partial differential equations

N/A
N/A
Protected

Share "On the asymptotic behavior of solutions of second order parabolic partial differential equations"

Copied!
12
0
0

Pełen tekst

(1)

n

i,j=1

ij

xixj

n

i=1

i

xi

t

α

α

N

β=1

αβ

β

α

α

n

i,j=1

αij

xixj

n

i=1

αi

xi

t

1

n

n

n

n

21

2n

n

n

i,j=1

ij

2

i

j

n

i=1

i

i

[223]

(2)

0

2

0

n

i=1

2

2i

2

2

n

2

n/2

2

n

4k1

2a+kk−2a

1

2

3

n

i,j=1

ij

i

j

1

2

−λ

2

1−µ

2

1

n

i

2

2

1/2

3

2

λ

2

µ

1

2

3

n

i,j=1

ij

i

j

1

2

1−λ

2

n

i

2

2

1/2

3

2

λ

(3)

1

2

3

n

i,j=1

ij

i

j

1

2

2

n

i

2

2

2

2

−1/2

3

2

µ

2

1

1

2

00

1

0

0

2

2−µ

n

α

α

N

β=1

αβ

β

α

n

i,j=1

αij

xixj

n

i=1

αi

xi

t

α

α

N

β=1

αβ

β

α

(4)

0

2

r→∞

(x,t)∈D

|x|=r

2

µ

r→∞

(x,t)∈D

|x|=r

2

µ

1

0

2

2

1

1

2

3

1

2

2

n

i,j=1

ij

i

j

1

2

2

n

i

2

2

2

2

−1/2

3

2

µ

3

r→∞

|x|=r 0≤t≤T

2

µ

00

2

µ

τ t

(5)

2

1

2

2

1

2

2

3

1

2

µ

τ t

3

00

3

0

2

µ

τ t

2

1

2

2

1

2

3

1

2

3

1

n

2

µ

τ t

n

i,j=1

ij

xixj

n

i=1

i

xi

t

2

2

2µ−2

0

2

2τ t

n

i,j=1

ij

i

j

µ−2

0

2

τ t

n

i,j=1

ij

i

j

µ−1

00

τ t

n

i,j=1

ij

i

j

µ−1

0

τ t

n

i=1

ii

µ−1

0

τ t

n

i=1

i

i

µ

τ t

1

2

3

00

2

1

2

2

2τ t

µ

1

2

τ t

2

µ

τ t

3

µ

µ

τ t

2

1

2

2

1

2

2

3

µ

2τ t

(6)

1

2

1

2

2

2τ t

µ

1

2

τ t

µ−1

3

1

τ t

2

µ

τ t

3

µ

µ

τ t

2

1

2

2

1

2

3

1

2

3

µ

2τ t

1

1

3

4

1

2

3

3

n

i,j=1

ij

i

j

1

2

2

n

i

2

2

2

2

−1/2

3

2

µ

3

2

µ

00

2

µ

τ t

1

2

2

3

2

2

µ

τ t

3

00

3

0

2

µ

τ t

1

2

1

3

2

3

2

n

2

µ

τ t

(7)

2

2

2µ−2

0

2

2τ t

n

i,j=1

ij

i

j

µ−2

0

2

τ t

n

i,j=1

ij

i

j

µ−1

00

τ t

n

i,j=1

ij

i

j

µ−1

0

τ t

n

i=1

ii

µ−1

0

τ t

n

i=1

i

i

µ

τ t

1

2

τ t

2

µ

τ t

3

µ

µ

τ t

1

2

2

3

µ

τ t

2

µ

τ t

2

1

2

τ t

1

µ−1

τ t

3

2

µ

τ t

3

µ

µ

τ t

1

2

1

3

2

3

τ t

µ

2

1

2

3

00

2

µ

τ t

2

1

2

2

2

3

3

2

µ

τ t

3

00

3

0

2

µ

τ t

(8)

2

1

2

2

3

1

2

3

3

n

1

3

4

00

2

µ

τ t

2

3

4

2

µ

τ t

3

00

3

0

2

µ

τ t

1

3

2

3

4

n

α

α

N

β=1

αβ

β

α

n

i,j=1

αij

xixj

n

i=1

αi

xi

t

5

α

α

α

N

β=1

αβ

β

α

0

2

6

α

αβ

1

1

2

3

1

2

2

n

i,j=1

αij

i

j

1

2

2

αi

2

2

2

2

−1/2

(9)

N

β=1

αβ

3

2

µ

7

r→∞

|x|=r

|t|<T

α

2

µ

00

α

2

µ

τ t

2

1

2

2

1

2

2

3

5

α

2

µ

τ t

3

00

3

0

α

2

µ

τ t

2

1

2

2

1

2

3

1

2

3

5

n

α

α

αβ

α

α

n

α

α

α

α

(10)

n

8

n

i,j=1

αij

i

j

1

2

2

n

αi

2

2

2

2

−1/2

αβ

n

β=1

αβ

3

2

µ

1

2

3

α

α

α

N

β=1

αβ

β

n

α

n

α

2

µ

n

α

n

8

N

β=1

αβ

3

α

t→∞

α

n

t→∞

α

n

α

n

α

(x,t)∈Rn×[0,∞)

α

±α

3

α

−h(t−δ)

α

3

α

±α

N

β=1

αβ

β

3

N

β=1

αβ

α

−h(t−δ)

N

β=1

αβ

α

−h(t−δ)

α

α

−h(t−δ)

3

(11)

n

±α

3

α

α

n

±α

n

3

α

−h(t−δ)

α

3

α

−h(t−δ)

n

3

t→∞

α

t→∞

α

3

, which proves our theorem.

(12)

R´evis´e le 31.8.1995

Cytaty

Powiązane dokumenty

In this paper, we present some results concerning the existence and the local asymptotic stability of solutions for a functional integral equation of fractional order, by using

[14] Hartogs type extension theorem of real analytic solutions of linear partial differential equations with constant coefficients, in: Advances in the Theory of Fr´ echet

The asymptotic behavior of the solutions of the n-th order differential equations have been considered by T.. Similar problems with regard to the second order

In other papers, there are considered some sufficient conditions in order that components of all nontrivial solutions o f systems o f differential equations have

( 0. The results obtained here overlap some results of E.. the successive zeros of an oscillatory solution x{t). This condition is a generalization of one given

Existence of weak solutions and an L ∞ -estimate are shown for nonlinear non- degenerate parabolic systems with linear growth conditions with respect to the gradient.. The L ∞

Key words and phrases: infinite systems of parabolic differential-functional equations, monotone iterative method, method of lower and upper functions...

Keywords: Banach space, difference equation, fixed point, measure of noncompactness, asymptotic behaviour of solutions.. 2000 Mathematics Subject Classification: