Энергетический метод

2u

(x)

div( p(x) u) q(x)u, x , t 0

t2

u

u(x,0) 0,

(x,0) 0,

x

t

u

(x)u(x,t) (x)

(x,t) 0, x , t 0

E(t)

1

u 2

dx

1

2

2

(x)

2

p(x) | u(x,t) | dx

t

1

1

( x)

2 q(x) | u(x,t) |2 dx

2 p(x) (x)

| u(x,t) |2 dS

E(0)

1

u

2

1

2

dx

(x)

t

(x,0)

dx

2

p(x) | u(x,0) |

2

1

1

(x)

2 q(x) | u(x,0) |2 dx 2 p(x)

(x) | u(x,0) |2 dS 0

0

E(t) E(0)

E(t) 0

1

(x)

u

2

1

2

dx

2

dx

2

p(x) | u(x,t) |

t

1

2

1

(x)

2

2 q(x) | u(x,t) |

dx

2 p(x) (x)

| u(x,t) |

dS 0

2u

T 2u

F(x,t)

, a

T

, f (x,t)

F(x,t)

t2

x2

t

(x0

,t0 )

Характеристический конус

D {(x,t) :| x x0 | a(t0 t)}

t T

DT {(x,t) D : t T}

единичная внешняя нормаль к DT

{ 1, 2 ,..., n , t}

xn

x1

{x Rn :| x x0 | at0}

DT S1 S2 S3

S1 {(x,t) : x ,t 0}

S2 {(x,t) :| x x0 | a(t0 T ),t T}

S3 {(x,t) : 0 t T,| x x0 | a(t0 t)}