able to generalized solution of problem is proved. 1. Problem statement − 2

( − 2 ( | | | ( | ) ||) The initial conditions are given as = ( , ), 0 < < ,

0 < < , ) = 0, 0 < < ,

0 < < . (0, ) = 0, (, ) +

(, ) = 0, (0, ) = (, ) = 0.

2 ( , 0) = 0( ), ( , 0) = 0( ), 0 ≤ ≤ . 0000-0002-3376-5064 (M.F. Pavlova); 0000-0003-3616-7665 (E.V. Rung) © 2021 Copyright for this paper by its authors. (1) (2) (3) (4) (5) (| |) = { 0, | | ≤ 0, 1, | | > 0.

In what follows, we assume that the functions ( ), (, ) satisfy the following conditions: 1. ( ), ≥ 0 is an absolutely continuous in , nonnegative, nondecreasing function and there exist 0 ≥ 0, , > 0, such that at ≥ 0 the following inequality holds 2. The function (, ) is continuous at (, ) ∈ , where = (0, ) × (0, ].

Conditions (6) imposed on the function g mean that the filtration rate will be zero for small values of the gradient modulus. 2. Defining a generalized solution and let ◦1 be the closure of smooth functions equal to zero on the boundary of the interval [0, ], in the norm of the same space.

Definition. By a generalized solution to problem (1)–(5), we imply functions (, ), for which the following conditions hold:

◦ ◦ ∈ 2(1)(0, ; ), ∈ 2(0, ; 1), (, 0) = 0( ), (, 0) = 0( ) and for any functions ∈ 2(1)(0, ; ◦ ), ∈ 2(0, ; ◦ 1) the following equality is true: almost everywhere on ∈ (0, ), 3. The discrete problem ̄ ℎ = { 0 = 0 < 1 < ... < = }. ̄ = { = , 0 ≤ ≤ , = },

Definition. By the approximate solution to the problem (1)–(5) constructed by the method of semiwhich the following conditions hold: discretization in combination with the finite element method, we imply the functions ( ̂ ( ), ̂ ( )) for ̂ ( ) ∈ , ̂ ( ) ∈ 1

∀ ∈ , almost everywhere on ∈ (0, ), and for any functions ∈ , ∈ 1 the following equality is true + ) − ̂ + ̂ + Theorem 1. Approxim ate solution of the problem (1)–(5) exists.

Proof. Obviously, it sufices to establish the existence of ̂ , ̂ satisfying (8), under the assumption Since the choice of the functions , is arbitrary, the equality (8) is equivalent to the following { that , are known.

system We will look for approximate solutions in the form ̂ − ∫ 0 { . ∫ 0

) { ̂ + − ̂ }

̂ 0 } conditions equations: { ∫

= 1, 2 are determined by the following system of , are linear on each element, continuous on the interval [0, ] functions satisfying the { ) ( ) − ̂ ( ) = ∫ ̂ (, ) ( ) ,

Let H ∶ 2 → 2 be a nonlinear operator such that the equation ∫ 0 { ̂ + ||) radius, on which We have

0 ∫ ( 0 (H( ), ) 2 = ∫ (

)

+ Here ∥ ∥12= ∫ (

2 )

.

Using the C0auchy-Bunyakovsky inequality from (15) it is easy to obtain the following estimate The first term on the right-hand side of equality (14) can be transformed to the form: + ) 1

1 = ( + 2 ) ∥ ̂ ∥21 − ( + 2 )∫ 1 2 ̂

1 ) ≥ ( + 2 − )∥ ̂ ∥21 − 4 ( 1 1 1 1

Using (16) and inequality (6), for the second term in equality (14) we have 0

∫ | 0 | | |

|∫ ̂ (, ) ⋅ | | | | | || ≤ ∥ ̂ here is a constant of the Friedrichs inequality, is a constant such that Substituting the estimates obtained in (14), we have where

Let ∗ be a constant such that for all 0 < ≤ ∗ the following inequality holds sphere. The proof of Theorem 1 is complete.

Lemma 1. For the approximate solution (8), the following a priori estimates are valid Note that ̂ = ̂ ( ̂ − ) = ⋅ 1 ̂ ( (18) (19) (20) (21) (22) ) + | ′ − and obtain 1

‖

‖ = =0 + ∑ ∫ 0 =0 0

= ∑ ∫ ̂ (, ) ⋅ .

(24) that ‖ ‖ ‖ ‖ ‖ ‖ ‖ we have estimate (21). The proof of Lemma 1 is complete.

Lemma 2. There exist function and sequences { }, { } such that at → 0, → ∞ ⇀

1 ◦ + } pactness of bounded sets in a reflexive Banach space. The proof of Lemma 2 is complete. Proof. Let the functions , satisfy relations (25)–(27), it is required to prove that , satisfy + (, ) =

1 tor Π+, can be written in the form ) +

+ Π − Π ̂ + Π + Π Π ̂ +

+ + +

| (|| || Π+ ̂ || Π ̂ Π ̂ + +

e r e p r e s e n t a s t h e s u m w h e r e =

, , , w e u s e e q u a l i t y ( 2 8 ) a t , a n d o b t a i n = − = − , { + + + + + + Π Π ( − Π Π Π Π

)

( 1 ) +

+ ̂ ̂ = − − Π + Π − + ∫

∫ , , ) Π ( − , w e m a k e t h e p a s s a g e t o t h e l i m i t a s , t a k i n g i n t o a c c o u n t ( 2 5 ) – ( 2 7 ) . → 0 → ∞ , A s a r e s u l t , w e o b t a i n { 2 2 2

2 ( −

) ( 1 ) → − − +

− ∫

∫ , . , t h e r i g h t h a n d s i d e o f r e l a t i o n ( 3 3 ) t a k e s t h e f o l l o w i n g f o r m {

}

2 ( − ) ( − ) ( 3 4 ) ( 1 ) → + , f r o m ( 2 5 ) – ( 2 7 ) , ( 2 9 ) f o r , w e o b t a i n → 0 →

∞ {

} 2 |

| ( − → − + , i t f o l l o w s f r o m t h e d e if n i t i o n o f t h a t , { } 2 |

| ( − ) ( − , w e c h o o s e w h e r e c o n s t a n d a r e a r b i t r a r y f u n c t i o n s f r o m = + , = + , w h e r e f o r

A s a r e s u l t w e o b t a i n ∞

∞ ( , ; ( , ) ) , ( , ) .

We divide inequality (37) by and pass to the limit as → 0, we obtain is an arbitrary function; therefore, we have = ( | | | | ||

The proof of theorem 2 is complete. 33 (1962) 1482–1498.