method for solving one nonlinear parabolic equation with double degeneration and nonlocal spatial operator.

Abstract

We consider the initial-boundary value problem for nonlinear parabolic equation. This type of equation can be classified as a parabolic equation with double degeneration: degeneration can be present in space operator, and a nonlinear function which is under the derivative sign with respect to the variable , may not be separated from zero. The space operator of the considered equation nonlinearly depends on the sought function, its gradient and the non-local (integral) solution characteristic. This problem has an applied nature. Such equations appear, for example, in modeling the process of bacteria population spreading. In the present paper we propose and investigate the explicit diferential scheme. A priori estimates are obtained, and the convergence of constructed algorithm is proved. The current work is a continuation of the research begun in the works [1], [2], [3], where the existence and uniqueness theorems for the generalized solution have been proved, the convergence of the finite-element method scheme and the explicit diference scheme in the case when nonlinearity is present only in the spatial operator have been investigated. In paper [4] for a problem with double degeneration, an approximate method has been studied. That method was constructed with the use of semidiscretization with respect to a variable and the finite element method in the space variable with lowering nonlocality to the lower layer, the existence of an approximate solution and the convergence of the constructed algorithms were proved. parabolic equation, nonlocal spatial operator, double degeneration, convergence Far Eastern Workshop on Computational Technologies and Intelligent Systems, March 2-3, 2021, Khabarovsk, Russia " [e-mail] (L.L. Glazyrina); [e-mail] (O.V. Glazyrina); [e-mail] (M.F. Pavlova) ~ https://kpfu.ru/Ludmila.Glazyrina (L.L. Glazyrina); https://kpfu.ru/olga.glazyrina (O.V. Glazyrina); https://kpfu.ru/Maria.Pavlova (M.F. Pavlova) 0000-0002-4499-1577 (L.L. Glazyrina); 0000-0002-9569-4425 (O.V. Glazyrina); 0000-0002-3376-5064 (M.F. Pavlova)

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( 1 ) ( 2 ) ( 3 ) 1. Statement of the problem consider the initial-boundary value problem Let the Ω be bounded domain in the space , Γ is its boundary, Ω, = Ω × (0, ). In the domain ( )

=1 ( ( , , ∇, ))= ,

∈ Ω, ∈ (0, ), ( , 0) = 0( ) ∈ Ω, ( , ) = 0, ∈ Γ, ∈ [0, ].

Here , 0 are known functions, is an operator of the form ( ) = ∫ ( , Ω ′ ( , )) , = − ∑ ( (, , the following inequalities for arbitrary ∈ 1,

We assume that function ( ) is an absolutely continuous, strongly increasing function and it satisfy ′ =

− 1 with respect to the gradient of the operator .

Lets note that the condition ( 7 ) implies that the operator , acting from 1 (Ω) into −′1(Ω), where , is bounded. The conditions (8), (9) provide, respectively, the coercivity and monotonicity measurable with respect to and satisfies the following condition

We assume that the function (, ) , defining the operator , is continuous with respect to , =1

here are constants such that following inequalities are correct to and for arbitrary ∈ Ω, 0, ∈ , 1 , 2, ∈ satisfy the following conditions functions (,

0, , ), = 1, … , , are continuous with respect to 0, and , measurable with respect Lets define the operator is a given function, Ω′ is a domain that is contained in Ω or coincides with it.

for almost all ∈ Ω, where 0 is a function integrable over Ω, ≥ 0.

Space operators with non-localities of the form ( 3 ) arise, for example, in the mathematical describing the difusion of bacteria population when it is assumed that the propagation speed at a point is specified by the global state of environment (e.g., see [5], [6]).

Lets define a generalized solution for a problem ( 1 )–( 2 ).

A function ∈ (0, ; 1(Ω)) ⋂ ∞(0, ; (Ω)) such that (, 0) = 0( ) almost everywhere in Ω, ∈ ′ (0, ; −′1(Ω)), (11) integral identity holds will be called a generalized solution of problem ( 1 ), ( 2 ), if for any function from (0, ; 1(Ω)) the ∫ ⟨ 0 , ⟩

+ ∫ ∫

∑ (, , ◦ { here ⟨, ⟩ is the value of a functional from −′1(Ω) on element from 1 (Ω). 2. Auxiliary results and notation 0 ≤ ≤ , = 1, 2, … , . In what follows, we will assume that the domain Ω is a -dimensional parallelepiped: Ω = ∈ ∶ }. On Ω construct a uniform mesh ̄ ℎ with a mesh step ℎ in the -th direction, ℎ⃗ = (ℎ1, … , ℎ ), ℎ = 1m≤ i≤n ℎ . We will assume that there is a constant such that ℎ ≤ ℎ, ℎ = 1m≤a≤x ℎ . ℎ = = ( 1, … , ) ∈ Ω ∶ = ℎ , = 0, … , , =

ℎ = ̄ ℎ ∩ Γ, ℎ = ℎ⧵ ℎ.

On [0, ] we construct a uniform mesh with a step : {

{ = ∈ [0, ] ∶ = , = 0, … , , , = ⧵{0}.

We denote by the set of mesh functions defined on , zero on . Let further is the -dimensional vector with coordinates ◦ are the functions from , that equal = ±1, ∇ ( ) = ( 1 ( ), 2 ( ), … , ( )), ( ) = { ( ), = +1, ̄ ( ), = −1. of mesh functions ◦ introduce the following norms and scalar products Let us denote by ( ) a mesh cell , , which contains all the mesh points participating in the notation of operator ∇ ( ), is the set of points ∈ , at which the operator ∇ ( ) is defined. In the space (, ) = ∑ ̃ ( ) ( ),

[, ] = (1/2 ) ∑(, ) , ∈ ∥ ∥ = [∣ ∣ , 1]1/ ,

∥ ∥+ = (1/2 ) ∑

∑(∣ ∣ , 1) , ∥ ∥− ′ = sup ≠0 ∥ ∥+ [, ] here ̃ = mes ( ).

For mesh functions, we define piecewise constant extensions and each Π ( ) = { ( ′), ′

∈ , ∈ ( ′)}, Π− ( ′) = { ( ), = , ( − 1) < ′ ≤ }, +

′ Π ( ) = { ( ), = ,

≤ ′ < ( + 1) }, + Π

+ = Π Π , Π = Π−Π .

− inequality holds

Lemma 1. (See [7]) If ( ) is an absolutely continuous increasing function, then the following Lemma 2. (See [7]) Let ≥ 2, function satisfies the condition ( 4 ) and besides ( ( ) − ( )) ≥ Φ( ) − Φ( ),

∀ , ∈ 1. ′( ) ≥ 6 ∣ ∣ −2,

Then for any constant > 1 there is ̄ = >

0, such that for any , ∈ 1 the inequality holds fying the conditions( 4 )–( 6 ). Then for any function such that

Lemma 3. (See [7]) Let ( ) be an absolutely continuous, monotonically increasing function satisthe following equality holds

∈ ′(0, ; −′1(Ω)), (, 0) ∈ (Ω) ⋂ (Ω), ∫ ⟨ 0 ( ) , ⟩ = lim →0 ∫ ∫ Φ( ( )) − ∫ Φ( (0)) .

Ω It is easy to check the validity of the following lemma.

Lemma 4. (See [7]) For any ∈ ◦ the inequality holds √

where = ℎ1+ ( − )/ , if ≥ and = ∥ ∥+ ≤ ∥ ∥ , √ ℎ

, if 1 < < . 3. Construction and investigation of an explicit diference scheme For the problem ( 1 ), ( 2 ), consider the explicit diference scheme Here is a diference operator acting from ◦ to ◦ , defined by the relation ( ) + (, ) = ℎ (, ), ∈ ℎ, ∈ ⧵{ }, (, 0) = 0( ),

∣ ℎ= 0. ∇ , ℎ ), ) , where ℎ ( ) = (2− ∑Π ( )), 0 a diference analog of 0 such that ℎ is a mesh function, that is an approximation of the original equation right side, which we define 0 ≡ , ℎ, ( ) = [ ℎ , ] = ∑

∑( ℎ, , ) ∀ ∈ , mes( ( ))∫ +

∫ ( )

◦ ( , ) .

Conditions ( 7 )–(8) on the coeficients provide continuity, boundedness: (22) (23) (24) (25) (26) (27) (28) (29) 1 [ ( ), ̂ − ( − 1) ] + [ ,

̂ − ( − 1) ] = [ ℎ , ̂ − ( − 1) ] ∥ ∥− ′≤ 0 ∥ ∥+−1 + ̄0, [ ,

] ≥ 2 ∥ ∥+ − 3, coercivity of the operator ∶

Lemma 5. Let ≥ 2, function satisfies the conditions ( 4 )–( 5 ) and besides diference scheme (21) follows from the condition that the function is strictly monotonic. with constants 2 > 0, 3 ≥ 0, 0 > 0, ̄0 ≥ 0, independent on ℎ̄ and . The unique solvability of the 0 ∈ (Ω), ∈ (0, ; −′1(Ω)), = max{ ′, ′}.

Then for any for the solution of the diference scheme (21) the following a priori estimates hold ≤ ⎨ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎪ / , ⎪

2 ℎ + ( − )/ ℎ 2

1 < < , ,

≥ , ′ =0 ∑ ∥ ∥+ ≤ , ′∈ ̄ max ∥ ( ′) ∥ ≤ , ′ =0 ∑ ∥ ∥ ≤ Using lemma 2, we estimate the first summand in the left-hand side of the equation ( 30) inequality and a diference analogue of the Friedrichs inequality, as a result we have [ ℎ , ] ≤ 2 [ ℎ , ] ≤ ≤ 1 ′ 1 ′

1 2 ′ ′

1 2 ′ ′ ∑ ∥ ℎ, =0 ∑ ∥ ℎ, =0 ∑ ∥ ℎ, =0 ′ ∥ ′ + ′ ∥ ′ + ′ ∥ ′ + 1 2 +1 2

+1 (1 + Ω) ∥ ∥+ , (∥ ∥+ + ∥ ∥

) ≤ (1 + Ω) ∥ ∥ + 1 , here Ω is the constant from the diference analog of the Friedrichs inequality. From (23) follows that 2 [ , ] ≤ ( 0 ∥ ∥+

+̄0) ∥ ∥+ ≡ + ̄0 ∥ ∥+ .

Further, using (31)–(34) and the coercivity of the operator from (30) is easy to obtain [Φ( ̂ ) − Φ( ), 1] + ̄

∥ ∥ + 2 ∥ ∥+ − 3 ≤ −1 ′ ∥ ′ + ′ ∥ ′ + 1 2 +1

, (1 + Ω) ∥ ∥+ + or (30) (31)

– (32) (33) (34) (35) (36) (37) ∑ ∥ ℎ, ( ) ∥ ′ +[Φ( (0)), 1] + 3. (2 + Ω) ∥ ∥ + + 1 .

Let ≥ .

We estimate

using Hölder’s inequality and lemma 2, as a result we obtain have Substituting (36) into (35) and summing the resulting inequalities over from 0 to ′ ∈ ̄ , we will 2

′ 3 ′ ≤ 1 ′ 1 ′

1 + 2 ′ ′

/ ′ ≤

′ ′ ∑ ∥ ℎ, ( ) ∥ ′ + 2 ′ ′ First, let us prove that (37) implies the estimate ∥ ( ′) ∥ ≤

∑ are constants independent of ℎ̄ and . For ′ = 0 estimate (38) holds. We assume that (38) (38) (39) (40) Choosing 1, 2, 3, ℎ̄ and so that [Φ( ( 1 + )), 1] + ( 2 − + ̄ − ( 1 1 2 (2 + Ω) − ( 0 ) ′ ∑ ∥ ℎ, ( ) ∥ ′ +[Φ( (0)), 1] + 3. ̄ − 2 (2 + Ω) − ( 0 ) ′ 3 ′ 3 − ≥ 1 > 0, ≥ 2 > 0 , over ̄ from 0 to − , as a result we will have and using the condition ( 4 ), of (39) is easy to obtain (38) for ′ = 1 + . Therefore, the estimate (38) will be valid for any ′ ∈ ̄ . From (37) and (38) the estimates (26)–(28) follow. Note that the constant in (25) is chosen so that the inequality (40) holds.

Similarly to the way above, it is easy to verify the validity of estimates(26)–(28) in the case 1 < < .

Let us further prove the validity of the estimate (29).To do this, we sum both sides (21) over from ̄ to ̄ + ( − 1) , then multiply the resulting equality scalarly in by ( (̄ + ) − (̄)) and again sum 1 1 Using the boundedness property of the operator , Hölder’s inequalities and (34), from (41) it is easy ∑ [ ( (̄ + )) − ( (̄)), (̄ + ) − (̄)] ≤ 1 ∑ ∥ (̄) ∥+ + From the last inequality and (26) it follows (29). The lemma is proved. weak compactness of bounded sets in reflexive spaces and the *-weak compactness of bounded sets

}∞ 1 and the element , which belongs Π ⇀ in ( ),

± ± Π ⇀

in ( ),

Using the estimates (27), (28), (30) and the mesh analogue of the compactness theorem (see [7], lemma 9), it is easy to confirm the existence of subsequences (42)– (44) the limit relation of the form below holds ⃗ ℎ { ( )}∞

{ =1, =1

}∞ , for which, along with Π → almost everywhere in .

± ⃗ ℎ { ( )}∞ {

}∞ =1, =1

such that ± Π (, , set { Further, the condition ( 7 ) and the estimate (26) imply the boundedness in the space ′ ( ) of the ∇ , ℎ )} for any ∈ {1, 2, … , }. Therefore, there are ∈ ′ ( ) and sequences For ≤ from (27), (45) and Lebesgue’s theorem on passage to the limit, it is easy to show that

Theorem 1. Let the functions , holds. Let, in addition, for , ℎ̄ → 0 → 0, if ≥ , → 0, if 1 < < .

Then for any function ∈ (0, ; −′1(Ω)), where = max{ ′ , ′}, and 0, ∈ (Ω) ⋂ subsequence of piecewise constant extensions of the solution to the diference scheme (21), defined by the relations (42)–(47), converges to a generalized solution of the problem ( 1 )–( 2 ).

Proof of this theorem is close to the proof of Lemma 3 from ([3]). Therefore, we present here only fragments of reasoning diferent from Lemma 3.

Let’s scalarly multiply the diference scheme (21) by ∞(0, ; 0∞(Ω)), ̄ (, ) = 0 and sum over from 0 to − .

As a result we get , where

– drift of the function ̄ from satisfy conditions ( 7 )–(9), (14), ≥ 2 and the inequality (25) ± Π (, ,

∇ , ℎ ) ⇀ in ′ ( ). Π± ( ) →

in 1(0, ). equality using piecewise constant extensions in the form of the integral identity 1 ∑ 2 {

− − ∫ ∫ Π ( )Π ( ̄ )

− 0 Ω

+ } = 1In what follows, for the selected subsequences we will keep the notation of the sequences themselves. (42) (43) (44) (45) (46) (47) (48) In the equality (49) , we pass to the limit as , ℎ → 0. As a result, we will have Following ([3], lemma 3), from (50) it is easy to obtain that − ∫ ∫ ( ) − ∫ ( 0) ̄ (, 0) + ∑ ∫ ∫ = ∫ ⟨ , ̄ ⟩. =1 − + 1 0 1 2 =1 0 Ω = ∑ ∑ ∫ ∫ Π ℎ,

+ Π

.

0 Ω ∫ ⟨ 0 and, besides, (, 0) = 0( ) almost everywhere in Ω. Let us prove further that for any function ̄ from (0, ; (Ω)). To do this, we consider the following inequality =1 + 1

2 ∑ ∫ ∫ {Φ(Π+ ̂ ) − Φ(Π+ )}. [ ( ̂ ) − ( ), ̂ ] + ∑ [( (, ,

∇, ℎ ) − (, ∇ ̂ , ℎ )), ( − ̂ )] ≥ [Φ( ̂ ) − Φ( ), 1], where function is the solution of the diference scheme (21), (, ) is the drift of the function ̄ (, ) ∈ ∞(0, ; 0∞(Ω)) to the points of the mesh ̄ × ̄ . The validity of (53) follows from (9) and the lemma 1. Considering that the function satisfies equality (21), we rewrite inequality (53) as follows [ ℎ , ̂ ] + [ , ] − ∑[ (, ∇ ̂ , ℎ ), ( − ̂ )] − ∑[ (, ∇, ℎ ), ̂ ] ≥ [Φ( ̂ ) − Φ( ), 1]. inequality over the segment [0, ′], ′ ∈ [0, ]. As a result we will have Using the extension Π+, we write the last inequality for all ∈ [0, ] and integrate the resulting Further, using the [3] methodology, when the condition (48) holds we establish the validity of the limit equality lim ,ℎ →0 =0 with respect to . Using the convexity of the function Φ, we have Let further ∗ be a mesh point , belonging to ( ′, ′ + ], ( ′) = ( ′ + − ∗)/ , Λ − linear extension ′+ ′ Ω = ∫ ( ′)Φ(Π ( ∗)) + (1 − ( ′))Φ(Π ( ∗ − )) ≥ everywhere in . In addition, from the estimate ( 7 ) it follows that Limit relations (45), (47), smoothness of the function and continuity of (, , , the arguments allow us to assert that the integrand function in (58) tends to 0 as ℎ, → 0 almost ) for each of | + ||Π ( (, , ∇ ̂ , ℎ )) − (, , ∇ ̄ , |

′ )| ≤ ( 0 ∑ | =1 { | ̄ || −1 | + || Let us prove further that

≥ ∫ Φ(Π ( ( ′) ( ∗) + (1 − ( ′)) ( ∗ − ))) = ∫ Φ(Λ Π ( ( ′))) .

+ Π ( (, , ∇ ̂ , ℎ )) → (, , The right-hand side of the last inequality, due to the smoothness of is a function integrable over , therefore, by the Lebesgue theorem on the passage to the limit → 0 for , ℎ → 0, it means From the inequalities (54)–(56) it follows that From the relations (42)–(47), (57) it follows that lim ( ′) ≥ ,ℎlim→0 ∫ Φ(Λ Π ( ( ′)) − ∫ Φ( 0( )).

,ℎ →0 lim ( ′) = lim ( ′) = ( ′) ≡ ∫ {⟨ , ⟩− ,ℎ →0 ′ Ω } Considering (51), we will obtain ∫ Φ( ( )).

Therefore (̄) = ∫ ∫ Φ( ( ′)) ′ − ∫ Φ( 0( )), = >

0, we will have Substituting (60), (61) in the inequality (59) and integrating the result over ′ from − to , Assuming in the inequality (64) first ̄ = + , and then ̄ = − , where = > 0, is an arbitrary function from (0, ; (Ω)), it is easy to obtain equality (52). The theorem is proved. (61) (62) (63) ∫ ( ′) ′ ≥ ∫ ,ℎlim→0 ∫ Φ(Λ Π ( ( )) ′ ′ − ∫ Φ( 0( )).

Ω The convexity of the function Φ( ) implies the weak lower semicontinuity on (Ω) of the functional ∫ ,ℎlim→0 ∫ Φ(Λ Π ( ( )) ′ ′ ≥ ∫ ∫ Φ( ( )) ′ ′

Considering (63), we will obtain We transform the left-hand side of inequality (62) using the mean value theorem. The application of this theorem is admissible, since the function ( ′) is absolutely continuous with respect to ′. here ̄ ∈ [ a result we get − , ]. We divide both sides of the last inequality by and pass to the limit as → 0, as ∫ ⟨ 0

The last inequality and lemma 3 imply ∑( − (, , ∇ ̄ , )) ( − ̄ )

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