Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

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Boundary value problems for a quasilinear parabolic equation with an unknown coefficient N.L. Gol’dman Science Research Computer Center, Moscow State University, Moscow, 119 992, Russia Received 17 March 2018; revised 3 July 2018

Abstract The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions. As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed. © 2018 Published by Elsevier Inc.

Keywords: Parabolic equations; Nonlinear problems; A priori estimates for Rothe method; Properties of solutions in Hölder spaces; Mathematical models of thermodestruction

E-mail address: [email protected]. https://doi.org/10.1016/j.jde.2018.10.015 0022-0396/© 2018 Published by Elsevier Inc.

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1. Introduction Some modern technologies both in high temperature fields (e.g., aircraft, astronautics, power engineering, etc.) and in hydrology, exploitation of oil-gas stratums, etc. are characterized by physical–chemical processes in which inner properties of materials are subjected to changes. The goal of the work is to investigate nonlinear parabolic models that arise in the mathematical modeling of such processes. In Section 2, these models are formulated as a system that involves a boundary value problem for a quasilinear parabolic equation with an unknown coefficient at the derivative with respect to time and, moreover, an additional relationship for a time dependence of this coefficient. Justification of mathematical statements for these parabolic models is an important task since such statements essentially differ from usual boundary value problems for parabolic equations (see the well known monographs [1,2]). Therefore, a considerable theoretical interest is to obtain conditions for existence and uniqueness of their smooth solutions. In the work in order to prove solvability, the Rothe method is applied which also provides the approximate solutions of the considered statements for a quasilinear parabolic equation. Thus, the proposed approach can be used for their various applications, especially in high temperature processes, where it is necessary to take into account the dependence of thermophysical characteristics upon the temperature. The proof of the solvability in a class of smooth functions will be split into several stages which are subjects of Sections 3–5. Their main aim is to establish a priori estimates in the corresponding function spaces for solutions of a differential-difference nonlinear system that approximates the original system by the Rothe method. In Section 5, as a result of these estimates, the conditions of unique solvability for the considered statements of the nonlinear boundary value problems are obtained. Moreover, in Section 5 the error estimates for the Rothe method are given. The approach that is proposed in the work allows one to avoid additional assumptions of the smoothness of the input data which have usually been imposed by the Rothe method (see, e.g., [1]). Thus we determine the faithful character of differential relations between the input data and the solution in the chosen function spaces. Section 6 contains examples of the parabolic models that are connected with use of composites in modern heat-protective systems in various technical fields. The presented models describe destruction of the heat-protective material, namely irreversible changes of the density and the concentration of components of the composite under high temperature action. Some our results of the numerical experiments are also discussed. Finally, a short conclusion in Section 7 summarizes the content of this work. The following remarks must be added. In our analysis we use standard definitions for the function spaces from [1]. In particular, the Hölder class H 2+λ,1+λ/2 (Q) (0 < λ < 1) is determined as the space of functions u(x, t) continuous on the closed set Q = {0 ≤ x ≤ l, 0 ≤ t ≤ T } together with their derivatives uxx , ut which satisfy the Hölder condition as functions of x, t with the corresponding exponents λ and λ/2. Moreover for a convenient presentation of the work, the following designations for the function spaces are also used. H 1,λ/2,1 (D) is the space of functions which are continuous for (x, t, u) ∈ D = Q × [−M0 , M0 ] together with their derivatives with respect to x, u and, moreover, satisfy the Hölder condition as functions of t with the exponent λ/2. O 1,0 (Q) is the space of functions continuous on the set Q and having bounded derivatives with respect to x.

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O 1,0,1 (D) is the space of functions continuous on the set D and having bounded derivatives with respect to x and u. O 2,0,2 (D) is the space of functions continuous on the set D together with their derivatives with respect to x, u and having bounded derivatives with respect to xx, xu, and uu. We also use the corresponding analogs of the Hölder classes in the case of the grid functions uˆ = (u0 , . . . , un , . . . , uN ) defined on the grid ωτ = {tn } = {nτ, n = 0, N, τ = T N −1 } and in the case of the grid-continuous functions u(x) ˆ = (u0 (x), . . . , un (x), . . . , uN (x)) defined on the set Qτ = {0 ≤ x ≤ l, tn ∈ ωτ }. In [3] these analogs are determined in the following way. 1+λ/2 Hτ (ωτ ) is the difference analog of the space H 1+λ/2 [0, T ] (see [1]) for the functions uˆ having a finite norm 1+λ/2

|u| ˆ ωτ

λ/2

= max |un | + max |unt | + uˆ t ωτ , 0≤n≤N

1≤n≤N

unt = (un − un−1 )τ −1 , n = 1, N, uˆ t ωτ = λ/2

max

1≤n
{|unt − un t ||tn − tn |−λ/2 }.

λ,λ/2

Hτ (Qτ ) is the difference-continuous analog of the space H λ,λ/2 (Q) (see [1]) for the functions u(x) ˆ continuous in x for (x, tn ) ∈ Qτ and having a finite norm |u(x)| ˆ

λ,λ/2 Qτ

λ u(x) ˆ x,Q

τ

λ/2

u(x) ˆ t,Q

τ

= =

λ max |un (x)| + u(x) ˆ

),(x  ,t

λ/2 , t,Qτ

+ u(x) ˆ

{|un (x) − un (x  )||x − x  |−λ },

sup (x,tn

=

x,Qτ

(x,tn )∈Qτ

n )∈Qτ

sup (x,tn ),(x,tn )∈Qτ

{|un (x) − un (x)||tn − tn |−λ/2 }.

1+λ, 1+λ

1+λ

2 Hτ (Qτ ) is the difference-continuous analog of the space H 1+λ, 2 (Q) (see [1]) for the functions u(x) ˆ continuous in x together with their derivatives with respect to x for (x, tn ) ∈ Qτ and having a finite norm

1+λ, 1+λ 2 Qτ

|u(x)| ˆ

=

λ,λ/2 Qτ

max |un (x)| + |uˆ x (x)|

(x,tn )∈Qτ

+ u(x) ˆ

1+λ 2

t,Qτ

,

where uˆ x (x) = (u0x (x), . . . , unx (x), . . . , uN x (x)). 2+λ,1+λ/2 Hτ (Qτ ) is the difference-continuous analog of the space H 2+λ,1+λ/2 (Q) for the functions u(x) ˆ continuous in x together with their derivatives uˆ xx (x) and uˆ t (x) for (x, tn ) ∈ Qτ and having a finite norm 2+λ,1+λ/2

|u(x)| ˆ Q

τ

=

max |un (x)| +

(x,tn )∈Qτ

λ,λ/2

max |unx (x)| + |uˆ xx (x)|Q

(x,tn )∈Qτ

τ

where uˆ xx (x) = (u0xx (x), . . . , unxx (x), . . . , uN xx (x)), = (u1t (x), . . . , unt (x), . . . , uN t (x)), uˆ t (x) unt (x) = (un (x) − un−1 (x))τ −1 , n = 1, N.

λ,λ/2

+ |uˆ t (x)|Q

τ

,

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2. Statements of the nonlinear boundary value problems with an unknown coefficient We formulate these statements as a system for determining the functions {u(x, t), ρ(x, t)} in the domain Q = {0 ≤ x ≤ l, 0 ≤ t ≤ T } that satisfy the boundary value problem for the quasilinear parabolic equation c(x, t, u)ρ(x, t)ut − (a(x, t, u)ux )x + b(x, t, u)ux + d(x, t, u)u = f (x, t),

(x, t) ∈ Q, (1)

a(x, t, u)ux − h(t, u)u|x=0 = g(t),

0 < t ≤ T,

(2)

a(x, t, u)ux + e(t, u)u|x=l = q(t),

0 < t ≤ T,

(3)

u|t=0 = ϕ(x),

0 ≤ x ≤ l,

(4)

and the additional relationship ρt (x, t) = γ (x, t, u), (x, t) ∈ Q,

ρ(x, t)|t=0 = ρ 0 (x), 0 ≤ x ≤ l,

(5)

where all the input data in equation (1), boundary conditions (2), (3), initial condition (4), and in relationship (5) are the known functions of their arguments; a ≥ amin > 0, c ≥ cmin > 0, ρ 0 ≥ 0 > 0, h ≥ 0, and e ≥ 0, a 0 ρmin min , cmin , ρmin = const > 0. In what follows, we assume that the function γ (x, t, u) is of constant signs for (x, t, u) ∈ D = Q × [−M0 , M0 ] (where M0 ≥ max(x,t)∈Q |u|, M0 is the constant from the maximum principle for boundary value problem (1)–(4)). In order to ensure the parabolic form of equation (1) the sought coefficient ρ(x, t) must satisfy some requirements depending on the sign of γ (x, t, u) for (x, t, u) ∈ D. These requirements have the form 0 0 < ρmin < ρ(x, t) ≤ max ρ 0 (x) + T 0≤x≤l

0 0 < ρmin −T

max γ (x, t, u) for γ (x, t, u) > 0,

max |γ (x, t, u)| ≤ ρ(x, t) ≤ max ρ 0 (x) for γ (x, t, u) ≤ 0.

(x,t,u)∈D

(6)

(x,t,u)∈D

0≤x≤l

(7)

If γ (x, t, u) ≤ 0 in the domain D, then condition (7) leads to the restriction to the time interval [0, T ], where the solution {u(x, t), ρ(x, t)} of system (1)–(5) is sought: 0 < T < 0 (max −1 ρmin (x,t,u)∈D |γ (x, t, u)|) . The represented statements for the quasilinear parabolic equation are especially important in the mathematical modeling of high temperature processes since they allow one to take into account the dependence of thermophysical characteristics upon the temperature. The possibility of defining the smooth solution {u(x, t), ρ(x, t)} of system (1)–(5) is ensured by the choice of the corresponding function spaces for the input data of this system. In what follows, this choice of the Hölder classes is justified both for the statement of the general form (1)–(5) and for a variant of this statement with a = a(x, t), h = 0, e = 0. To this end, in next section we apply the Rothe method and study a differential-difference nonlinear system that approximates the original system.

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3. Differential-difference approximation of system (1)–(5) We approximate system (1)–(5) by using the following discretization procedure of the Rothe method on the uniform grid ωτ = {tn } ∈ [0, T ] with time-step τ = T N −1 : cn ρn unt − (an unx )x + bn unx + dn un = fn ,

(x, tn ) ∈ Qτ = {0 < x < l} × ωτ ,

(8)

an unx − hn un |x=0 = gn ,

0 < tn ≤ T ,

(9)

an unx + en un |x=l = qn ,

0 < tn ≤ T ,

(10)

u0 (x) = ϕ(x), ρnt = γn−1 , (x, tn ) ∈ Qτ ,

0 ≤ x ≤ l,

ρn (x)|n=0 = ρ 0 (x),

(11) 0 ≤ x ≤ l.

(12)

The approximating system can be formulated as follows: Find {un (x), ρn (x)} — approximate values of the functions u(x, t) and ρ(x, t) for t = tn — satisfying conditions (8)–(12) in which an , bn , cn , and dn are the values of the corresponding coefficients at the point (x, tn , un ); fn = f (x, tn ), hn = h(tn , un (0)), en = e(tn , un (l)), gn = g(tn ), qn = q(tn ), and γn−1 = γ (x, tn−1 , un−1 ). In system (8)–(12) the following designations are also used: unt = (un (x) − un−1 (x))τ −1 , unx = dun (x)/dx, ρnt = (ρn (x) − ρn−1 (x))τ −1 . The proof of solvability of system (1)–(5) by the Rothe method involves several principal stages which are subjects of our next analysis. First, we investigate the differential-difference boundary value problem (8)–(11) in the 2+λ,1+λ/2 (Qτ ) under assumption that ρn (x) is the known difference-continuous Hölder space Hτ function. This stage is to prove unique solvability of problem (8)–(11) and to drive the corresponding a priori estimates for the solution un (x) (independent of x, τ , n). The second stage is to obtain a priori estimates for the solution {un (x), ρn (x)} (independent of x, τ , n) of the differential-difference system (8)–(12) by using the results of the first stage. The aim of this stage is to prove the compactness of the set {un (x), ρn (x)} thanks to the obtained estimates. Finally, the third stage involves the passage to the limit as time-step τ goes to 0 (i.e., n → ∞) in conditions (8)–(12). The aim of this last stage is to show that original system (1)–(5) has at least one solution in the corresponding Hölder spaces. If the justification of the Rothe method in these stages must take into account specific properties of system (1)–(5), then we show the proof in details. Otherwise, we only sketch the proof referring to the known results. 4. Differential-difference boundary value problem (8)–(11) At first we assume that ρn (x) in equation (8) is the given function continuous in x together with the derivative ρnx (x) on the domain Qτ and satisfying the Hölder condition as a function of tn with the exponent λ/2. By using this assumption we derive a priori estimates for the solution un (x) of problem (8)–(11). 4.1. A priori boundness of the solution un (x) Begin with finding the constant in the maximum principle for un(x).

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Lemma 1. Assume that for (x, t) ∈ Q and any u, |u| < ∞, the input data of the boundary value problem (1)–(4) are bounded functions of their arguments, where the coefficient a(x, t, u) has the bounded derivatives ax (x, t, u) and au (x, t, u). Moreover, assume that 0 < amin ≤ a ≤ amax , 0 < cmin ≤ c ≤ cmax , 0 < ρmin ≤ ρ ≤ ρmax , h ≥ 0, e ≥ 0. Then the solution un (x) of the approximating problem (8)–(11) in the domain Qτ for any timestep τ sufficiently small satisfies the estimate max |un (x)| ≤ M0 ,

(x,tn )∈Qτ

  l M0 = K2 T exp(K1 T ) + K3 l 1 + , 4

(13)

in which K1 , K2 , and K3 are positive constants, τ ≤ τ0 = εK1−1 , ε > 0 is arbitrary, K1 ≥ −1 −1 ρmin , (1 + ε)dmax cmin K2 K3

≥ ≥

−1 −1 cmin ρmin {fmax + 2K3 amax + K3 l(ax max + K3 lau max + bmax + (1 + 4l )dmax )}, −1 −1 max(l −1 ϕmax , l −1 amin gmax , l −1 amin qmax ).

Proof. We introduce an auxiliary function vn (x) of the form un (x) = vn (x)(1 + K1 τ )n − ψ(x),

(14)

where K1 > 0 is a constant, ψ(x) is a function from C 2 [0, l], their choice will be indicated later. There hold the following relationships unt

(an unx )x

= τ −1 {vn (1 + K1 τ )n − vn−1 (1 + K1 τ )n−1 } = (1 + K1 τ )n−1 {(1 + K1 τ )vnt − K1 τ vnt + K1 vn } = (1 + K1 τ )n−1 vnt + K1 (1 + K1 τ )n−1 vn , = (1 + K1 τ )n (an vnx )x − 2(1 + K1 τ )n anu vnx ψx − (an ψxx + anx ψx − anu ψx2 ).

Taking into account these relationships we consider an operator Lvn of the form Lvn ≡ cn ρn vnt −(1+K1 τ )(an vnx )x +(1+K1 τ )(bn +2anu ψx )vnx +{cn ρn K1 +(1+K1 τ )dn }vn . Thanks to (8)–(11) the function vn (x) satisfies the following differential-difference boundary value problem Lvn = (1 + K1 τ )−(n−1) Fn (x),

(x, tn ) ∈ Qτ ,

(15)

an vnx − hn vn |x=0 = (1 + K1 τ )−n (gn + an ψx − hn ψ)|x=0 ,

0 < tn ≤ T ,

(16)

an vnx + en vn |x=l = (1 + K1 τ )−n (qn + an ψx + en ψ)|x=l ,

0 < tn ≤ T ,

(17)

v0 (x) = ϕ(x) + ψ(x),

0 ≤ x ≤ l,

(18)

where Fn (x) = fn (x) −an (x)ψxx (x) −anx (x)ψx (x) +anu (x)ψx2 (x) +bn (x)ψx (x) +dn (x)ψ(x).

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Now we introduce another auxiliary function wn (x) wn (x) = vn (x) − K2 τ n,

(19)

−1 −1 where K2 is a positive constant, K2 ≥ cmin ρmin max(x,tn )∈Qτ |Fn (x)|. It easily follows from (15) that

Lwn = (1 + K1 τ )−(n−1) Fn − cn ρn K2 − {cn ρn K1 + (1 + K1 τ )dn }K2 τ n. −1 −1 Let us choose the constant K1 (see (14)) from the condition K1 ≥ (1 + ε)dmax cmin ρmin with −1 arbitrary ε > 0. Then for any τ sufficiently small (τ ≤ τ0 = εK1 ) there holds

Lwn (x) < 0,

(x, tn ) ∈ Qτ .

(20)

Moreover, by (16)–(18) the function wn (x) satisfies the relations an wnx − hn wn |x=0 = hn |x=0 K2 τ n + (1 + K1 τ )−n (gn + an ψx − hn ψ)|x=0 ,

0 < tn ≤ T , (21)

an wnx + en wn |x=l = −en |x=l K2 τ n + (1 + K1 τ )−n (qn + an ψx + en ψ)|x=l ,

0 < tn ≤ T , (22)

w0 (x) = ϕ(x) + ψ(x),

0 ≤ x ≤ l.

(23)

Note that wn (x) ≤ 0 for all inner points (x, tn ) ∈ Qτ . In fact, if we use a contradiction argument and assume that wn (x) > 0 in Qτ , then the following relationships hold at the positive maximum point (x ∗ , tn∗ ) ∈ Qτ wn∗ t (x)|x=x ∗ ≥ 0, (an∗ (x)wn∗ xx (x))|x=x ∗ ≤ 0, wn∗ x (x)|x=x ∗ = 0, Lwn∗ (x)|x=x ∗ ≥ 0. This contradicts inequality (20). Now we choose the function ψ(x) (see (14)) in the following way   l 2 ψ(x) = −K3 x − − K3 l, 2

(24)

−1 −1 where K3 > 0 is a constant, K3 ≥ max(l −1 ϕmax , l −1 amin gmax , l −1 amin qmax ). Thanks to such a choice of ψ(x) the function wn (x) ≤ 0 not only for the inner points (x, tn ) but also at the boundary of the domain Qτ . In fact, if we assume, for example, that wn (x)|x=0 > 0, then the derivative wnx (x)|x=0 < 0 since wn (x) ≤ 0 for 0 < x < l. Hence the left-hand side of boundary condition (21) is negative. But thanks to the choice of the constant K3 , the right-hand side of (21) satisfies the relationship

hn |x=0 K2 τ n + (1 + K1 τ )−n (gn + an ψx − hn ψ)|x=0 ≥ 0,

n = 1, N.

Thus, our assumption leads to contradiction. By repeating the similar arguments for boundary condition (22) we show that wn (x)|x=l ≤ 0. At last, initial condition (23) and the choice of the constant K3 in (24) allow one to conclude that for n = 0 there holds w0 (x) ≤ 0, 0 ≤ x ≤ l. Thus, the function wn (x) ≤ 0 everywhere in the domain Qτ . Thanks to (14) and (19) this leads to the following upper bound for un (x)

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  l un (x) ≤ (1 + K1 τ )n K2 τ n + max |ψ(x)| ≤ K2 T exp(K1 T ) + K3 l 1 + . 0≤x≤l 4 The similar arguments for the auxiliary functions wn (x) and ψ(x) of the form wn (x) = vn (x) + K2 τ n,

  l 2 ψ(x) = K3 x − + K3 l, 2

lead to the lower bound for un (x) un (x) = ≥

(1 + K1 τ )n (wn (x) − K2 τ n) − ψ(x) ≥ −(1 + K1 τ )n K2 τ n − max |ψ(x)| 0≤x≤l   −K2 T exp(K1 T ) − K3 l 1 + 4l .

Lemma 1 is completely proved. 2 4.2. A priori boundness of the derivative unx (x) Our next analysis is connected with obtaining a priori estimates for unx (x) on the closed domain Qτ . 4.2.1. Case of the input data of the alternative form At first we consider the original boundary value problem (1)–(4) under the assumption that a = a(x, t), h(t, u) = 0, and e(t, u) = 0. Lemma 2. Let for (x, t) ∈ Q and any u, |u| < ∞, the coefficients and the right-hand side of equation (1) be bounded functions of their arguments. Moreover, assume that the coefficient a(x, t) has the bounded derivatives ax (x, t), at (x, t) for (x, t) ∈ Q and the following conditions hold 0 < amin ≤ a ≤ amax , 0 < cmin ≤ c ≤ cmax , 0 < ρmin ≤ ρ ≤ ρmax , h = 0, e = 0. In addition, let the functions ϕ(x), g(t), and q(t) be, respectively, in O 1 [0, l] and O 1 [0, T ], a(x, 0)ϕx |x=0 = g(0), a(x, 0)ϕx |x=l = q(0). Then for any time-step τ ≤ τ0 (τ0 > 0 is the constant defined by Lemma 1) the derivative unx (x) of the solution to the differential-difference boundary value problem (8)–(11) satisfies the estimate max |unx (x)| ≤ M1 ,

(25)

(x,tn )∈Qτ

in which M1 is a positive constant independent of x, τ , and n. Proof. We carry out a substitution vn (x) = un (x) − x 2 ψnl + (x − l)2 ψn0 , ψn0 = gn (2lan |x=0 )−1 ,

(x, tn ) ∈ Qτ ,

ψnl = qn (2lan |x=l )−1 ,

n = 1, N,

(26)

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that reduces the boundary conditions at x = 0 and x = l to homogeneous ones vnx (x) = unx (x) − 2xψnl + 2(x − l)ψn0 ,

vnx (x)|x=0 = 0,

vnx (x)|x=l = 0.

Next we use even extension of the function vn (x) into domains Q− τ = {−l < x < 0} × ωτ and Q+ τ = {l < x < 2l} × ωτ :  vn (x) =

un (−x) − x 2 ψnl + (x + l)2 ψn0 , (x, tn ) ∈ Q− τ , un (2l − x) + (l − x)2 ψn0 − (2l − x)2 ψnl , (x, tn ) ∈ Q+ τ .

Such a function (we remain the preceding notation vn (x) for it) has a continuous derivative + vnx (x) for all (x, tn ) ∈ Q0τ = Q− τ ∪ Qτ ∪ Qτ , moreover, vnx (x)|x=−l = 0, vnx (x)|x=2l = 0, n = 0, N . There holds the following relationship cn ρn vnt − (an vnx )x + bn vnx + dn vn = Fn0 (x),

(x, tn ) ∈ Q0τ , x = 0, x = l,

(27)

+ in which the functions an , bn , cn , dn , ρn , and fn are extended into the domains Q− τ and Qτ in the even way. The right-hand side of equation (27) has the form

⎧ ⎨ Fn (x), (x, tn ) ∈ Qτ , Fn0 (x) = Fn− (x), (x, tn ) ∈ Q− τ , ⎩ + Fn (x), (x, tn ) ∈ Q+ τ ,

(28)

in which Fn (x) = Fn− (x) = Fn+ (x) =

  fn (x) + 2an (ψnl − ψn0 ) + 2(anx − bn ) xψnl − (x − l)ψn0

   l − (x − l)2 ψ 0 − d x 2 ψ l − (x − l)2 ψ 0 , (x, t ) ∈ Q , −cn ρn x 2 ψnt n n τ n n nt   l l 0 0 fn (x) + 2an (ψn − ψn ) + 2(anx − bn ) xψn − (x + l)ψn

   l − (x + l)2 ψ 0 − d x 2 ψ l − (x + l)2 ψ 0 , (x, t ) ∈ Q− , −cn ρn x 2 ψnt n n n n τ nt   fn (x) + 2an (ψnl − ψn0 ) + 2(anx − bn ) (x − 2l)ψnl − (x − l)ψn0

   l − (x − l)2 ψ 0 − d (2l − x)2 ψ l − (x − l)2 ψ 0 , −cn ρn (2l − x)2 ψnt n n n nt (x, tn ) ∈ Q+ τ .

Our next step is to estimate max(x,t )∈Q0 |vnx (x)|. The known technique [4] allows one to derive n τ bounds for derivatives of solutions to boundary value problems of the first kind for parabolic equations. We apply the discrete analog of this technique modifying it in the corresponding way for differential-difference equation (27) in the domain Q0τ with the boundary conditions vnx (x)|x=−l = 0, vnx (x)|x=2l = 0, n = 0, N . Indeed, we introduce an additional space variable z and consider the function Wn (x, z) = vn (x) − vn (z) in the domain τ = {(x, z, tn ) : −l ≤ z < x ≤ 2l, 0 ≤ tn ≤ T }. Our aim is to show that for all (x, z, tn ) ∈ τ there holds |Wn (x, z)| ≤ M 1 |x − z| (M 1 = const > 0). This inequality leads to the estimate for vnx and, consequently, to the desired estimate (25) for unx thanks to (26).

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First of all we note that Wnx = vnx (x), Wnz = −vnz (z), Wnxx = vnxx (x), Wnzz = −vnzz (z), i.e., the function Wn (x, z) satisfies the following relationship for (x, z, tn ) ∈ τ , x, z = 0, x, z = l (see (27)): Wnt − (cn (x)ρn (x))−1 {an (x)Wnxx − (bn (x) − anx )Wnx } − (cn (z)ρn (z))−1 {an (z)Wnzz − (bn (z) − anz )Wnz }

= (cn (x)ρn (x))−1 F 0 (x) − dn (x)vn (x)

− (cn (z)ρn (z))−1 F 0 (z) − dn (z)vn (z) .

(29)

Consider auxiliary functions in τ wn± (x, z) = η2 (x, z) {exp(±K4 Wn (x, z)) − 1} ,

(30)

where K4 > 0 is a positive constant chosen later, η(x, z) = ζ (x)ζ (z), ζ (x) is a test function for the interval [−l, 2l], belongs to C 2 [−l, 2l] and has the form ⎧ ⎨ 0, −l ≤ x ≤ −l + σ2 , ζ (x) = 1, −l + σ ≤ x ≤ 2l − σ, ⎩ 0, 2l − σ2 ≤ x ≤ 2l,

(31)

in which σ > 0 is a sufficiently small value. It is easily seen that wn± |x=z = 0, wn± |z=−l = 0, wn± |x=2l = 0 for n = 0, N . Now we introduce the operator Lwn± (x, z) of the form ± ± ± Lwn± (x, z) ≡ Cn wnt − An (x)wnxx − An (z)wnzz ,

in which An (x) Cn

(cn (x)ρn (x))−1 an (x) exp (∓K4 Wn ) , 1 −1 = exp (±K4 {θ Wn + (1 − θ) Wn−1 }) dθ .

=

0

Expressions for the derivatives of the functions wn± can be obtained from (30), in particular, ± wnx ± wnxx

= =

± wnt

=

±K4 Wnx η2 (x, z) exp (±K4 Wn ) + 2ηηx {exp (±K4 Wn ) − 1} ,

2 2 2 K4 Wnx η (x, z) ± K4 Wnxx η2 (x, z) ± 4ηηx (x, z)K4 Wnx exp (±K4 Wn )

+ 2ηx2 (x, z) + 2ηηxx (x, z) {exp (±K4 Wn ) − 1} , 1   ±K4 Wnt η2 (x, z) exp ± K4 {θ Wn + (1 − θ )Wn−1 } dθ. 0

Taking into account these expressions and equation (29) for Wn (x, z) we conclude that for (x, z, tn ) ∈ τ , x, z = 0, x, z = l

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2 Lwn± = (cn (x)ρn (x))−1 −K42 η2 (x, z)an (x)Wnx ± K4 η2 (x, z) (anx (x) − bn (x)) Wnx

2 ∓ 4K4 η(x, z)ηx an (x)Wnx + (cn (z)ρn (z))−1 −K42 η2 (x, z)an (z)Wnz

± K4 η2 (x, z) (anz (z) − bn (z)) Wnz ∓ 4K4 η(x, z)ηz an (z)Wnz + Fn (x) + Fn (z).

(32)

The functions Fn (x) and Fn (z) in the right-hand side of equation (32) have the form Fn (x) = Fn (z)

=

  (cn (x)ρn (x))−1 ±K4 η2 (x, z) Fn0 (x) − dn (x)vn (x)  2 

−2 ηx + η(x, z)ηxx an (x) {1 − exp (∓K4 Wn )} ,   (cn (z)ρn (z))−1 ∓K4 η2 (x, z) Fn0 (z) − dn (z)vn (z)  

−2 ηz2 + η(x, z)ηzz an (z) {1 − exp (∓K4 Wn )} .

Thanks to Cauchy inequality ’with ε’ (ab ≤ 2ε a 2 +

1 2 2ε b ,

ε > 0 is arbitrary) one can show that

1 2 ±K4 η2 (x, z) (anx (x) − bn (x)) Wnx ≤ K4 η2 (x, z)Wnx + K4 η2 (x, z) (anx (x) − bn (x))2 , 4 2 2 ∓4K4 η(x, z)ηx (x, z)an (x)Wnx ≤ K4 η (x, z)Wnx + 4K4 ηx2 (x, z)an2 (x). The corresponding terms with Wnz in the right-hand side of (32) are similarly processed. Then −1 we infer that setting K4 ≥ 2amin 2 K4 η2 (x, z) (−K4 an (x) + 2) Wnx ≤ 0, 2 K4 η2 (x, z) (−K4 an (z) + 2) Wnz ≤ 0.

This allows one to state the following inequality Lwn± (x, z) ≤ K5 ,

(x, z, tn ) ∈ τ , x, z = 0, x, z = l,

(33)

where K5 is a positive constant depending, in particular, on max(x,z,tn )∈ τ |Fn |, amax , ax max , and bmax . It is not difficulty to see that the functions Fn (x) and Fn (z) are uniformly bounded in the domain τ . Including, this assertion is a consequence of definition (28) for Fn0 . Moreover, it is obvious that the bound for |Fn | depends on the bounds of the input data in (1)–(4) and also on  thanks to (26). We especially note the dependence of this bound on

bound (13) 0 l maxtn ∈ωτ |ψnt |, |ψnt | , i.e., on at max , gt max , and qt max (see (26), (28) and definitions of Fn , Fn− , Fn+ ). Now we use auxiliary functions in τ

vn± (x, z) = wn± (x, z) + K6 {exp(z − x) − 1},

K6 = const > 0,

(34)

and show that the corresponding choice of K6 ensures the inequality vn± (x, z) < 0 everywhere in

τ except for x = z, where vn± |x=z = 0. First of all, from (30) and (33) it easily follows that for (x, z, tn ) ∈ τ , x, z = 0, x, z = l, Lvn± (x, z) = Lwn± (x, z) − K6 (An (x) + An (z)) exp(z − x).

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Hence there holds Lvn± (x, z) < 0,

−l < z < x < 2l, 0 < tn ≤ T

(35)

  −1 for K6 > K7 = K5 amin cmax ρmax exp 2K4 M 0 + 3l with M 0 = max(x,t )∈Q0 |vn (x)|. n τ Moreover, it follows from (30), (31), and (34) that vn± |x=z = 0,

vn± |z=−l,z =x < 0,

vn± |x=2l,x =z < 0,

n = 0, N.

(36)

Then we consider the functions vn± (x, z) for n = 0 and choose the constant K6 in order to ensure the following inequalities ± v0x (x, z) < 0 for − l < z < x,

± v0z (x, z) > 0 for z < x < 2l.

(37)

± ± To do so we consider expressions for v0x (x, z) and v0z (x, z) ± v0x (x, z) = ±W0x η2 (x, z) exp (±K4 W0 )+2η(x, z)ηx (x, z) {exp (±K4 W0 ) − 1}−K6 exp(z−x), ± v0z (x, z) = ±W0z η2 (x, z) exp (±K4 W0 )+2η(x, z)ηz (x, z) {exp (±K4 W0 ) − 1}+K6 exp(z −x),

and define K6 from the condition  K6 > K8 =

   max |W0x | + 4 max |ηx |M 0 exp 2K4 M 0 + 3l ,

0≤x≤l

0≤x≤l

in which max0≤x≤l |W0x | > 0 is a constant depending on ϕx max , gmax , qmax , and amin (see (26)). Such a choice of K6 allows one to conclude that inequalities (37) are valid. This together with conditions (36) for n = 0 leads to the desired assertion v0± (x, z) < 0 for −l ≤ z < x ≤ 2l, t0 = 0. Now we assume that the inequality vj± (x, z) < 0 for −l ≤ z < x ≤ 2l is established for each time layer t = tj , j = 1, n − 1. In order to prove it for j = n we use the maximum principle taking into account conditions (35) and (36). In fact, properties of uniformly elliptic equations in boundary extreme points allow one to conclude that vn±(x, z) < 0 everywhere for −l ≤ z < x ≤ 2l including x = 0, l, z = 0, l. The obtained inequality and definitions (30), (34) of the functions vn± (x, z) and wn± (x, z) supply the estimate for Wn (x, z) = vn (x) − vn (z) |Wn (x, z)| ≤ M 1 |x − z|,

M 1 = K4−1 K6 exp(2K4 M 0 ),

K6 ≥ max(K7 , K8 ).

We have already noted that this estimate leads to the desired estimate (25) for the derivative unx with the constant M1 depending on M 1 , gmax , qmax , and amin . Lemma 2 is proved. 2 Remark 1. The method of Lemma 2 for estimating unx allows one to avoid differentiating equation (8) with respect to x. Hence this method does not require additional smoothness of the input data of problem (1)–(4) if a = a(x, t), h = 0, and e = 0. But we can not apply the proposed method for problem (1)–(4) of the general form, i.e., for a = a(x, t, u), h(t, u) > 0, and e(t, u) > 0. In this case reduction of the boundary conditions (9) and (10) to homogeneous ones leads to the functions ψn0 and ψnl of the form (compare with (26))

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ψn0 = (gn + hn un |x=0 )(2lan |x=0 )−1 ,

13

ψnl = (qn − en un |x=l )(2lan |x=l )−1 .

0 and ψ l with unknown Hence the right-hand side Fn0 (x) of equation (27) contains the terms ψnt nt estimates for unt |x=0 and unt |x=l .

4.2.2. Case of the input data of the general form For the boundary value problem (1)–(4) under the assumption that a = a(x, t, u), h(t, u) ≥ 0, and e(t, u) ≥ 0 the corresponding estimate for unx in Q is supplied by the next lemma. Lemma 3. Assume that the input data of problem (1)–(4) satisfy, in addition to the hypotheses of Lemma 1, the following conditions: for (x, t, u) ∈ D = Q × [−M0 , M0 ] (M0 > 0 is the constant defined by (13)) a(x, t, u) is in O 2,0,2 (D), b(x, t, u), c(x, t, u), and d(x, t, u) are in O 1,0,1 (D), the functions ρ(x, t) and f (x, t) are bounded in Q together with their derivatives with respect to x. Moreover, let the initial function ϕ(x) be in O 1 [0, l] and the matching conditions be fulfilled at t = 0: a(x, 0, ϕ)ϕx |x=0 − h(0, ϕ)ϕ|x=0 = g(0),

a(x, 0, ϕ)ϕx |x=l + e(0, ϕ)ϕx |x=l = q(0).

Then for any time-step τ ≤ τ0 (τ0 > 0 is the constant defined by Lemma 1) the derivative unx (x) of the solution to the differential-difference boundary value problem (8)–(11) satisfies the estimate max |unx (x)| ≤ M1 ,

(38)

(x,tn )∈Qτ

in which M1 is a positive constant independent of x, τ , and n. Proof. In order to derive estimate (38) we use the known method and therefore only sketch the proof, for details see, e.g., [1]. Namely, we apply a nonlinear substitution un (x) = (vn (x)), where (v) is a function bounded together with the derivatives v (v), vv (v), vvv (v) and satisfying the conditions v (v) > 0, vv (v) < 0, and vvv (v) < 0. Thanks to this substitution each function un ∈ [−M0 , M0 ] is one-to-one correspondence of the new function vn ∈ (−1, 1). Then we differentiate equation (8) with respect to x and replace the derivatives unx (x), unxx (x), unxxx (x), and unxt (x) by the corresponding expressions for vnx (x), vnxx (x), vnxxx (x), and vnxt (x), in particular, unx (x) = v vnx (x),

2 unxx (x) = v vnxx (x) + vv vnx (x),

3 (x). unxxx (x) = v vnxxx (x) + 3vv vnx (x)vnxx (x) + vvv vnx

It is not difficult to see that for the function wn (x) = vnx (x) the following relationship can be obtained v wnt − (cn ρn )−1 an v wnxx + Cn0 wnx wn + Cn1 wnx + Cn2 wn3 + Cn3 wn2 + Cn4 wn + Cn5 = 0,

(x, tn ) ∈ Qτ ,

(39)

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in which all the coefficients (in particular, Cni , i = 0, 5) are uniformly bounded in domains of their definition thanks to assumptions of Lemma 3 and the properties of (v). In order to complete the proof it remains to construct such a function (v) which allows one to apply the maximum principle to the boundary value problem of the first kind for wn (x). In fact, the desired estimate (38) is an obvious consequence of the bound max(x,tn )∈Qτ |wn (x)|. 2 λ,λ/2

4.3. A priori estimates in Hτ

for un (x) and unx (x)

The next step in proving solvability of problem (8)–(11) is to derive a priori estimates of the Hölder norms for un (x) and unx (x). The estimates that are proved above allow us to pass to this step. To do so we consider some special classes B2τ that are embedded in the differenceλ,λ/2 continuous Hölder spaces Hτ . These classes introduced in [3] as discrete analogs of the known classes B2 (see [1]) are defined as follows. B2τ (Qτ , M, ν, , δ) is the set of the functions wn (x) that belong to W21 [0, l] for any given node tn of the grid ωτ ∈ [0, T ], have vraimax(x,tn )∈Qτ |wn (x)| ≤ M and satisfy the inequalities ⎛ ⎜ ⎝

⎞



⎟ (wn − k) ζ (x) dx ⎠ + ν

 2 2 wnx ζ (x) dx ≤

2 2

Ak,r (tn )

≤

⎧ ⎪ ⎨ 

⎫ ⎪ ⎬ (wn − k)2 ζx2 dx + measAk,r (tn ) ⎪ ⎭ t

⎪ ⎩

Ak,r (tn )

Ak,r (tn )



for all k such that k ≥ max maxx∈Kr ∩[0,l] wn (x) − δ, wn (0), wn (l) , and respectively, ⎛ ⎜ ⎝

⎞



⎟ (wn − k) ζ (x) dx ⎠ + ν



2 2

Bk,r (tn )

≤

⎧ ⎪ ⎨  ⎪ ⎩

Bk,r (tn )

2 2 wnx ζ (x) dx ≤

⎫ ⎪ ⎬ (wn − k)2 ζx2 dx + measBk,r (tn ) ⎪ ⎭ t

Bk,r (tn )



for all k such that k ≤ min minx∈Kr ∩[0,l] wn (x) + δ, wn (0), wn (l) . Here ζ (x) is a test function for the interval Kr = {x : |x − x0 | ≤ r}, 0 < r < l, x0 is any point on the segment [0, l], Ak,r (tn ) = {x : x ∈ Kr ∩ [0, l], wn (x) > k}, Bk,r (tn ) = {x : x ∈ Kr ∩ [0, l], wn (x) < k}, the parameters ν,  , and δ are positive constants independent of τ and n. λ,λ/2 Embedding of B2τ (Qτ , M, ν, , δ) in the difference-continuous Hölder space Hτ means that for any function wn (x) of this set and for any rectangle Qστ 1 ,σ2 ,σ3 = {σ1 ≤ x ≤ l − σ2 , σ3 ≤ tn ≤ T },

σi > 0,

i = 1, 2, 3,

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λ,λ/2

there holds |w(x)| ˆ σ ,σ ,σ ≤ K, where the constants K > 0 and λ > 0 depend only on the paQτ 1 2 3 rameters determining the class B2τ . For σi = 0 this estimate depends, moreover, on the Hölder norms of the function wn (x) for x = 0, x = l and n = 0. The proof of these claims is similar to that of the corresponding claims relative to the classes B2 in [1]. In [3] it is shown for differential-difference equations of the form unt − (An (x, un )unx )x + Bn (x, un , unx ) = 0,

(x, tn ) ∈ Qτ ,

that their solutions un (x) with max(x,tn )∈Qτ |un (x)| ≤ M belong to the class B2τ (Qτ , M, ν, , δ). The parameters ν,  , and δ depend only on values ν1 > 0 and μ1 > 0, where ν1 ≤ An (x, u) ≤ μ1 , |Bn (x, u, p)| ≤ μ1 (1 + |p|2 ) for (x, tn ) ∈ Qτ and arbitrary u, p. If, moreover, the solutions un (x) have the bounded derivatives unx (x) (namely, max(x,tn )∈Qτ |unx (x)| ≤ M1 , M1 = const > 0) then unx (x) belong to the class B2τ (Qτ , M1 , ν, , ∞). The parameters ν and  depend only on values M, M1 , ν1 , μ1 , and μ2 , where |Anx (x, u), Anu (x, u), Bn (x, u, p)| ≤ μ2 , μ2 = const > 0, (x, tn ) ∈ Qτ , |u| ≤ M, |p| ≤ M1 . It is not difficult to see that these assertions are valid for the functions un (x) and their derivatives unx (x) which solve the differential-difference boundary value problem (8)–(11). Indeed, equation (8) can be reduced to the corresponding form in which  −1 An = an cn ρn ,     Bn = an cn−1 ρn−1 x unx + an cn−1 ρn−1 u u2nx −1

 (bn − anx )unx − anu u2nx + dn un − fn . + cn ρ n These reasonings allow one to estimate the Hölder norms for un (x) and unx (x) in the case of the input data of the alternative form a = a(x, t), h(t, u) = 0, and e(t, u) = 0 by taking into account Lemmas 1, 2. Lemma 4. Assume that the conditions of Lemma 2 hold and, moreover, the functions c(x, t, u) and ρ(x, t) are in O 1,0,1 (D) and O 1,0 (Q), respectively. σ ,σ ,σ Then for any rectangle Qτ 1 2 3 = {σ1 ≤ x ≤ l − σ2 , σ3 ≤ tn ≤ T } ∈ Qτ the solution un (x) of problem (8)–(11) and its derivative unx (x) satisfy the bounds λ,λ/2 σ ,σ ,σ Qτ 1 2 3

|u(x)| ˆ

≤ M2 ,

λ,λ/2 σ ,σ ,σ Qτ 1 2 3

|uˆ x (x)|

≤ M3 ,

(40)

in which the positive constants M2 > 0, M3 > 0, and λ > 0 depend only on the corresponding parameters of the classes B2τ and the values of σi for σi > 0 (i = 1, 2, 3). Remark 2. If σi = 0 then the values of M2 , M3 , and λ depend, in addition, on the corresponding initial and boundary functions. In particularly, if σ3 = 0, then M2 , M3 , and λ depend on the Hölder norms |ϕ(x)|ε[0,l] and |ϕx (x)|ε[0,l] with ε > 0. If σ1 = 0 and σ2 = 0, there exists the   dependence of M3 on the bounds max0≤t≤T |gt (t)|, |qt (t)| . This follows from the boundary conditions a(x, t)ux (x)|x=0 = g(t), a(x, t)ux (x)|x=l = q(t).

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For problem (1)–(4) of the general form, i.e., for a = a(x, t, u), h(t, u) > 0, and e(t, u) > 0 λ,λ/2 for un (x) and unx (x) are valid under the assumptions of analogous estimates (40) in Hτ Lemma 3. The corresponding rectangles Qστ 1 ,σ2 ,σ3 are considered for σ1 > 0, σ2 > 0, and σ3 ≥ 0, where for σ3 = 0 there exists the dependence of M2 , M3 , and λ on the Hölder norm |ϕ(x)|1+ε [0,l] . 4.4. Unique solvability of the differential-difference boundary value problem (8)–(11) A priori estimates derived by Lemmas 1–4 allow one to prove the existence and uniqueness 2+λ,1+λ/2 for problem (8)–(11). At first we formulate the corretheorems in the Hölder classes Hτ sponding claim for the input data of the alternative form a = a(x, t), h(t, u) = 0, and e(t, u) = 0. Theorem 1. Let the following conditions be satisfied. 1. For (x, t) ∈ Q and any u, |u| < ∞, the input data of problem (1)–(4) are uniformly bounded functions of their arguments, where the coefficient a(x, t) — together with the derivatives ax (x, t) and at (x, t); moreover, there hold 0 < amin ≤ a(x, t) ≤ amax , 0 < cmin ≤ c(x, t, u) ≤ cmax , 0 < ρmin ≤ ρ(x, t) ≤ ρmax , h(t, u) = 0, e(t, u) = 0. 2. For (x, t) ∈ Q the functions ax (x, t) and f (x, t) are in H λ,λ/2 (Q), the coefficient ρ(x, t) is in H 1,λ/2 (Q). 3. For (x, t, u) ∈ D = Q × [−M0 , M0 ] (M0 > 0 is the constant defined by (13)) the functions b(x, t, u) and d(x, t, u) are in H λ,λ/2 (Q) with respect to x and t and have the uniformly bounded derivatives with respect to u; the function c(x, t, u) is in H 1,λ/2,1 (D). 4. The functions ϕ(x), g(t), and q(t) are in H 2+λ [0, l] and O 1 [0, T ], respectively; there hold the matching conditions a(x, 0)ϕx |x=0 = g(0), a(x, 0)ϕx |x=l = q(0). Then for any time-step τ ≤ τ0 (τ0 > 0 is the constant defined by Lemma 1) there exists one and only one the solution to the differential-difference boundary value problem (8)–(11) in the 2+λ,1+λ/2 (Qτ ) which satisfies the estimate class Hτ 2+λ,1+λ/2

|u(x)| ˆ Q

τ

≤ M4 ,

(41)

where M4 is a positive constant independent of x, τ , and n. The next theorem formulates the conditions of unique solvability for problem (8)–(11) with the input data of the general form a = a(x, t, u), h(t, u) ≥ 0, and e(t, u) ≥ 0. Theorem 2. Let the following assumptions be satisfied. 1. For (x, t) ∈ Q and any u, |u| < ∞, the input data of problem (1)–(4) are uniformly bounded functions in their definition domains, where the coefficient a(x, t, u) has the uniformly bounded derivatives ax (x, t, u) and au (x, t, u); moreover, there hold 0 < amin ≤ a(x, t, u) ≤ amax , 0 < cmin ≤ c(x, t, u) ≤ cmax ,

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0 < ρmin ≤ ρ(x, t) ≤ ρmax , h(t, u) ≥ 0, e(t, u) ≥ 0. 2. For (x, t) ∈ Q the functions ρ(x, t) and f (x, t) have the uniformly bounded derivatives with respect to x and Hölder continuous in t with the exponent λ/2. 3. For (x, t, u) ∈ D = Q × [−M0 , M0 ] (M0 > 0 is the constant defined by (13)) the functions ax (x, t, u), au (x, t, u), b(x, t, u), c(x, t, u), and d(x, t, u) are in H 1,λ/2,1 (D), the functions e(t, u) and h(t, u) have the uniformly bounded derivatives with respect to t and u. 4. The functions ϕ(x), g(t), and q(t) are in H 2+λ [0, l] and O 1 [0, T ], respectively; there hold the matching conditions a(x, 0, ϕ)ϕx |x=0 − h(0, ϕ)ϕ|x=0 = g(0), a(x, 0, ϕ)ϕx |x=l + e(0, ϕ)ϕx |x=l = q(0). Then for any time-step τ ≤ τ0 (τ0 > 0 is the constant defined by Lemma 1) the differential2+λ,1+λ/2 difference boundary value problem (8)–(11) has a solution un (x) in the class Hτ (Qτ ) which satisfies the bound 2+λ,1+λ/2

|u(x)| ˆ Qτ

≤ M5 ,

(42)

where the constant M5 > 0 does not depend on x, τ , and n. This solution is unique under the additional condition hu max M0 + au max M1 ≤ hmin , eu max M0 + au max M1 ≤ emin ,

(43)

where M1 > 0 is the constant determined by Lemma 3. The proof of Theorems 1, 2 is supplied by the Leray–Schauder principle on the existence of the fixed points of the completely continuous transforms (the formulations of this principle can be found, e.g., in [1,5]). Such a technique requires, besides the estimates of Lemmas 1–4, a priori 2+λ,1+λ/2 estimates in Hτ for solutions of a linear differential-difference boundary value problem Ln un ≡ unt − An (x)unxx + Bn (x)unx + Cn (x)un = Fn (x), unx |x=0 = gn ,

unx |x=l = qn ,

0 < tn ≤ T ,

(x, tn ) ∈ Qτ ,

u0 (x) = ϕ(x), 0 ≤ x ≤ l.

λ,λ/2

If An (x), Bn (x), Cn (x), and Fn (x) are in Hτ and, moreover, ϕ(x), gn , and qn are in H 2+λ [0, l] and Oτ1 [0, T ], respectively, then these bounds have the following form (see [3]) 2+λ,1+λ/2

|u(x)| ˆ Q

τ

λ,λ/2 2+λ 1 1 ˆ , ≤ K |F(x)| + |ϕ(x)| + | g| ˆ + | q| ˆ ω ω [0,l] τ τ Q

(44)

τ

 λ,λ/2 ˆ where the constant K > 0 is determined by the values of λ (0 < λ < 1), max |A(x)| , Qτ  λ,λ/2 λ,λ/2 ˆ ˆ |B(x)| and does not depend on x, τ , and n. , |C(x)| Qτ Qτ We discuss the main steps in achieving the proof of Theorem 2. Similar propositions hold for the proof of Theorem 1 with the corresponding simplifications.

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1+λ, 1+λ 2

Proof of Theorem 2. For any function ωn (x) ∈ Hτ 

1+λ, 1+λ 2

 = ωn (x) ∈ Hτ

(Qτ ),

max |ωx | ≤ M 1 + ε,

(x,tn )∈Qτ

(Qτ ) from the convex bounded set

max |ω| ≤ M 0 + ε,



(x,tn )∈Qτ λ,λ/2 Qτ

|ω|

λ,λ/2 Qτ

+ |ωx |

≤ M2 + M3 + ε

and for any real θ , 0 ≤ θ ≤ 1, we consider vn = (ωn , θ ), where vn (x) solves the linear differential-difference boundary value problem Lθn vn ≡ Cn (x, ω)ρn (x)vnt − An (x, ω)vnxx + θ Fn (x, ω) = 0,

(x, tn ) ∈ Qτ ,

(45)

An (x, ω)vnx − {θ hn (ωn ) + (1 − θ )amin }vn (x)|x=0 = θgn + (1 − θ )amin (ϕx (x) − ϕ(x))|x=0 ,

0 < tn ≤ T ,

(46)

An (x, ω)vnx + {θ en (ωn ) + (1 − θ )amin }vn (x)|x=l = θ qn + (1 − θ )amin (ϕx (x) + ϕ(x))|x=l , v0 (x) = ϕ(x),

0 ≤ x ≤ l,

0 < tn ≤ T ,

(47) (48)

where ⎧ An (x, ω) = θ a(x, tn , ωn ) + (1 − θ )amin , ⎪ ⎪ ⎪ ⎪ ⎨ Fn (x, ω) = {b(x, tn , ωn ) − ax (x, tn , ωn )}ωnx − au (x, tn , ωn )ω2 + d(x, tn , ωn )ωn − f (x, tn ), nx ⎪ Cn (x, ω) = θ c(x, tn , ωn ) + (1 − θ )cmin , ρn (x) = ρ(x, tn ), ⎪ ⎪ ⎪ ⎩ hn (ωn ) = h(tn , ωn ), en (ωn ) = e(tn , ωn ), gn = g(tn ), qn = q(tn ). The possibility of determining the operator (ωn , θ ) follows from unique solvability of problem (45)–(48) which can be considered for every n = 1, N as a boundary problem for the linear elliptic equation with the coefficient at vn (x) of the form −Cn (x, ω)ρn (x)τ −1 and with the righthand side θ Fn (x, ω) − Cn (x, ω)ρn (x)τ −1 vn−1 (x). The fixed points of the transform (ωn , θ ), i.e., the points uθn = (uθn , θ ) are the solutions of the quasilinear differential-difference boundary value problem for which all the assumptions of Lemmas 1 and 3 hold. This results in the estimates max(x,tn )∈Qτ |uθn (x)| ≤ M 0 , max(x,tn )∈Qτ |uθnx (x)| ≤ M 1 and, respectively, in the estimates of the Hölder norms λ,λ/2

λ,λ/2

|uˆ θ (x)|Qτ ≤ M 2 , |uˆ θx (x)|Qτ ≤ M 3 (see (40)). Hence, all the possible fixed points uθn of (ωn , θ ) are strongly interior points of the set . Note that for θ = 1 the considered problem (45)–(48) coincides with problem (8)–(11). It is not difficult to see that the operator (ωn , θ ) is completely continuous on  × [0, 1] 1+λ, 1+λ

2 mapping the set  (bounded in Hτ (Qτ )) into the compact set. Indeed, by estimates (44) the solutions vn (x) of the linear differential-difference boundary value problem (45)–(48) are 2+λ,1+λ/2 (Qτ ) for any ωn ∈  and 0 ≤ θ ≤ 1. uniformly bounded in Hτ It easily follows from estimates (44) that the operator (ωn , θ ) is also uniformly contin(1) (2) uous on  × [0, 1]. Indeed, let ωn (x) and ωn (x) be elements of the set  near in the

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(2)

(1)

2 sense of the norm of Hτ (Qτ ). Then the difference vn (x) = vn (x) − vn (x), where 2+λ,1+λ/2 (2) (2) (1) (1) vn = (ωn , θ ) and vn = (ωn , θ ), is small in the sense of the norm of Hτ (Qτ ). This result is an immediate consequence of estimates (44) applied to the corresponding linear boundary value problem for vn (x). In order to use the Leray–Schauder principle it remains to prove that for θ = 0 the transform (ωn , θ ) has only one fixed point interior to the set  and there exists the inverse transform in a vicinity of this point. Indeed, (ωn , 0) maps the set  into a unique element vn0 = (ωn , 0) which solves the linear differential-difference boundary value problem

L0n vn ≡ cmin ρn (x)vnt − amin vnxx = 0,

(x, tn ) ∈ Qτ ,

amin vnx (x) − amin vn (x)|x=0 = amin (ϕx (x) − ϕ(x))|x=0 ,

0 < tn ≤ T ,

amin vnx (x) + amin vn (x)|x=l = amin (ϕx (x) + ϕ(x))|x=l ,

0 < tn ≤ T ,

v0 (x) = ϕ(x),

0 ≤ x ≤ l.

This problem considered for every n = 1, N as a boundary problem for the linear elliptic equation has one and only one solution because of the coefficient at vn(x) of the form −cmin ρn (x)τ −1 . Thus the Leray–Schauder principle is applicable and allows one to conclude that for any θ ∈ 1+λ, 1+λ

2 [0, 1] there exists at least one fixed point of the transform (ωn , θ ) in the space Hτ (Qτ ). We noted above that for θ = 1 such a point is a solution of problem (8)–(11). From esti2+λ,1+λ/2 (Qτ ) since the coefficients of mates (44) it easily follows that this solution is in Hτ θ equation (45) with ωn (x) replaced by un (x) for θ = 1 are the elements of the corresponding difference-continuous Hölder spaces as functions of (x, tn ). Because of the initial function 2+λ,1+λ/2 everywhere for 0 < x < l, 0 ≤ tn ≤ T . ϕ(x) ∈ H 2+λ [0, l], the solution un (x) is in Hτ Moreover, the derivative unx (x) is continuous on the closed domain Qτ , i.e., limx→0 unx (x) = unx (0), limx→l unx (x) = unx (l), n = 1, N . The proof of this assertion is based on the classical barrier method (see, e.g. [1,2]) which is applied to equation (39) for wn (x) in Lemma 3. Namely, we consider the operator Ln wn

Ln wn ≡ v wnt − (cn ρn )−1 an v wnxx + Cn0 wnx wn + Cn1 wnx ,

(x, tn ) ∈ Qτ .

Since the coefficients Cni (i = 0, 5) in (39) are uniformly bounded then one can conclude thanks to estimate (38) that Ln wn ≤ K for (x, tn ) ∈ Qτ . Here the constant K > 0 depends, in particular, on M1 (see (38)) and does not depend on x, τ , and n. Next the usual way of barrier functions is applied in order to show that limx→0 wn (x) = wn (0), limx→l wn (x) = wn (l), n = 1, N . It remains to prove that under the additional condition (43) the solution un (x) of problem (8)–(11) is unique in the function class sup

|un (x), unx (x), unxx (x), unt (x)| < ∞.

(x,tn )∈Qτ

Assume that un (x) also solves problem (8)–(11). Then the difference vn (x) = un (x) − un (x) satisfies the relationships cn (x, un )ρn (x)vnt − (an (x, un )vnx )x + A0n vnx + A1n vn = 0,

(x, tn ) ∈ Qτ ,

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an (x, un )vnx − A2n vn |x=0 = 0,

0 < tn ≤ T ,

an (x, un )vnx + A3n vn |x=l = 0,

0 < tn ≤ T ,

v0 (x) = 0,

0 ≤ x ≤ l,

where the coefficients A0n and A1n depend on the derivatives au , axu , auu , bu , cu , and du at the point (x, tn , σ un + (1 − σ )un ) (0 < σ < 1). Moreover, A0n and A1n depend appropriately on the solution un (x) and its derivatives unx (x), unxx (x), and unt (x). The coefficients A2n and A3n in the boundary conditions depend appropriately on au , hu , and eu , where A2n ≥ 0, A3n ≥ 0 thanks to (43). Since all the input data of this linear differential-difference boundary value problem are uniformly bounded in Qτ as functions of (x, tn ), then the maximum principle allows one to conclude that max(x,tn )∈Qτ |vn (x)| = 0. Theorem 2 is proved. Remark 3. If the input data of problem (8)–(11) satisfy the conditions of Theorem 1, then the 2+λ,1+λ/2 everywhere on the closed set Qτ . Indeed, moreover a priori essolution un (x) is in Hτ timate max(x,tn )∈Qτ |unx (x)| ≤ M1 , in this case there hold the estimates for max0≤tn ≤T |unxt (x)| λ,λ/2 Qτ

at x = 0 and x = l. This means that the estimate of the Hölder norm |uˆ x (x)| everywhere for (x, tn ) ∈ Qτ and, hence,

2+λ,1+λ/2 |u(x)| ˆ Qτ

≤ M3 is valid

≤ M4 .

5. Existence and uniqueness of the solution {u(x, t), ρ(x, t)} of the nonlinear boundary value problems (1)–(5) We have already noted that obtaining the existence result for the considered system (1)–(5) by the Rothe method involves investigation of the differential-difference approximating system (8)–(12) in order to find {un (x), ρn (x)} — approximate values of u(x, t) and ρ(x, t) for t = tn . The values of ρn (x) are beforehand unknown and simultaneous determined with un(x). This requires additional reasonings for proving the solvability of system (8)–(12). 5.1. Unique solvability of the approximating system (8)–(12) The main result of this subsection is the following Lemma 5. Assume that the hypotheses of Theorem 2 hold and, moreover, let the function γ (x, t, u) be of constant signs in H 1,λ/2,1 (D), the initial function ρ 0 (x) in condition (5) be 0 ≤ ρ 0 (x) ≤ ρ 0 , ρ 0 , ρ 0 = const > 0. in C 1 [0, l] and such that 0 < ρmin max min max Then for any time-step τ ≤ τ0 (τ0 > 0 is the constant defined by Lemma 1) there exists a unique solution {un (x), ρn (x)} of the differential-difference system (8)–(12) having the properties 2+λ,1+λ/2

un (x) ∈ Hτ

(Qτ ),

0 < ρ (x) ≤ ρ 0 < ρmin n max , 0 −T 0 < ρmin

max (x,t,u)∈D

0 +T ρmax = ρmax

≤ M5 ,

M5 = const > 0 from (42),

max γ (x, t, u) for γ (x, t, u) > 0,

(x,t,u)∈D 0 for γ (x, t, u) ≤ 0, |γ (x, t, u)| ≤ ρn (x) ≤ ρmax

max |ρnx (x)| +

(x,tn )∈Qτ

2+λ,1+λ/2

|u(x)| ˆ Qτ

max |ρnt (x)| +

(x,tn )∈Qτ

(49) λ/2 t,Qτ

max |ρnxt (x)| + ρˆt (x)

(x,tn )∈Qτ

≤ M,

(50)

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where M is a positive constant independent of x, τ , and n. Its value is determined, in particular, by max0≤x≤l |ρx0 (x)|, max(x,t,u)∈D (|γx (x, t, u)|, |γu (x, t, u)|), and values of the constants M1 and M2 from estimates (38), (40). Proof. Starting with the initial conditions for t0 = 0, we assume that for each of time layers tj (j = 1, n − 1) the solutions {uj (x), ρj (x)} are found and the corresponding estimates are established. Then for t = tn the assumptions of Lemma 5 allow one to conclude that for 0 ≤ x ≤ l |ρnt (x)| ≤ γmax , γmax = max(x,t,u)∈D |γ (x, t, u)|, λ/2 λ/2 ρˆt (x)t,Q ≤ γ t,D + max(x,t,u)∈D |γu (x, t, u)| M2 , τ

|ρnxt (x)| ≤ max(x,t,u)∈D |γx (x, t, u)| + max(x,t,u)∈D |γu (x, t, u)| M1 .

These inequalities are an easy consequence of (12) since λ/2 t,Qτ

ρˆt (x)

λ/2 t,D τ

≤ γˆ (x, tn−1 , un−1 )

+ |γu (x, tn−1 , un−1 )| M2 ,

ρnxt (x) = γx (x, tn−1 , un−1 ) + γu (x, tn−1 , un−1 ) un−1x (x). Moreover, from (12) it follows that ρn (x) = ρn−1 (x) + τ γn−1 (x, un−1 ) = ρ 0 (x) +

n−1 

τ γj (x, uj ),

(51)

j =0

i.e., depending on the sign of the function γ (x, t, u) there hold 0 0 0 < ρmin < ρn (x) ≤ ρmax + tn−1 γmax for γ (x, t, u) > 0, (x, t, u) ∈ D, 0 0 − tn−1 γmax ≤ ρn (x) ≤ ρmax for γ (x, t, u) ≤ 0, (x, t, u) ∈ D. 0 < ρmin

From here it is not difficult to obtain the required estimates (49) for ρn (x). Next we note by (51) that there holds the following representation ρnx (x) = ρx0 (x) +

n−1 

τ {γx (x, tj , uj ) + γu (x, tj , uj ) uj x (x)},

j =0

which leads to the bound |ρnx (x)| ≤ max |ρx0 (x)| + tn−1 0≤x≤l



max |γx (x, t, u)| + max |γu (x, t, u)| M1 .

(x,t,u)∈D

(x,t,u)∈D

λ/2

Thus, for t = tn the bounds of |ρnx (x)|, |ρnt (x)|, |ρnxt (x)|, and ρˆt (x) are received. This t,Qτ allows one to prove estimate (50) since we assume that the corresponding estimates for tj (j = 1, n − 1) are already known. As a result of (49), (50) the grid-continuous function ρn (x), which is determined from (12) by using the given values of ρn−1 (x) and un−1 (x), satisfies the conditions of Theorem 2. Hence the differential-difference boundary value problem (8)–(11) with this coefficient ρn (x) has a unique 2+λ,1+λ/2 (Qτ ) for which bound (42) holds. Lemma 5 is proved. 2 solution un (x) in Hτ

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Remark 4. If the input data of the original system (1)–(5) satisfy the conditions of Theorem 1, then assumptions of Lemma 5 relative to γ (x, t, u) and ρ 0 (x) allow one to infer that the approximating system (8)–(12) has a unique solution {un (x), ρn (x)} in a class of more smooth functions. 2+λ,1+λ/2 2+λ,1+λ/2 on the closed set Qτ and |u(x)| ˆ ≤ M4 , where M4 > 0 Namely, un (x) is in Hτ Qτ is the constant from estimate (41). 5.2. Conditions of unique solvability of system (1)–(5) by the Rothe method Now we complete the investigation of the statements in the Hölder spaces for the nonlinear boundary value problems (1)–(5). The corresponding result for the input data a = a(x, t), e(t, u) = 0, and h(t, u) = 0 is established by the following theorem. Theorem 3. Let the following conditions be satisfied. 1. For (x, t) ∈ Q and any u, |u| < ∞, the input data of problem (1)–(4) are uniformly bounded functions of their arguments, where the coefficient a(x, t) — together with the derivatives ax (x, t) and at (x, t), moreover, ax (x, t) and f (x, t) are in H λ,λ/2 (Q), there hold 0 < amin ≤ a(x, t) ≤ amax , 0 < cmin ≤ c(x, t, u) ≤ cmax , h(t, u) = 0, e(t, u) = 0. 2. For (x, t, u) ∈ D = Q × [−M0 , M0 ] (M0 > 0 is the constant defined by (13)) the functions b(x, t, u) and d(x, t, u) are Hölder continuous in x, t with the corresponding exponents λ, λ/2 and have the uniformly bounded derivatives with respect to u; the function c(x, t, u) is in H 1,λ/2,1 (D). 3. The functions γ (x, t, u) and ρ 0 (x) in condition (5) are, respectively, in H 1,λ/2,1 (D) 0 ≤ ρ 0 (x) ≤ ρ 0 , and C 1 [0, l], moreover γ (x, t, u) is of constant signs in D, 0 < ρmin max 0 0 ρmin , ρmax = const > 0. 4. The functions ϕ(x), g(t), and q(t) are in H 2+λ [0, l] and O 1 [0, T ], respectively, and satisfy the matching conditions a(x, 0)ϕx |x=0 = g(0), a(x, 0)ϕx |x=l = q(0). Then there exists a unique solution {u(x, t), ρ(x, t)} of the nonlinear boundary value problem (1)–(5) which has properties u(x, t) ∈ H 2+λ,1+λ/2 (Q), ρ(x, t) ∈ C(Q),

2+λ,1+λ/2

|u(x, t)|Q

≤ M4 ,

ρx (x, t) ∈ C(Q),

M4 = const > 0 from (41),

ρt (x, t) ∈ H λ,λ/2 (Q),

satisfies restrictions (6), (7) depending on the sign of γ (x, t, u), and which is the limit of the solution {un (x), ρn (x)} of the approximating system (8)–(12) as the time-step τ of the grid ωτ goes to 0. Conditions of unique solvability of problem (1)–(5) for the general case a = a(x, t, u), h(t, u) ≥ 0, and e(t, u) ≥ 0 are formulated by the following theorem. Theorem 4. Let the following conditions be satisfied.

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1. For (x, t) ∈ Q and any u, |u| < ∞, the input data of problem (1)–(4) are uniformly bounded functions in their definition domains, where the coefficient a(x, t, u) — together with the derivatives ax (x, t, u) and au (x, t, u), moreover, the function f (x, t) has the uniformly bounded derivative fx (x, t) and is Hölder continuous in t with the exponent λ/2; there hold 0 < amin ≤ a(x, t, u) ≤ amax , 0 < cmin ≤ c(x, t, u) ≤ cmax , h(t, u) ≥ 0, e(t, u) ≥ 0. 2. For (x, t, u) ∈ D = Q × [−M0 , M0 ] (M0 > 0 is the constant defined by (13)) the functions a(x, t, u), ax (x, t, u), au (x, t, u), b(x, t, u), c(x, t, u) and d(x, t, u) are in H 1,λ/2,1 (D), the functions h(t, u) and e(t, u) have the uniformly bounded derivatives with respect to t and u. 3. The functions γ (x, t, u) and ρ 0 (x) in condition (5) are, respectively, in H 1,λ/2,1 (D) 0 ≤ ρ 0 (x) ≤ ρ 0 , and C 1 [0, l], moreover γ (x, t, u) is of constant signs in D, 0 < ρmin max 0 0 ρmin , ρmax = const > 0. 4. The functions ϕ(x), g(t), and q(t) are in H 2+λ [0, l] and O 1 [0, T ], respectively, and satisfy the matching conditions a(x, 0, ϕ)ϕx |x=0 − h(0, ϕ)ϕ|x=0 = g(0),

a(x, 0, ϕ)ϕx |x=l + e(0, ϕ)ϕx |x=l = q(0).

Then the nonlinear boundary value problem (1)–(5) has at least one smooth solution {u(x, t), ρ(x, t)} which has properties u(x, t) ∈ C(Q),

ux (x, t) ∈ C(Q), 2+λ,1+λ/2

|u(x, t)|Q

ρ(x, t) ∈ C(Q),

u(x, t) ∈ H 2+λ,1+λ/2 (Q) for 0 < x < l, 0 ≤ t ≤ T ,

≤ M5 ,

M5 = const > 0 from (42),

ρx (x, t) ∈ C(Q),

ρt (x, t) ∈ H λ,λ/2 (Q),

satisfies restrictions (6), (7) depending on the sign of γ (x, t, u), and which is the limit of the solution {un (x), ρn (x)} of the approximating system (8)–(12) as the time-step τ of the grid ωτ goes to 0. If the input data satisfy the additional condition (43), then the solution {u(x, t), ρ(x, t)} is unique in this class of smooth functions. Let us to pass at once to proving Theorem 4 since the proof of Theorem 3 is similar (with the corresponding simplifications). Proof of Theorem 4. At first we note that the uniform estimates (42), (49), and (50) mean the compactness of the set {un (x), ρn (x)} in the corresponding spaces. By taking the limit as τ goes to 0 (i.e., as n → ∞) in conditions (8)–(12), we can show in a standard way that the original problem (1)–(5) has at least one solution {u(x, t), ρ(x, t)} such that u(x, t) ∈ H 2+λ,1+λ/2 (Q) for 0 < x < l, 0 ≤ t ≤ T , ρ(x, t) ∈ C(Q), ρt (x, t) ∈ H λ,λ/2 (Q). Moreover, estimates (49) allow one to establish that ρ(x, t) satisfies inequalities (6) and (7) depending on the sign of γ (x, t, u) in condition (5). Next we note that under the assumptions of Theorem 4 u(x, t) and ρ(x, t) have more smooth properties. Indeed, the derivative ux (x, t) is continuous everywhere on the closed set Q, i.e., limx→0 ux (x, t) = ux (0, t), limx→l ux (x, t) = ux (l, t). This assertion is based on the barrier method thanks to the bounds max(x,t)∈Q |u(x, t)| ≤ M0 , max(x,t)∈Q |ux (x, t)| ≤ M1 (see similar reasonings carried out in Theorem 2).

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Moreover, the supposed smoothness of the functions γ (x, t, u) and ρ 0 (x) in condition (5) allows one to prove that ρ(x, t) has the derivative ρx (x, t) continuous everywhere on Q. Indeed, from (5) it follows that for 0 ≤ x ≤ l, 0 ≤ t ≤ T t ρ(x, t) =

γ (x, τ, u(x, τ )) dτ + ρ 0 (x),

(52)

0

t ρx (x, t) =

{γx (x, τ, u(x, τ )) + γu (x, τ, u(x, τ ))ux (x, τ )} dτ + ρx0 (x). 0

Since ux (x, t) ∈ C(Q), γ (x, t, u) ∈ H 1,λ/2,1 (D), and ρ 0 (x) ∈ C 1 [0, l], from here it is easily seen that ρx (x, t) ∈ C(Q). Now we prove that under the additional condition (43) the solution {u(x, t), ρ(x, t)} is unique in the class of smooth functions sup |u, ux , uxx , ut | < ∞, (x,t)∈Q

sup |ρ, ρx , ρt | < ∞. (x,t)∈Q

Assume that for t ∈ [0, t 0 ], 0 ≤ t 0 < T , the uniqueness is already proved. Let us show the uniqueness result for t ∈ [t 0 , t 0 + t], where t > 0 is a sufficiently small but bounded time interval that allows us exhaust all the segment [0, T ] by a fixed number of steps. We will use a contradiction argument. Assume that for t ∈ [t 0 , t 0 + t] there exist two solutions of system (1)–(5) {u(x, t), ρ(x, t)} and {u(x, t), ρ(x, t)}. By (52) expressions for ρ(x, t) and ρ(x, t) have the form t ρ(x, t) =

t γ (x, τ, u(x, τ )) dτ + ρ(x, t ), 0

ρ(x, t) =

t0

γ (x, τ, u(x, τ )) dτ + ρ(x, t 0 ). t0

By taking into account that ρ(x, t 0 ) = ρ(x, t 0 ), the differences v(x, t) = u(x, t) − u(x, t),

ζ (x, t) = ρ(x, t) − ρ(x, t)

satisfy the following estimate in the domain Qt 0 = {0 ≤ x ≤ l, t 0 ≤ t ≤ t 0 + t} max |ζ (x, t)| ≤ t

(x,t)∈Qt 0

max |γu (x, t, u)| max |v(x, t)|.

(x,t,u)∈D

(53)

(x,t)∈Qt 0

Moreover, thanks to (1)–(4) v(x, t) and ζ (x, t) satisfy the relationships c(x, t, u)ρ(x, t)vt − (a(x, t, u)vx )x + A0 vx + A1 v = c(x, t, u)ut ζ (x, t), a(x, t, u)vx − A2 v|x=0 = 0,

t 0 < t ≤ t 0 + t,

a(x, t, u)vx + A3 v|x=l = 0,

t 0 < t ≤ t 0 + t,

v(x, t 0 ) = 0,

0 ≤ x ≤ l,

(x, t) ∈ Qt 0 ,

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where the coefficients A0 and A1 depend in the corresponding way on the derivatives au , axu , auu , bu , cu , and du at the point (x, t, σ u + (1 − σ )u) (0 < σ < 1). Moreover, A0 and A1 depend on u(x, t), ρ(x, t), and the derivatives ux (x, t), uxx (x, t), and ut (x, t). The coefficients A2 and A3 in the boundary conditions depend in the corresponding way on the u-derivatives of the functions a(x, t, u), h(t, u), e(t, u), and moreover, on u(x, t) and ux (x, t) at the points x = 0, x = l. All the input data of this linear boundary value problem are uniformly bounded in the domain Qt 0 as functions of (x, t). Moreover, in view of condition (43) there hold A2 ≥ 0, A3 ≥ 0. This allows one to apply the maximum principle that leads to the following estimate max |v(x, t)| ≤ K0 max |ζ (x, t)|,

(x,t)∈Qt 0

(x,t)∈Qt 0

K0 = const > 0.

From here by taking into account (53) we obtain max |v(x, t)| ≤ t K0 max |γu (x, t, u)| max |v(x, t)|.

(x,t)∈Qt 0

(x,t,u)∈D

(54)

(x,t)∈Qt 0

Choosing then t > 0 such that t K0 max |γu (x, t, u)| ≤ 1 − μ,

0 < μ < 1,

(x,t,u)∈D

we output from (54) the following relationship max |v(x, t)| ≤ (1 − μ) max |v(x, t)|,

(x,t)∈Qt 0

(x,t)∈Qt 0

i.e., max(x,t)∈Q 0 |v(x, t)| = 0. Thanks to (53) this also means that max(x,t)∈Q 0 |ζ (x, t)| = 0. t

t

Thus, the uniqueness result is completely proved for t ∈ [t 0 , t 0 + t]. By repeating the analogous arguments for t ∈ [t 1 , t 2 ] (t 1 = t 0 + t , t 2 = t 1 + t ), t ∈ [t 2 , t 3 ], etc., up to the final time T , we drive the uniqueness result for problem (1)–(5) on all the segment [0, T ]. Theorem 4 is proved. 5.3. Error estimates of the Rothe method for approximate solutions of problems (1)–(5) with the input data of the general and alternative form In order to complete the investigation of the solvability of nonlinear boundary value problems (1)–(5) by the Rothe method, it remains to obtain the error estimates for this method. Our aim to estimate the differences wn (x) = un (x) − u(x, tn ),

ξn (x) = ρn (x) − ρ(x, tn ),

where {u(x, tn ), ρ(x, tn )} solves the original problem (1)–(5) for t = tn , {un (x), ρn (x)} solves the approximating system (8)–(12). Denote by  = max(x,tn )∈Qτ n (x), ψ = max(x,tn )∈Qτ ψn (x), where n (x) is the discretization error for the differential-difference boundary value problem (8)–(11) and ψn (x) is the discretization error for equation (12).

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Theorem 5. Assume that the input data satisfy the conditions of Theorem 4. Then for any sufficiently small time-step τ of the grid ωτ there hold the error estimates for the Rothe method max |wn (x)| ≤ K1 ( + ψ),

(x,tn )∈Qτ

max |ξn (x)| ≤ K2 ( + ψ),

(55)

(x,tn )∈Qτ

where K1 and K2 are positive constants independent of x, t, τ , and n. The proof repeats — with the corresponding modification — the above proof of the uniqueness result in Theorem 4. We only note that estimates (55) are shown step by step for the bounded time intervals [0, tn0 ], [tn0 , tn1 ], [tn1 , tn2 ], etc., up to the final time tN = T . Existence of such estimates allows one to apply the Rothe method for construction of approximate solutions for the nonlinear problem (1)–(5) with the input data of the general form. Analogous claims are valid for the input data of the alternative form satisfying the conditions of Theorem 3. 6. Mathematical models of the physical–chemical process of the thermodestruction Various statements of problem (1)–(5) arise in the modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. As an example of such applications, one can indicate a change of filtration properties of oil-gas stratums in modern technologies of their exploitation. We turn our attention to the other applications which are connected with a use of composites in heat-protective systems for high temperature processes in various technical fields (power engineering, aircraft, astronautics, etc.). Under the influence of high temperature heating, composites are subjected to various physical–chemical transformations and destruction, i.e., irreversible changes of inner parameters (such as the density, the concentrations of components, etc.). Let us consider the mathematical model describing destruction of the heat-protective plate under the influence of heat flux [6,7]. In this model it is assumed that the thermodestructible composite of the plate has several stages (reactions) with their kinetic parameters. The corresponding statement consists of finding the temperature u(x, t) and the concentration ρ(x, t) of the stages that satisfy the relationships c(u)ρ(x, t)ut (x, t) − (λ(u)ux )x = 0, λ(u)ux |x=0 = 0,

(x, t) ∈ Q = {0 < x < l, 0 < t ≤ T },

λ(u)ux + σ u |x=l = q(t), 4

u|t=0 = ϕ(x), 0 ≤ x ≤ l,   t S  s s s s ρ(x, t) = ρ (x, t), ρ (x, t) = ρ0 (x) exp − A exp − s=1

0 < t ≤ T,

(56) (57) (58)

  Es dτ , (x, t) ∈ Q, (59) u(x, τ )

0

where c(u) and λ(u) are the thermophysical characteristics of the plate (respectively, heat capacity and thermal conductivity), q(t) is the heat flux, the action of which is a cause of destruction of the plate, σ is the Stefan–Bolzmann constant,  is the reduced emissivity, S is the number of stages (reactions) of the composite, As and E s are the kinetic parameters of the s-th stage, ρ s (x, t) is the concentration of the s-th stage, ϕ(x) > 0 and ρ0s (x) > 0 are the initial temperature and concentration of the s-th stage for t = 0, l is the thickness of the plate, and T is the time of the heat action.

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The mathematical models which describe the process of destruction of the other inner parameters of the composite are distinguished by the corresponding form of condition (59). For example, if ρ(x, t) is the density of the heat-protective material, then condition (59) is given by the following way 

t

ρ(x, t) = ρ0 (x) exp − A

 exp −

  E dτ , u(x, τ )

ρ0 (x) > 0,

0

where ρ0 (x) is the initial density of the composite up to the heat action. In connection with the aerospace technologies, all these models arise in the investigation of the heat-protective system for the re-entry vehicle. Returning to problem (56)–(59) we show our results of the numerical modeling of the thermodestruction carried out for the composite with the given thermophysical and kinetic characteristics (the number of stages S = 5). The heat flux acting on the surface of the pattern (the thickness l = 3.5 · 10−3 m, the time of action T = 20 sec) has the form ⎧ 1.8t, 0 ≤ t ≤ 5, ⎪ ⎪ ⎨ 1.8(10 − t), 5 ≤ t ≤ 10, q(t) = 10 ≤ t ≤ 15, ⎪ 3(t − 10), ⎪ ⎩ 3(20 − t), 15 ≤ t ≤ 20. As a result of this heat action, the temperature regime in the pattern changes from 300 °C at the initial time t = 0 to almost 600 °C at the final time T = 20 sec. The process is accompanied by the destruction of components of the composite. At it is shown below, two stages (s = 2 and s = 3) have been subjected to especially essential destruction. Concentration of stages on surface of pattern for t = 0 and t = T s-th stage

ρ 1 (l, t)

ρ 2 (l, t)

ρ 3 (l, t)

ρ 4 (l, t)

ρ 5 (l, t)

For t = 0

3.811 · 10−2

6.302 · 10−1

2.561 · 10−1

4.132 · 10−2

2.124 · 10−2

For t = T

3.552 · 10−2

3.635 · 10−2

0.

3.762 · 10−3

4.721 · 10−3

7. Conclusion In this work, the nonlinear parabolic models are considered which arise in the investigation of physical–chemical processes with changeable inner characteristics. The corresponding mathematical statements essentially differ from usual boundary value problems for parabolic equations. They are formulated as a system that involves a boundary value problem for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional relationship for a time dependence of this coefficient. For such statements we justify conditions of unique solvability in a class of smooth functions for the input data of the general and alternative form. To this end, the Rothe method is applied which provides not only the proof of solvability but also the approximate solutions for the considered statements. We turn our main attention to the nonlinear differential-difference system that approximates the original system by the Rothe method. For this approximating system we establish a priori estimates in the corresponding difference-continuous Hölder spaces which allow one to prove the existence of the

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smooth solutions and to obtain the error estimates for the Rothe method. Moreover we avoid additional assumptions of the smoothness of the input data (which have usually been imposed by the Rothe method) and determine the faithful character of differential relations for the nonlinear boundary value problems of the considered type. These results are similar to the ones obtained in [1] for the boundary value problems in the case of quasilinear parabolic equations with the given coefficients. As an example of important applications of the nonlinear parabolic models that are considered in this work, we analyze the problem of the destruction of the heat-protective composite under the effect of high temperature heating. The proposed approach allows one to take into account the dependence of thermophysical characteristics upon the temperature that it is especially important for such processes. Acknowledgments This work was supported by M.V. Lomonosov Moscow State University (the program Main Directions of Scientific Researches). References [1] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc., Providence, RI, 1968. [2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, New York, 1964. [3] N.L. Gol’dman, Inverse Stefan Problems, Kluwer Academic, Dordrecht, 1997. [4] S.N. Kruzhkov, A priori estimates of the derivative of a solution for a parabolic equation, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1967) 41–48. [5] L.V. Kantorivich, G.P. Akilov, Functional Analysis, Pergamon, New York, 1982. [6] A.K. Alekseev, On the restoration of the heating history of a plate made of a thermodestructible material from the density profile in the final state, High Temp. 6 (1993) 897–901. [7] S.F. Gilyazov, N.L. Gol’dman, Regularization of Ill–Posed Problems by Iteration Methods, Kluwer Academic, Dordrecht, 2000.