T< ∞ , S isthesetofpositiveintegers, f :Ω × CB (Ω) 3 ( t,x,s ) → f ( t,x,s ) ∈ where id istheidentityoperator,( t,x ) ∈ Ω:= { ( t,x ): t ∈ (0 ,T ) ,x ∈ R } , ∂x ∂x F := −A , A := a ( t,x ) + b ( t,x ) + c ( t,x ) id, ∂∂t ∂ ∂∂x X X Here u (0 ,x )= ϕ ( x )f

EXISTENCE OF SOLUTIONS OF THE CAUCHY PROBLEM FOR SEMILINEAR INFINITE SYSTEMS OF PARABOLIC

DIFFERENTIAL–FUNCTIONAL EQUATIONS

by Anna Pude lko

Abstract. We consider the Cauchy problem for a countable system of weakly coupled semilinear differential–functional equations of parabolic type. The right-hand sides of the system are functionals of unknowns.

The object of this paper is to transform the system of parabolic differential equations into the associated system of integral equation in order to prove the existence of the solution of the latter problem with use of the Schauder fixed point theorem.

1. Introduction. We consider a countable system of weakly coupled semi- linear differential–functional equations of the form

(1) Fⁱ[uⁱ](t, x) =fⁱ(t, x, u), i∈S, with the initial condition

(2) u(0, x) =ϕ(x) for x∈R^m.

Here Fⁱ:= ∂

∂t − Aⁱ,Aⁱ:=

m

X

j,k=1

aⁱ_jk(t, x) ∂²

∂xj∂x_k +

m

X

j=1

bⁱ_j(t, x) ∂

∂xj

+cⁱ(t, x)id, where id is the identity operator, (t, x)∈Ω :={(t, x) :t∈(0, T), x∈R^m}, T <∞,Sis the set of positive integers,fⁱ: Ω×CB^∞_S (Ω)3(t, x, s)→fⁱ(t, x, s)∈

2000Mathematics Subject Classification. 35R10, 35K55.

Key words and phrases. Infinite systems, parabolic differential–functional equations, Schauder fixed point theorem, Cauchy problem.

Part of this work is supported by local Grant No.11.420.04.

R, i∈S, u stands for the mapping

u:S×Ω3(i, t, x)→uⁱ(t, x)∈R, composed of unknowns uⁱ and ϕ= (ϕ¹, ϕ², . . .).

A functionuis said to be aC–solutionof differential problem (1), (2) in Ω if u ∈CB_S^∞(Ω) and satisfies the system of integral equations associated with the differential problem in Ω.

This paper can be considered as a continuation of the author’s study of the certain infinite systems of parabolic differential–functional equations. Now the author considers a more general form of operator with lower order of terms with respect to the derivatives in x–variables. There have appeared papers devoted to an analysis of the initial-boundary value problem, e.g. [2], [3]. The infinite systems of parabolic type and properties of their solutions are treated, e.g., in [7] and [14].

The goal of this paper is to prove the existence of a solution of Cauchy problem (1), (2) for a countable systems using the Schauder fixed point theorem under weaker assumptions than in [11]. In paper [11], to solve the above problem the Banach fixed point theorem was used. This approach allowed us to prove the existence and uniqueness of the solution under rather restrictive assumptions which were previously imposed on the initial data (namely equi- boundeness of all components) and the right-hand sides (among other things, equi-boundeness of all components and the Lipschitz condition with respect to functional argument).

Now, resigning from that and applying the Schauder fixed point theorem (see [4], [10] or [15]), we will get the existence of the considered problem in a layer [0, τ)×R^m,where τ is a sufficiently small number. Unfortunately, this theorem does not help us to obtain the uniqueness of the problem (1), (2).

In order to apply the Schauder fixed point theorem, we have to check, among other things, that certain operator is compact. We obtained it by using the well-known Fr´echet compactness theorem. It is in some way an extension of the idea we have first come across in [10].

Let us stress that the result is obtained without using the weighted-spaces.

Instead, we assume that the truncated operator F_N of the operator F (ge- nerated by right-hand sides) uniformly smothers the functions at the infinity.

An infinite system of equations was first considered by M. Smoluchowski as a model for coagulation of colloids moving according to a Brownian mo- tion. A system of infinite number of reaction–diffusion equations related to the system of ODE derived by Smoluchowski is investigated in [1]. A nonlocal discrete model of cluster coagulation and fragmentation expressed in terms of an infinite system of integro–differential bilinear equations was studied in [8].

The coagulation–fragmentation local model which add spatial diffusion to the

classical coagulation equations is considered in [12]. For a treatment of infinite systems of parabolic differential–functional equations with the initial-boundary value problem, where Schauder’s method of fixed point is also applied, we refer the reader to [2].

This paper is organized as follows. In the next section the necessary no- tations and definitions are introduced. In section 3 we impose the necessary assumptions and formulate the main result of this paper. In section 4 we state and prove the auxiliary lemmas. The last section contains a proof of the main theorem.

2. Notations and definitions. Throughout the paper, we use the fol- lowing notation. By Ω^τ we denote the set (0, τ)×R^m for eachτ ∈(0, T].The set Ω^T is denoted for short by Ω. The norm in R^N we note by | · |^N. B(x, δ) denotes an open ball with center at x and radiusδ >0.

Here, by CB^∞(Ω) we unusually denote the space of functionsh ∈C(Ω), such that h vanishes uniformly at infinity, i.e.

∀ >0 ∃R >0∀t∈[0, T]∀x∈R^m\B(0, R) : |h(t, x)|< . The author hopes that it causes no misunderstanding.

The space CB_S^∞(Ω) comprises all functions h = (h¹, h², . . .) such that hⁱ ∈CB^∞(Ω), i∈S with the finite norm

khk^Σ_Ω =

∞

X

i=1

1

Qⁱkhⁱk_Ω, where khⁱk_Ω= sup

(t,x)∈Ω

|hⁱ(t, x)|fori∈S,

and Q is an arbitrary real number. Obviously, the space CB_S^∞(Ω) endowed with the norm k · k^Σ_Ω is a Banach space.

For fixedN ∈ N letCB_N^∞(Ω) be the space of all functionsh= (h¹, . . . , h^N) such that hⁱ ∈ CB^∞(Ω) for i = 1, . . . , N. We endow this space with the following norm: khk^N_Ω = max

i=1,...,N sup

(t,x)∈Ω

|hⁱ(t, x)|.

We understand the spacesCB^∞(R^m), CB_S^∞(R^m), CB_N^∞(R^m) analogously.

Now we introduce the following (?)–condition.

The set K⊂CB_N^∞(Ω) is said to satisfy(?)–conditionif and only if

∀ >0 ∃R_,N >0 ∀h∈K ∀t∈[0, T]∀x∈R^m\B(0, R,N)

|hⁱ(t, x)|< fori= 1, . . . , N.

Let η∈CB_S^∞(Ω). We define the following operatorF= (F¹,F², ...) F:η→F[η],

setting

Fⁱ[η](t, x) :=fⁱ(t, x, η), i∈S.

3. Assumptions and the main result. Now we formulate the assump- tions necessary for obtaining the result which is given at the end of this section.

We will assume that:

(H) : the coefficients aⁱ_jk(t, x), bⁱ_j(t, x), cⁱ(t, x), i∈ S, j, k = 1, . . . , m are bounded continuous functions such that aⁱ_jk(t, x) =aⁱ_kj(t, x) and satisfy the followingH¨older continuous condition with exponent α (0< α61) in Ω :

∃H >0 ∀i∈S ∀ j, k= 1, . . . , m∀ t∈[0, T]∀ x, x⁰ ∈R^m :

|aⁱ_jk(t, x)−aⁱ_jk(t, x⁰)|6H|x−x⁰|^α

|bⁱ_j(t, x)−bⁱ_j(t, x⁰)|6H|x−x⁰|^α

|cⁱ(t, x)−cⁱ(t, x⁰)|6H|x−x⁰|^α.

We suppose as well that the operators Fⁱ, i∈ S,are uniformly parabolic in Ω (the operators Aⁱ are uniformly elliptic in Ω), i.e.

∃ µ₁, µ₂>0∀ ξ= (ξ₁, . . . , ξ_m)∈R^m ∀ (t, x)∈Ω∀ i∈S: µ₁

m

X

j=1

ξ_j²

m

X

j,k=1

aⁱ_jk(t, x)ξ_jξ_k6µ₂

m

X

j=1

ξ_j².

The crucial assumptions related to the initial data ϕ= (ϕ¹, ϕ², . . .) are:

(ϕ₁) : ϕⁱ ∈CB^∞(R^m) for each i∈S (i.e. ϕ∈CB_S^∞(R^m))

(ϕ₂) : there exist Q⁰ ∈(0, Q) and ¯M > 0 such that kϕⁱk_Rm 6 M(Q¯ ⁰)ⁱ⁻¹ for each i∈S.

Our main requirements concerning the right-hand sides are as follows: Let the function f = (f¹, f², . . .) generating the operator Fbe such that for each τ ∈(0, T]

(F1) : F: CB_S^∞(Ω^τ)→CB_S^∞(Ω^τ) is continuous;

(F₂) : if K is a closed, bounded set inCB^∞_S (Ω^τ) then there exist Q⁰ ∈(0, Q) and ¯M >0 such that sup

w∈K

kFⁱ[w]k_Ωτ 6M(Q¯ ⁰)ⁱ⁻¹ for each i∈S;

(F₃) : for eachN ∈ N,the mapping F_N := (F¹, . . . ,F^N), F_N :CB_S^∞(Ω^τ) → CB_N^∞(Ω^τ) transforms a bounded set in CB_S^∞(Ω^τ) into a set which sat- isfies(?)–condition;

(V) : functions fⁱ satisfy the Volterra condition, i.e. for arbitrary (t, x) ∈ Ω and arbitrary η,η˜ ∈ CB_S^∞(Ω) such that η^j(¯t, x) = η˜^j(¯t, x), for 06¯t6t, j ∈S there isfⁱ(t, x, η) =fⁱ(t, x,η), i˜ ∈S.

The examples of the F operator which satisfy the imposed conditions in- clude:

Fⁱ[u](t, x) = ( Pi

n=1b_n(t, x)uⁿ(t, x) for (t, x)∈[0, T]×B(0,1)

Pi

n=1bn(t,x)uⁿ(t,x)

|x|^1/i for (t, x)∈[0, T]×(R^m\B(0,1)), where bn∈C(Ω) satisfykb_nk_Ω < M

Q⁰ Q

i−1

for someM >0, Q⁰ ∈(1, Q) or Fⁱ[u](t, x) =

Z t

0

bi(τ, x)uⁱ(τ, x)uⁱ⁻¹(τ, x) dτ,

where b_i∈C(Ω) satisfykb_ik_Ω < M

Q⁰ Q²

i−1

for someM >0, Q⁰∈(1, Q).

Now, let us state the main result of the paper.

Theorem. Let all the above assumptions hold and τ^∗ ∈ (0, T] be a suf- ficiently small number. Then the Cauchy problem (1), (2) has at least one C–solution u∈CB_S^∞(Ω^τ),where 0< τ 6τ^∗ 6T.

4. Auxiliary lemmas. To get the proof of the foregoing theorem, we first formulate the following lemmas.

Lemma 1. ([5], Th. 2.1, p. 71 [6], Th. 10, p. 23)

If the operators Fⁱ (i ∈ S) are uniformly parabolic in Ω with the constants µ₁, µ₂ and the coefficients aⁱ_jk(t, x), bⁱ_j(t, x), cⁱ(t, x), i∈S, j, k= 1, . . . , m sa- tisfy the condition (H) in Ω then there exist the fundamental solutions Γⁱ(t, x;τ, ξ) of the equations

Fⁱ[uⁱ](t, x) = 0, i∈S and the following inequalities hold

|Γⁱ(t, x;τ, ξ)|6C(t−τ)^−m2 exp

− µ^∗|x−ξ|² 4(t−τ)

, i∈S

for any µ^∗ < µ, where µ depends on µ1, µ2, H and C depends on µ1, µ2, α, T and the nature of continuity aⁱ_jk(t, x) in t.

Owing to the fact that Fⁱ (i ∈ S) are uniformly parabolic in Ω and the coefficients aⁱ_jk(t, x) (i ∈ S, j, k = 1, . . . , m), satisfy condition (H_a) in Ω, we notice that R

R^m|Γⁱ(t, x;τ, ξ)|dξ, i∈S are equi-bounded. ByC we denote the infimum of their upper bounds.

Using the fundamental solutions Γⁱ(t, x;τ, ξ), i ∈ S, for the equations Fⁱ[uⁱ](t, x) = 0, we consider the following system of integral equations as- sociated with differential problem (1), (2)

(3) uⁱ(t, x) = Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ+

t

Z

0

Z

R^m

Γⁱ(t, x;τ, ξ)Fⁱ[u](τ, ξ)dξdτ i∈S fort >0, x∈R^m. Now, we define T:CB_S^∞(Ω)−→CB_S^∞(Ω) in the following way

T[z] := (T¹[z¹],T²[z²], . . .), where

Tⁱ[zⁱ] = Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ+ Zt

0

Z

R^m

Γⁱ(t, x;τ, ξ)zⁱ(τ, ξ)dξdτ.

By T_N we denote the operatorT_N = (T¹, . . . ,T^N).

We will now prove the following lemmas.

Lemma 2. Let N ∈ N be fixed, ϕⁱ ∈ CB^∞(R^m), i = 1, . . . , N. If G ⊂ CB_N^∞(Ω)is a bounded set which satisfies (?)–condition, then

T_N(G)⊂CB_N^∞(Ω) is a family of equicontinuous functions.

Proof. To begin with, we notice that ϕⁱ, i = 1, . . . , N, are uniformly continuous. We denote M⁰ := sup

z∈G

kzk^N_Ω.

Fix η > 0. Let (t, x), (t⁰, x⁰) ∈ Ω. First of all, we consider the easiest case t=t⁰ = 0.Since ϕⁱ, i= 1, . . . N are uniformly continuous, there exists δ1 >0 such that

(2.1)

Tⁱ[zⁱ](0, x)−Tⁱ[zⁱ](0, x⁰)

=|ϕⁱ(x)−ϕⁱ(x⁰)|< η if only |x−x⁰|^m < δ1.Now lett > t⁰ = 0.It is obvious that

Tⁱ[zⁱ](t, x)−Tⁱ[zⁱ](0, x⁰)

=

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ+

t

Z

0

Z

R^m

Γⁱ(t, x;τ, ξ)zⁱ(τ, ξ)dξdτ −ϕⁱ(x⁰)

6

t

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ+

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x⁰) .

We estimate the first term in the following way (2.2)

t

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ 6M⁰

t

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)|dξdτ =CM⁰t < η 2 if only t < δ₂ := _2CM^η 0.In order to estimate the second one, we select R¹_η >0 such that for x∈R^m\B(0, R¹_η)

(2.3) |ϕⁱ(x)|< η

6C fori= 1, . . . , N.

Now, we choose R¹_η > R¹_η such that for all x ∈ R^m\B(0, R¹_η), ξ ∈B(0, R¹_η), t∈[0, T]

(2.4) |Γⁱ(t, x; 0, ξ)|< η

6kϕⁱk_R^mm(B(0, R¹_η)), where m stands for the Lebesgue measure.

Let us remark that it is enough to considerx, x⁰ such that|x−x⁰|^m< δ2.Thus, if x∈B(0, R¹_η+δ2),thenx, x⁰ ∈B(0, R¹_η+ 2δ2).According to the definition of the fundamental solution,

t&0lim Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ=ϕⁱ(x)

follows for each x ∈ R^m. Since B(0, R¹_η + 2δ2) is a compact set and ϕⁱ, i = 1, . . . , N are continuous functions, R

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ converges to ϕⁱ(x) uniformly on B(0, R¹_η + 2δ₂), that is, there exists δ₃ > 0 independent of x such that fort∈(0, δ₃), x∈B(0, R¹_η + 2δ₂)

(2.5)

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x)

< η 4

follows. Next, we choose 0< δ4 < δ3 such that for all|x−x⁰|^m < δ4

(2.6)

ϕⁱ(x)−ϕⁱ(x⁰)

< η

4 fori= 1, . . . , N.

Then, (2.5) and (2.6) yield the estimate

(2.7)

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ −ϕⁱ(x⁰)

6

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x)

+|ϕⁱ(x)−ϕⁱ(x⁰)|< η 2 for t∈(0, δ4), x, x⁰∈B(0, R¹_η+ 2δ2), |x−x⁰|< δ4.

If x /∈B(0, R¹_η+δ2) thenx, x⁰ ∈/ B(0, R¹_η).Certainly,

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x⁰) 6

Z

R^m

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ+|ϕⁱ(x⁰)|.

We see from (2.3) that the second term is less than ^η₆, (as it is less than _6C^η ) and we present the first one as follows

Z

R^m

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ

= Z

R^m\B(0,R¹_η)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ+ Z

B(0,R¹_η)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ.

To estimate the first integral, we employ (2.3):

(2.8)

Z

R^m\B(0,R¹_η)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ < η 6C

Z

R^m

|Γⁱ(t, x; 0, ξ)|dξ < η 6, and the second one is a directly estimated by (2.4) as follows

(2.9) Z

B(0,R¹_η)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ <

Z

B(0,R¹_η)

η

6kϕⁱk_R^mm(B(0, R_η¹))kϕⁱk_Rmdξ= η 6. Hence (2.2), (2.7), (2.8) and (2.9) lead to the assertion that

if max{t, |x−x⁰|^m}< δ5 := min{δ₁, δ2, δ4}then (2.10)

Tⁱ[zⁱ](t, x)−Tⁱ[zⁱ](0, x⁰)

< η.

In the next step of our proof we consider the case t, t⁰ > 0 the sake of simplicity let us now assume that min{t, t⁰}=t⁰.We estimate

Tⁱ[zⁱ](t, x)−Tⁱ[zⁱ](t⁰, x⁰)

6

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ− Z

R^m

Γⁱ(t⁰, x⁰; 0, ξ)ϕⁱ(ξ)dξ

+

t

Z

0

Z

R^m

Γⁱ(t, x;τ, ξ)zⁱ(τ, ξ)dξdτ−

t⁰

Z

0

Z

R^m

Γⁱ(t⁰, x⁰;τ, ξ)zⁱ(τ, ξ)dξdτ

=:I1+I2. To estimate I1,we select R²_η >0 such that, for eachx∈R^m\B(0, R_η²) (2.11) |ϕⁱ(x)|< η

8C fori= 1, . . . , N

takes place. Next, we choose R²_η > R²_η such that, for all x ∈ R^m\B(0, R²_η), ξ ∈B(0, R²_η), t∈(0, T]

(2.12) |Γⁱ(t, x; 0, ξ)|< η

8kϕⁱk_R^mm(B(0, R²_η)) fori= 1, . . . , N

follows. As previously, let us remark that it is enough to consider x, x⁰ such that |x−x⁰|^m< δ5.

Obviously, if x ∈ B(0, R²_η +δ5), then x, x⁰ ∈ B(0, R²_η + 2δ5). By the same argument as before, there exists 0 < δ₆ < δ₅ independent of x such that for t∈(0, δ₆), x∈B(0, R²_η+ 2δ₅),the following inequality holds

(2.13)

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x)

< η

8 fori= 1, . . . , N.

If we select 0< δ7< δ6 such that (2.14) |ϕⁱ(x)−ϕⁱ(x⁰)|< η

4 fori= 1, . . . , N,

takes place for all |x−x⁰|^m < δ₇ then for t < δ₆, x, x⁰ ∈B(0, R²_η) such that

|x−x⁰|^m < δ7,using (2.13) and (2.14) we immediately obtain

(2.15) I₁ 6

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x)

+

Z

R^m

Γⁱ(t⁰, x⁰; 0, ξ)ϕⁱ(ξ)dξ−ϕⁱ(x⁰)

+|ϕⁱ(x)−ϕⁱ(x⁰)|< η 2.

Now, let t ∈ [δ6, T]. We remark that it is enough to consider t⁰ such that

|t−t⁰|< δ₇. Then certainly t, t⁰ ∈[δ₆−δ₇, T].Since Γⁱ(·,·,0, ξ), i= 1, . . . , N are uniformly continuous on [δ₆−δ₇, T]×B(0, R²_η+2δ₅) there exists 0< δ₈ < δ₇ such that, if max{|t−t⁰|, |x−x⁰|^m}< δ8,then

(2.16) |Γⁱ(t, x; 0, ξ)−Γⁱ(t⁰, x⁰; 0, ξ)|< η

4kϕⁱk_R^mm(B(0, R²_η)). We estimate the term I₁ in the following way

(2.17)

I1 6 Z

R^m

|Γⁱ(t, x; 0, ξ)−Γⁱ(t⁰, x⁰; 0, ξ)||ϕⁱ(ξ)|dξ

= Z

B(0,R²_η)

|Γⁱ(t, x; 0, ξ)−Γⁱ(t⁰, x⁰; 0, ξ)||ϕⁱ(ξ)|dξ

+ Z

R^m\B(0,R²_η)

|Γⁱ(t, x; 0, ξ)−Γⁱ(t⁰, x⁰; 0, ξ)||ϕⁱ(ξ)|dξ=:I11+I12.

Hence, applying (2.16) and (2.11), we easily see that for (t, x), (t⁰, x⁰)∈[δ₆− δ₇, T]×B(0, R²_η+ 2δ₅) such that max{|t−t⁰|, |x−x⁰|^m}< δ₈ there is

(2.18) I₁₁<

Z

B(0,R²_η)

η

4kϕⁱk_R^mm(B(0, R²_η))kϕⁱk_R^mdξ= η 4 as well as

(2.19) I12< η 8C

Z

R^m

|Γⁱ(t, x; 0, ξ)|+|Γⁱ(t⁰, x⁰; 0, ξ)|dξ < η 4.

If, however, x /∈ B(0, R²_η +δ5), then x, x⁰ ∈/ B(0, R²_η). Consequently, using (2.17), (2.11) and (2.12) we estimate I₁ as follows

(2.20)

I₁ 6I₁₁+I₁₂<kϕⁱk_R^m Z

B(0,R²_η)

|Γⁱ(t, x; 0, ξ)|+|Γⁱ(t⁰, x⁰; 0, ξ)|dξ

+ η 8C

Z

R^m\B(0,R²_η)

|Γⁱ(t, x; 0, ξ)|+|Γⁱ(t⁰, x⁰; 0, ξ)|dξ6 η 2.

Let us now recall (2.15), (2.18), (2.19) and (2.20) to state that for all (t, x), (t⁰, x⁰) ∈ Ω such that max{|t−t⁰|, |x−x⁰|^m} < δ8 the following es- timate follows

(2.21) I1=

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ− Z

R^m

Γⁱ(t⁰, x⁰; 0, ξ)ϕⁱ(ξ)dξ

< η 2. Next, let us demonstrate thatI₂< ^η₂ as well. To confirm this assertion we note that

I₂ 6

t⁰

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)||zⁱ(τ, ξ)|dξdτ

+ Zt

t⁰

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ =:I₂₁+I₂₂. There exists δ₉ := min{_4CM^η 0, δ₈}, where M⁰ := sup

z∈G

kzk^N_Ω, such that for all x∈R^m and t, t⁰ such thatt−t⁰ < δ₉

(2.22) I226CM⁰(t−t⁰)< CM⁰δ9 < η 4.

Since Gsatisfies(?)–condition, there existsR³_η >0 such that for all t∈[0, T], x∈R^m\B(0, R³_η) the inequality

(2.23) |zⁱ(t, x)|< η

16CT for i= 1, . . . , N.

holds. We present I₂₁ as follows I₂₁ =

t⁰

Z

0

Z

R^m\B(0,R³_η)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)||zⁱ(τ, ξ)|dξdτ

+

t⁰

Z

0

Z

B(0,R³_η)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)||zⁱ(τ, ξ)|dξdτ :=I₂₁₁+I₂₁₂.

Applying (2.23) we can establish the following estimate (2.24) I211 < η

16CT

t⁰

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)|+|Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ 6 η 8.

Obviously

(2.25) I212 6M⁰

t⁰

Z

0

Z

B(0,R_η³)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ.

Let us notice that in view of the integrability of fundamental solution, there exists 0< δ10< δ9 such that if (t, x),(t⁰, x⁰)∈Ω and t⁰ 6t, τ ∈(0, t⁰) then for each x∈R^m

(2.26)

t⁰

Z

0

Z

B(x,δ10)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξ < η

16M⁰ fori= 1, . . . , N follows. In order to estimate (2.25) we present it as follows

(2.27)

t⁰

Z

0

Z

B(0,R³_η)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ

=

t⁰

Z

0

Z

B(0,R³_η)∩B(^x+x₂ ⁰,δ10)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ

+

t⁰

Z

0

Z

B(0,R³_η)\B(^x+x0

2 ,δ10)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ.

(2.26) yields that the first term of (2.27) is less than _16M^η 0. Now, we assume that max{|t−t⁰|, |x−x⁰|^m} < δ11 < δ10 for some δ11 > 0. By virtue of the mean-value theorem there exists (t0, x0) belonging to the segment connecting (t, x), (t⁰, x⁰),such that

(2.28)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|

=

(t−t⁰)∂Γⁱ

∂t (t0, x0, τ, ξ) +

m

X

k=1

(xk−x⁰_k)∂Γⁱ

∂xk

(t0, x0, τ, ξ)

< δ11

∂Γⁱ

∂t (t0, x0, τ, ξ)

+

m

X

k=1

∂Γⁱ

∂x_k(t0, x0, τ, ξ)

.

If|x₀−ξ| ≥ ^δ10₂ ,then the derivatives appearing in (2.28) are esimated as below (cf. [9], p. 427)

(2.29)

∂Γⁱ

∂t (t0, x0, τ, ξ)

< C⁰(t0−τ)^−m+22 exp

−C⁰⁰|x₀−ξ|² t0−τ

6C⁰(t₀−τ)^−m+22 exp

−C⁰⁰ δ₁₀² 4(t0−τ)

6M_t,

∂Γⁱ

∂x_k(t₀, x₀, τ, ξ)

< C⁰(t₀−τ)^−m+12 exp

−C⁰⁰|x₀−ξ|² t₀−τ

6C⁰(t0−τ)^−m+12 exp

−C⁰⁰ δ₁₀² 4(t0−τ)

6Mx fork= 1, . . . , m.

Owing to (2.29), we discover that for all (t, x), (t⁰, x⁰) ∈ Ω such that max{|t−t⁰|, |x−x⁰|^m}< δ₁₁ we obtain

(2.30)

t⁰

Z

0

Z

B(0,R³_η)\B(^x+x₂ ⁰,δ10)

|Γⁱ(t, x;τ, ξ)−Γⁱ(t⁰, x⁰;τ, ξ)|dξdτ

< δ₁₁(M_t+kM_x)m(B(0, R³_η))T < η 16M⁰ if only δ11< ^η

16(Mt+kMx)m(B(0,R³_η))M⁰T.

Thus, combining (2.22), (2.24), (2.25), (2.26) and (2.30), we see that for all (t, x), (t⁰, x⁰)∈Ω such that max{|t−t⁰|, |x−x⁰|^m}< δ₁₁,

(2.31) I2 =

t

Z

0

Z

R^m

Γⁱ(t, x;τ, ξ)zⁱ(τ, ξ)dξdτ −

t⁰

Z

0

Z

R^m

Γⁱ(t⁰, x⁰;τ, ξ)zⁱ(τ, ξ)dξdτ

< η 2. And finally, owing to (2.1), (2.10), (2.21) and (2.31) we conclude that for all (t, x), (t⁰, x⁰)∈Ω, z ∈G, if max{|t−t⁰|, |x−x⁰|^m}< δ11,then

Tⁱ[zⁱ](t, x)−Tⁱ[zⁱ](t⁰, x⁰)

< η, which completes the proof of Lemma 2.

Next we will show a certain property of the operatorT_N.Roughly speaking, we will check that the operator TN transfers the(?)–condition.

Lemma 3. Let N ∈ N be fixed. If ϕⁱ ∈CB^∞(R^m), i= 1, . . . N, then the operator TN transforms a bounded set in CB_N^∞(Ω), satisfying (?)–condition into a set satisfying (?)–condition in CB_N^∞(Ω).

Proof. Let K ⊂ CB^∞_N(Ω) be a bounded set which satisfies the (?)–

condition. Fix >0.There existsR¹ >0 such that forx∈R^m\B(0, R¹)

(3.1) |ϕⁱ(x)|<

4C fori= 1, . . . , N

follows. We selectR¹ > R¹ such that forx∈R^m\B(0, R¹), ξ∈B(0, R¹), t∈ [0, T] we can obtain the inequality

(3.2) |Γⁱ(t, x; 0, ξ)|<

4kϕⁱk_R^mm(B(0, R¹)) fori= 1, . . . , N.

It is obvious that

Z

R^m

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ=

= Z

B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ+ Z

R^m\B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ.

Therefore, by (3.1) and (3.2) for x ∈ R^m \ B(0, R¹), we easily obtain the following estimates

(3.3)

Z

R^m\B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ <

4C Z

R^m\B(0,R¹)

|Γⁱ(t, x; 0, ξ)|dξ6 4, (3.4)

Z

B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ <

Z

B(0,R¹)

4kϕⁱk_R^mm(B(0, R¹))kϕⁱk_R^mdξ= 4. From the (?)–conditionforK,it follows that there existsR² >0 such that for each function z∈K for all t∈[0, T], x∈R^m\B(0, R²),we can assert (3.5) |zⁱ(t, x)|<

4T C fori= 1, . . . , N.

Now consider

Z t

0

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ

= Z t

0

Z

R^m\B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ+

Z t

0

Z

B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ.

Making use of (3.5) we estimate the first term as follows (3.6)

Z t

0

Z

R^m\B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ <

4T C Z t

0

Z

R^m

|Γⁱ(t, x;τ, ξ)|dξdτ 6 4. In order to estimate the second one, we choose R² > R² such that for all x∈R^m\B(0, R²), ξ∈B(0, R²), t, τ ∈[0, T] : τ < t, we can get the estimate (3.7) |Γⁱ(t, x;τ, ξ)|<

4T M⁰m(B(0, R²)) fori= 1, . . . , N, where M⁰ := sup

z∈K

kzk^N_Ω.Then using (3.7) we readily assert (3.8)

Z t

0

Z

B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ <

4.

Thus, for R = max{R¹, R²}, z ∈K, t∈[0, T], x∈R^m\B(0, R) combining (3.3), (3.4), (3.6) and (3.8), we find

|(Lⁱ)⁻¹[zⁱ](t, x)|6 Z

R^m

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ+ Z t

0

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ

= Z

B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ+ Z

R^m\B(0,R¹)

|Γⁱ(t, x; 0, ξ)||ϕⁱ(ξ)|dξ

+ Z t

0

Z

B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ

+ Z t

0

Z

R^m\B(0,R²)

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)|dξdτ < fori= 1, . . . , N.

The Lemma 3 is proved.

Let us now prove the next auxiliary lemma. We adopt the Arzela–Ascoli compactness theorem, developing it for the case of functions defined on an unbounded domain but satisfying(?)–condition. The diagonal method will be our main tool in a proof of this lemma, like in the proof of the Arzela–Ascoli compactness criterion.

Lemma 4. Let Y ⊂ CB_N^∞(Ω). If Y is a closed set of equicontinuous, uniformly bounded functions andY satisfies(?)–condition, thenY is a compact set in CB_N^∞(Ω).

Proof. Let {v_n}_n∈N be a sequence in Y. Let1= 1.

From the (?)–condition it follows that there exists a constant R₁ > 0 such that |v_n(t, x)|^N < ₂¹ follows for each n∈ N, t∈[0, T], x∈R^m\B(0, R₁).

Thus, for each t∈[0, T], x∈R^m\B(0, R₁), n, k∈ N,there is

|v_n(t, x)−v_k(t, x)|^N 6|v_n(t, x)|^N +|v_k(t, x)|^N 6.

Now, let us remark that [0, T]× B(0, R₁) =: B₁ is separable. Let Ξ1 :={(t₁, x1),(t2, x2), . . .} be a countable and dense subset in B1.

In view of the boundness of Y,the sequence{v_n(ti, xi)}_n∈N is bounded in R^N for each (ti, xi)∈Ξ1.

In particular,{v_n(t₁, x₁)}_n∈N is a bounded sequence inR^N.Therefore there exists a convergent subsequence {v_n,1(t1, x1)}_n∈N. Obviously, {v_n,1}_n∈N is a subsequence of the sequence {v_n}_n∈N.

Since {v_n,1(t2, x2)}_n∈N is a bounded sequence in R^N, then there exists a convergent subsequence {v_n,2(t₂, x₂)}_n∈N and {v_n,2}_n∈N is a subsequence of the sequence{v_n,1}_n∈N.

Proceeding in this way, we will define an infinite matrix as follows:

v_1,1(t₁, x₁) v_2,1(t₁, x₁) v_3,1(t₁, x₁) . . . v1,2(t2, x2) v2,2(t2, x2) v3,2(t2, x2) . . . v_1,3(t₃, x₃) v_2,3(t₃, x₃) v_3,3(t₃, x₃) . . . . . . .

From the Cantor diagonal procedure it follows that the sequence{v_k,k}_k∈N is characterized by the fact that the sequence {v_k,k(ti, xi)}_k∈N converges in every point (t_i, x_i)∈Ξ₁

Hereafter we denote v_k¹:=v_k,k.

Now, we take ₂ = ¹₂. There exists a constant R₂ > 0 ≥ R₁ such that

|v_k¹(t, x)|^N < ₂² for each k ∈ N, t ∈ [0, T], x ∈ R^m \B(0, R2). In [0, T]× B(0, R₂) =:B₂,we take a dense and countable subset Ξ₂ such that Ξ₂∩B₁ = Ξ1.Proceeding as above, we choose such a subsequence{v_k²}_k∈N of the sequence {v¹_k}_k∈N which converges in every point of Ξ2.

Repeating the previous procedure with respect to _i = ¹_i for i= 3,4, . . . , we obtain an infinite matrix of functions

v₁¹ v¹₂ v₃¹ . . . v₁² v²₂ v₃² . . . v₁³ v³₂ v₃³ . . . . . . .

Using the Cantor procedure once again, we choose a subsequence {v^k_k}_k∈N of the sequence {v_n}n∈N at every point of S

n∈N Ξ_n.

Now, we prove that the sequence {v^k_k}_k∈N is a Cauchy sequence in C_N^∞(Ω).

Let >0 be arbitrary. By virtue of an assumption that{v_n}_n∈N are equicon- tinuous, there exists δ >0 such that for all k = 1,2, . . .

|v_k^k(t, x)−v_k^k(s, y)|^N <

3 if only max{|t−s|, |x−y|^m}< δ.

There exists j0 such that j0 = _j¹

0 < .

From the (?)–condition it follows that there exists a constant R_j

0 >0 such that |v_n(t, x)|^N < ₂¹ follows for each n∈ N, t∈[0, T], x∈R^m\B_j

0. Thus, for each t∈[0, T], x∈R^m\B_j

0, p, q > k₀,there is

(4.1) |v^p_p(t, x)−v^q_q(t, x)|^N 6|v_p^p(t, x)|^N +|v^q_q(t, x)|^N 6j0 < .

ByBi we denote an open ball with center at (ti, xi)∈Ξj0 and radiusδ >0.

Then Bj0 ⊂S∞

i=1B_i^δ and, moreover

(4.2) |v^k_k(t, x)−v_k^k(t_i, x_i)|^N 6

3 for (t, x)∈B_i^δ.

By the Borel-Lebesque theorem there exists i₀ ∈ N such thatB_j0 ⊂Si0

i=1B_i^δ. Since the sequence {v^k_k(t_i, x_i)}_k∈N converges for i= 1,2, . . . , i₀,it follows that there exists a constant k₀ such that

(4.3) |v_p^p(ti, xi)−v_q^q(ti, xi)|^N 6

3 forp, q > k0, i= 1, . . . , i0.

If x ∈ B(0, Rj0), then there exists i 6 i0, such that (t, x) ∈ B_i^δ. Hence, applying (4.2) and (4.3), we get

(4.4) |v^p_p(t, x)−v^q_q(t, x)|^N 6

|v_p^p(t, x)−v_p^p(t_i, x_i)|^N +|v_p^p(t_i, x_i)−v^q_q(t_i, x_i)|^N +|v^q_q(t_i, x_i)−v_q^q(t, x)|^N <

for p, q > k₀,where k₀ is independent of (t, x).

Then owing to (4.1) and (4.4), we discover that {v^k_k}_k∈N is a Cauchy sequence in C_N^∞(Ω), which in view of the completeness of the space implies the convergence of {v_k^k}_k∈N inCB_N^∞(Ω).Thus, since Y is closed, the limit of {v^k_k}_k∈N belongs to Y,which completes the proof of Lemma 4.

5. Proof of the theorem. In this section let us finally give a proof of the main result of the paper.

To begin with, we remark that proving that there exists a C–solution of problem (1), (2) is equivalent to showing that the operator TF:CB_S^∞(Ω^τ)→ CB_S^∞(Ω^τ) has a fixed point. We will prove it on the basis of the Schauder fixed point theorem.

For fixed M > 0 in CB_S^∞(Ω^τ) we consider a closed ball B^τ(0, M) =:B^τ with center at 0 and radius M. Obviously, B^τ is a closed and convex set τ ∈(0, T].

Let us first demonstrate that TF(B^τ) ⊂B^τ for a sufficiently smallτ. Let w∈B^τ. It is easy to see that

|TⁱFⁱ[w](t, x)|=

Z

R^m

Γⁱ(t, x; 0, ξ)ϕⁱ(ξ)dξ+

t

Z

0

Z

R^m

Γⁱ(t, x;τ⁰, ξ)Fⁱ[w](τ⁰, ξ)dξdτ⁰

6kϕⁱk_R^m Z

R^m

|Γⁱ(t, x; 0, ξ)|dξ+kFⁱ[w]k_Ω^τ

t

Z

0

Z

R^m

|Γⁱ(t, x;τ⁰, ξ)|dξdτ⁰ 6Ckϕⁱk_Rm+CτkFⁱ[w]k_Ωτ.

This immediately implies

kTⁱFⁱ[w]k_Ω^τ 6Ckϕⁱk_R^m+CτkFⁱ[w]k_Ω^τ. Next, we note that kϕk^Σ

R^m <∞ and kF[w]k^Σ_Ωτ <∞ for eachw∈B^τ.This is a direct consequence of assumptions (F₁), (F₂), (ϕ₁) and (ϕ₂).These facts instantly lead to the following estimate

kTF[w]k^Σ_Ωτ =

∞

X

i=1

1

QⁱkTⁱFⁱ[w]k_Ωτ

6C

∞

X

i=1

1

Qⁱkϕⁱk_Rm+Cτ

∞

X

i=1

1

QⁱkFⁱ[w]k_Ω^τ =Ckϕk^Σ

R^m+CτkF[w]k^Σ_Ωτ. If

0< τ 6min

M−Ckϕk^Σ

R^m

CM1

, T

:=τ^∗, where M₁ := sup

w∈B^T

kF[w]k^Σ_Ωτ (due to assumption (F₂),we see that M₁ <∞), then we readily obtain kTF[w]k^Σ_Ωτ 6M.

Now, let us check thatTF:B^τ →B^τ is the continuous operator. SinceF is a continuous operator in B^τ, it is enough to prove thatT is continuous on F(B^τ). To verify this assertion, fix > 0 and z = (z¹, z², . . .) ∈ F(B^τ). Let z= (z¹, z², . . .)∈F(B^τ) be such thatkz−zk^Σ_Ωτ < δ:= _Cτ .Then

|Tⁱ[zⁱ](t, x)−Tⁱ[zⁱ](t, x)|

6

t

Z

0

Z

R^m

|Γⁱ(t, x;τ, ξ)||zⁱ(τ, ξ)−zⁱ(τ, ξ)|dξdτ 6Cτkzⁱ−zⁱk_Ω^τ