Opuscula Math. 35, no. 6 (2015), 935–956
http://dx.doi.org/10.7494/OpMath.2015.35.6.935
Opuscula Mathematica
EXISTENCE, UNIQUENESS AND ESTIMATES
OF CLASSICAL SOLUTIONS
TO SOME EVOLUTIONARY SYSTEM
Lucjan Sapa
Communicated by Mirosław Lachowicz
Abstract. The theorem of the local existence, uniqueness and estimates of solutions in
Hölder spaces for some nonlinear differential evolutionary system with initial conditions is
formulated and proved. This system is composed of one partial hyperbolic second-order
equation and an ordinary subsystem with a parameter. In the proof of the theorem we
use the Banach fixed-point theorem, the Arzeli-Ascola lemma and the integral form of the
differential problem.
Keywords: hyperbolic wave equation, telegraph equation, system of nonlinear equations,
existence, uniqueness and estimates of solutions, Hölder space.
Mathematics Subject Classification: 35M31, 35A09, 35A01, 35A02, 35B45.
1. INTRODUCTION
Let functions f : [0, T ] × R2+n → R, g = (g1 , . . . , gn ) : [0, T ] × R2+n → Rn ,
u0 , u1 : R → R, v0 = (v01 , . . . , v0n ) : R → Rn and a constant c ≥ 0 be given. Consider
a nonlinear second-order partial differential system of (1 + n) equations of the form
(
utt − uxx + cut = f (t, x, u, v) ,
vt = g (t, x, u, v) ,
(t, x) ∈ [0, T ] × R,
(t, x) ∈ [0, T ] × R
(1.1)
with the initial conditions
c AGH University of Science and Technology Press, Krakow 2015
935
936
Lucjan Sapa


u (0, x) = u0 (x), x ∈ R,
v (0, x) = v0 (x), x ∈ R,


ut (0, x) = u1 (x), x ∈ R,
(1.2)
where v = (v1 , . . . , vn ). The equations are weakly coupled. If c > 0, then the first
hyperbolic wave equation in (1.1) is called the telegraph equation. The others equations in (1.1) are of first-order with a space parameter x. Our results are true if we
consider the first equation in (1.1) only with f not depending on v (see examples in
Remark 4.4).
Some information especially about the maximum principles and the existence of
time-periodic bounded weak solutions for the wave or telegraph equations is given
in [12, 16].
Physical motivation of system (1.1) with c = 1 and f, g of a special form together
with a construction of the solitary waves solutions and their stability are given in [13].
The existence, uniqueness and continuous dependence on the initial values of global
classical solutions to a similar system but with the parabolic leading equation instead
of our telegraph or wave equations were studied by J. Evans and N. Shenk [6, 7].
Moreover, J. Evans in [7–10] considered a stability in the suitable sense of stationary
and traveling wave classical solutions to such systems. Those systems describe for
example the dynamics of a nerve impulse in axons and they cover in particular the
Hodgkin-Huxley system. Similar evolutionary systems appear also in [15]. In [11]
there is described a connection between fast and slow waves in the FitzHugh-Nagumo
system. Realistic view of wave mechanics was first proposed by de Broglie [1]. In his
inspiring work Madelung related the linear Schrödinger equation to the hydrodynamic
type system [14]. The various aspects of the hydrodynamic [2] and mechanistic [5]
formulations of the nonlinear Schrödinger equation are still at the center of interest.
These formulations lead to systems composed of the second or the third order partial
differential evolution equations and the first order ones.
In this paper we consider local in time bounded together with their first derivatives
1+α
1+α
solutions (u, v), u ∈ Cloc
([0, τ ] × R, R) ∩ C 2 ([0, τ ] × R, R), v ∈ Cloc
([0, τ ] × R, Rn )
1+α
1+α
of the initial problem (1.1), (1.2), where Cloc ([0, τ ]× R, R) and Cloc ([0, τ ] × R, Rn )
1+α
(α ∈ (0, 1]) are some Hölder spaces (see Section 2). If α = 1, then Cloc
= C 1+α and
they are global Lipschitz spaces. The main result of the paper is a theorem on the
existence, uniqueness and estimates of such solutions. The estimates can be given a
priori and they depend on the estimates of the initial conditions and the right-hand
sides of equations. In the proof of the theorem we use the Banach fixed-point theorem,
the Arzeli-Ascola lemma and the integral form of the differential problem. The similar
construction of the proof was used by T. Człapiński [3, 4] for quasilinear hyperbolic
partial differential functional equations of the first order.
The paper is organized in the following way. In Section 2 a notation is introduced
and some definitions and assumptions are formulated. The integral system equivalent
under some assumptions to the initial differential problem (1.1), (1.2) is given in
Section 3. In Section 4 the theorem on the existence, uniqueness and estimates of
solutions to (1.1), (1.2) is proved.
937
Existence, uniqueness and estimates of classical solutions. . .
2. NOTATION, DEFINITIONS AND ASSUMPTIONS
Let k · k be the maximum norm in Rd , i.e.
kyk = maxi=1,...,d |yi |,
(2.1)
d
where y ∈ Rd . In the space of bounded continuous functions Cb Ω, R we define the
supremum norm
kzk0 = sup {kz(ω)k : ω ∈ Ω} ,
(2.2)
where z ∈ Cb Ω, Rd , Ω = R or Ω ⊂ R2 . Obviously Cb Ω, Rd , k · k0 is the Banach
space. The space of bounded continuous functions together with their first derivatives we denote
by Cb1 Ω, Rd and together with their first and second derivatives by
2
d
Cb Ω, R . By the symbol
kzk1 = kzk0 + kzt k0 + kzx k0
we denote the norm in Cb1 Ω, Rd .
For any z ∈ Cb Ω, Rd and α ∈ (0, 1], let
[z]H,α = sup z(t, x) − z(t, x) [|t − t| + |x − x|]−α : (t, x), (t, x) ∈ Ω .
(2.3)
(2.4)
Note that [z]H,α < ∞ is the smallest Hölder constant for the function z and the
exponent α, it is usually called the Hölder coefficient.
If α = 1, then it is the Lips
k+α
d
chitz coefficient.
The
Hölder
space
C
Ω,
R
,
k
=
0,
1, is the space of functions
k
d
z ∈ Cb Ω, R with the finite norm
kzk0+α = kzk0 + [z]H,α
if k = 0,
kzk1+α = kzk1 + [zt ]H,α + [zx ]H,α
if k = 1.
(2.5)
(2.6)
If Ω = R, then t and zt in the above definitions do not appear.
Let ∆(x, τ ) ⊂ R × [0, τ ], x ∈ R be any isosceles triangle with the vertices (x − τ, 0),
1+α
(x, τ ), (x + τ, 0). We denote by Cloc
([0, τ ] × R, Rd ) the Hölder space of functions
1
d
z ∈ Cb ([0, τ ] × R, R ) such that z ∈ C 1+α (∆(x, τ ), Rd ) for any ∆(x, τ ) with the
constants
[zt ]H,α,∆(x,τ )
= sup zt (t, x) − zt (t, x) [|t − t| + |x − x|]−α : (t, x), (t, x) ∈ ∆(x, τ ) ,
[zx ]H,α,∆(x,τ )
= sup zx (t, x) − zx (t, x) [|t − t| + |x − x|]−α : (t, x), (t, x) ∈ ∆(x, τ )
(2.7)
independent of ∆(x, τ ). We say in this case that zt , zx are uniformly Hölder continuous
1+α
in [0, τ ] × R. It is clear that if α = 1, then Cloc
([0, τ ] × R, Rd ) = C 1+α ([0, τ ] × R, Rd ).
The following assumption on the functions u0 , v0 , u1 will be needed.
Assumption H1 [u0 , v0 , u1 ]. Suppose that u0 ∈ C 1+α (R, R) ∩ C 2 (R, R), v0 ∈
C 1+α (R, Rn ), u1 ∈ C 0+α (R, R) ∩ C 1 (R, R) and that
938
Lucjan Sapa
ku0 k0 ≤ Λ1 ,
(1)
(2)
[u1 ]H,α ≤ Λ1 ,
kv0 k0 ≤ Λ2 ,
(1)
ku1 k0 ≤ Λ1 ,
k(u0 )x k0 ≤ Λ1 ,
(2.8)
(2)
[(u0 )x ]H,α ≤ Λ1 ,
(1)
kg (0, ·, u0 (·), v0 (·))k0 ≤ Λ2 ,
(2)
(1)
k(v0 )x k0 ≤ Λ2 ,
(2)
[g (0, ·, u0 (·), v0 (·))]H,α ≤ Λ2 ,
[(v0 )x ]H,α ≤ Λ2 ,
(j)
where Λi , Λi , i, j = 1, 2, are some non-negative constants.
Definition 2.1. Suppose that Assumption H1 [u0 , v0 , u1 ] is satisfied and given
(j)
(j)
(j)
non-negative constants Qi , Qi , i, j = 1, 2, such that Qi ≥ Λi , Qi ≥ Λi
1,α
we will denote by Cb,τ (Q), where τ ∈ (0, T ], the set of functions (u, v) ∈
1+α
Cloc
[0, τ ] × R, R1+n , v = (v1 , . . . , vn ), with the properties:
u (0, x) = u0 (x),
v (0, x) = v0 (x),
kuk0 ≤ Q1 ,
ut (0, x) = u1 (x),
(1)
kut k0 ≤ Q1 ,
(2)
[ut ]H,α,∆(x,τ ) ≤ Q1 ,
kvk0 ≤ Q2 ,
(1)
kux k0 ≤ Q1 ,
(2.10)
(2)
(1)
(2)
(2.9)
[ux ]H,α,∆(x,τ ) ≤ Q1 ,
kvt k0 ≤ Q2 ,
[vt ]H,α,∆(x,τ ) ≤ Q2 ,
x ∈ R,
(1)
kvx k0 ≤ Q2 ,
(2)
[vx ]H,α,∆(x,τ ) ≤ Q2 .
We will use the following assumptions on the functions f , g and their first derivatives.
Assumption H2 [f, g]. The functions f , g and the derivatives fx , fp , fr =
(fr1 , . . . , frn ), gx , gp , gr = (gr1 , . . . , grn ) are continuous. Moreover, there exist
(1)
(1)
non-decreasing functions Mk , Mk , H2 , H2 : R2+ → R+ , k = 1, 2, such that for
all t, t ∈ [0, T ], x, x ∈ R, (q1 , q2 ) ∈ R2+ , |p|, |p| ≤ q1 , krk, krk ≤ q2 , i = 1, . . . , n we
have
|f (t, x, p, r)| ≤ M1 (q1 , q2 ),
(2.11)
(1)
|fx (t, x, p, r)| , |fp (t, x, p, r)| , |fri (t, x, p, r)| ≤ M1 (q1 , q2 ),
(2.12)
kg(t, x, p, r)k ≤ M2 (q1 , q2 ),
kgx (t, x, p, r)k , kgp (t, x, p, r)k , kgri (t, x, p, r)k ≤
kg(t, x, p, r) − g t, x, p, r k ≤ H2 (q1 , q2 )|t − t|α ,
kgp (t, x, p, r) − gp (t, x, p, r)k ≤
(1)
H2 (q1 , q2 ) [|x
(1)
H2 (q1 , q2 ) [|x
kgri (t, x, p, r) − gri (t, x, p, r)k ≤
(1)
H2 (q1 , q2 ) [|x
kgx (t, x, p, r) − gx (t, x, p, r)k ≤
(2.13)
(1)
M2 (q1 , q2 ),
(2.14)
(2.15)
α
− x| + |p − p| + kr − rk] ,
α
− x| + |p − p| + kr − rk] ,
α
− x| + |p − p| + kr − rk] .
(2.16)
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Existence, uniqueness and estimates of classical solutions. . .
(1)∗
(1)
(1)∗
Put Mk∗ = Mk (Q1 , Q2 ), Mk
= Mk (Q1 , Q2 ), H2∗ = H2 (Q1 , Q2 ), H2
=
(j)
(Q1 , Q2 ), k = 1, 2. Some inequalities concerned Qi , Qi , i, j = 1, 2, have to be
satisfied.
Assumption H3 [Q]. Suppose that
(1)
H2
Q1 > Λ1 ,
(1)
(1)
Q1 > Λ1 ,
(2)
(2)
α ∈ (0, 1) ,
Q1 > 2Λ1 ,
(2)
Q1 > M1∗ +
c2
3 (1)
c2
(2)
Q1 + Λ1 + cΛ1 + 2Λ1 ,
4
4
2
(2.17)
α = 1,
Q2 > Λ2 ,
(1)
Q2 ≥ M2∗ ,
(2)
Q2 > H2∗ ,
(2)
Q2
(2)
Q2
(1)
(1)
(2)
(2)
Q2 > Λ2 ,
Q2 > Λ2 ,
α ∈ (0, 1) ,
h
i
(1)∗
(1)
(1)
≥ M2
1 + 2 Q1 + nQ2
+ H2∗ , α = 1,
(1)∗
(1)
(1)
(2)
> M2
1 + Q1 + nQ2 + Λ2 ,
α = 1.
(2.18)
Remark 2.2. The conditions (2.11)–(2.14) in Assumption H2 [f, g] are equivalent to
n
boundedness of f, g and fx , fp , fri , gx , gp , gri on the sets [0, T ]×R×[−q1 , q1 ]×[−q2 , q2 ]
2
for any (q1 , q2 ) ∈ R+ , i.e. globally with respect to t, x and locally with respect to p, r.
If f, g are globally Lipschitz continuous with respect to x, p, r or f, g do not depend
on t, x and are locally Lipschitz continuous with respect to p, r, then (2.12), (2.14)
hold. But the global Lipschitz continuity of f, g with respect to t, x and local one with
respect to p, r not always imply (2.12), (2.14), e.g. f (t, x, p, r) = x sin p. On the other
hand the function f (t, x, p, r) = p2 sin x is locally Lipschitz continuous with respect
to x, p only and (2.12) is true. The form of (2.11)–(2.14) is useful in applications
because we do not have to calculate the norms. It is important for example if we
consider differential equations with the polynomial right-hand sides. An analogous
situation appears in Assumption H1 [u0 , v0 , u1 ]. An analysis of the local or global
Hölder continuity of g, gx , gp , gri in conditions (2.15), (2.16) is similar as the above
one concerning the local or global Lipschitz continuity of f, g (see Remark 4.4 and
examples (4.23), (4.24)).
(j)
Remark 2.3. Note that in Assumption H3 [Q] a choice of the constants Qi , Qi ,
(j)
i, j = 1, 2, is possible for any given Λi , Λi , i, j = 1, 2. So we may give a priori the
estimates of the solution of the differential problem (1.1), (1.2), its first derivatives
and the Hölder constants of its first derivatives.
940
Lucjan Sapa
Define the constants
hc
i
1
c2
(1)
∗
S1τ =
M1 + Q1 τ 2 + Λ1 + Λ1 τ + Λ1 ,
2
4
2
c2
c (1)
c
c2
c2
(1.t)
M1∗ + Q1 τ 2 + M1∗ + Q1 + Λ1 + Λ1 τ
S1τ =
4
4
4
4
2
(2)
(1.x)
S1τ
(2.t)
S1τ
(2.x)
S1τ
(1)
+ 2−1+α Λ1 τ α + Λ1 ,
c (1)
c2
(2)
(1)
= M1∗ + Q1 + Λ1 τ + 2−1+α Λ1 τ α + Λ1 ,
4
2
2
c4
c
∗
M + Q1 τ 3−α
=
8 1
32
h
(1)∗ (1)
(1)
+ c 1 + 2−α M1∗ + 1 + 21−α M1
1 + Q1 + nQ2
(1)
c2
c3
1 + 2−α Q1 +
1 + 21−α Q1
+
4
4
(1) 2−α
c3
c2
1−α
+ Λ1 +
1+2
Λ1 τ
8
4
c (2)
c2
c2
3 (1) 1−α
(2)
∗
+ Λ1 τ + M1 + Q1 + Λ1 + cΛ1 τ
+ 2Λ1 ,
2
4
4
2
c3
(1)∗ c ∗
(1)
(1)
=
M1 + 1 + 21−α M1
1 + Q1 + nQ2 + Q1
2
8
2
(1)
c
1 + 21−α Q1 τ 2−α
+
4
c2
c2
31−α
(1)
(2)
∗
+ M1 + Q1 + Λ1 + c 1 +
Λ1 τ 1−α + 2Λ1 ,
4
4
2
(2.19)
S2τ = M2∗ τ + Λ2 ,
(1.t)
= M2∗ ,
(1.x)
= M2
S2τ
S2τ
(2.t)
S2τ
(2.x)
S2τ
(1)∗
h
(1)
(1)
1 + Q1 + nQ2
i
(1)
τ + Λ2 ,
h
i
(1)
(1)
(1)∗
21−α + 1 + 21−α Q1 + nQ2
τ 1−α + H2∗ ,
= M2
α
h
(1)
(1)
(1)∗
(1)
(1)
1 + Q1 + Q2
= H2
1 + Q1 + nQ2
i
(2)
(1)∗
(2)
Q1 + nQ2
τ
+M2
(2)
(1)∗
(1)
(1)
+ M2
1 + Q1 + nQ2 τ 1−α + Λ2 .
(2.20)
941
Existence, uniqueness and estimates of classical solutions. . .
Remark 2.4. Note that since
lim S1τ = Λ1 ,
τ →0+
(2.t)
(2.x)
lim S1τ
= lim+ S1τ
τ →0+
(2.t)
lim+ S1τ
τ →0
(1.t)
lim S1τ
τ →0+
τ →0
(2.x)
= M1∗ +
= lim+ S1τ
τ →0
lim S2τ = Λ2 ,
(2.t)
lim S2τ
(2.x)
lim S2τ
τ →0+
= H2∗ ,
τ →0
(2)
= 2Λ1
(1)
= Λ1 ,
if α ∈ (0, 1) ,
c2
3 (1)
c2
(2)
Q1 + Λ1 + cΛ1 + 2Λ1
4
4
2
τ →0+
τ →0+
(1.x)
= lim+ S1τ
(1.x)
lim S2τ
τ →0+
(2.x)
if α = 1,
(1)
= Λ2 ,
(2)
= Λ2
if α ∈ (0, 1) ,
(1)∗
(1)
(1)
(2)
= M2
1 + Q1 + nQ2 + Λ2
if α = 1,
lim S2τ
τ →0+
we may by Assumption H3 [Q] choose τ ∈ (0, T ] sufficiently small in order that
(j.t)
(j)
(j.x)
(j)
Siτ ≤ Qi , Siτ ≤ Qi , Siτ ≤ Qi , i, j = 1, 2.
3. INTEGRAL SYSTEM
In this section we give a lemma which will be crucial in our future considerations.
Consider a nonlinear integral system of (1 + n) equations of the form

i
h

Rt x+(t−s)
R

1
c2
− 2c (t−s)

u(t,
x)
=
u(s,
y)
dyds
e
f
(s,
y,
u(s,
y),
v(s,
y))
+

2
4


0 x−(t−s)



x+t
x+t

R
R
c
c

+ 12 e− 2 t
u1 (y)dy + 4c e− 2 t
u0 (y)dy
(3.1)
x−t
x−t

c

1 −2t


+2e
[u0 (x + t) + u0 (x − t)] ,
(t, x) ∈ [0, T ] × R,



Rt



(t, x) ∈ [0, T ] × R.
v(t, x) = v0 (x) + g (s, x, u(s, x), v(s, x)) ds,
0
Lemma 3.1. Under Assumptions H1 [u0 , v0 , u1 ] and H2 [f, g], the differential initial
problem (1.1), (1.2) and the integral system (3.1) are equivalent in the sense that
any solution (u, v) ∈ C 2 ([0, T ] × R, R) × C 1 ([0, T ] × R, Rn ) of (1.1), (1.2) is a solution of (3.1) and any solution (u, v) ∈ C 1 ([0, T ] × R, R1+n of (3.1) belongs to
C 2 ([0, T ] × R, R) × C 1 ([0, T ] × R, Rn ) and fulfills (1.1), (1.2).
Proof. The first equation in (1.1) is equivalent to the equation
utt − uxx + cut +
c
c2
c2
u = f (t, x, u, v) + u,
4
4
(t, x) ∈ [0, T ] × R.
Hence, the anzats u = we− 2 t transforms problem (1.1), (1.2) to the equivalent differential system
(
2
c
c
wtt − wxx = e 2 t f t, x, we− 2 t , v + c4 w, (t, x) ∈ [0, T ] × R,
(3.2)
c
vt = g t, x, we− 2 t , v ,
(t, x) ∈ [0, T ] × R
942
Lucjan Sapa
with the equivalent initial conditions


w (0, x) = u0 (x),
x ∈ R,



v (0, x) = v0 (x),
x ∈ R,




wt (0, x) = 2c u0 (x) + u1 (x), x ∈ R.
(3.3)
The continuity of f, g, u0 , v0 , u1 , the use of Riemann’s method (see [16, p. 196]) for
the first equation in (3.2) and integration of the second one imply that any solution (w, v) ∈ C 2 ([0, T ] × R, R) × C 1 ([0, T ] × R, Rn ) of (3.2), (3.3) is a solution of a
nonlinear integral system of (1 + n) equations of the form

Zt x+(t−s)
Z


c2
c
1

s
− 2c s

2
e f s, y, w(s, y)e
, v(s, y) + w(s, y) dyds

w(t, x) = 2

4


0

x−(t−s)






x+t
Z

hc
i


1


+
u
(y)
+
u
(y)
dy
0
1

2
2
x−t





1



+ [u0 (x + t) + u0 (x − t)] ,
(t, x) ∈ [0, T ] × R,


2





Zt



v(t, x) = v (x) + g s, x, w(s, x)e− 2c s , v(s, x) ds,

(t, x) ∈ [0, T ] × R.
0


(3.4)
0
c
Multiplying the first equation in (3.4) by e− 2 t we have that any solution (u, v) ∈
C 2 ([0, T ] × R, R) × C 1 ([0, T ] × R, Rn ) of the differential initial problem (1.1), (1.2)
is a solution of the integral system (3.1).
On the other hand, let (u, v) ∈ C 1 [0, T ] × R, R1+n be a solution of (3.1). Put
for simplicity
A(t, x, s) = (s, x + (t − s), u(s, x + (t − s)), v(s, x + (t − s))) ,
B(t, x, s) = (s, x − (t − s), u(s, x − (t − s)), v(s, x − (t − s))) .
From the regularity of f, g, u0 , v0 , u1 and differentiation of the integrals in (3.1) we
have
943
Existence, uniqueness and estimates of classical solutions. . .
c
ut (t, x) = −
4
x+(t−s)
Z
Zt
c
c2
e− 2 (t−s) f (s, y, u(s, y), v(s, y)) + u(s, y) dyds
4
0 x−(t−s)
1
+
2
Zt
c
c2
e− 2 (t−s) f (A(t, x, s)) + u(s, x + (t − s)) ds
4
0
1
+
2
Zt
c
c2
e− 2 (t−s) f (B(t, x, s)) + u(s, x − (t − s)) ds
4
0
x+t
Z
1 c
u1 (y)dy + e− 2 t [u1 (x + t) + u1 (x − t)]
2
c c
− e− 2 t
4
x−t
c2 c
− e− 2 t
8
x+t
Z
1 c
u0 (y)dy + e− 2 t [(u0 )x (x + t) − (u0 )x (x − t)] ,
2
x−t
c2
utt (t, x) =
8
x+(t−s)
Z
Zt
e
− 2c (t−s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y) dyds
4
0 x−(t−s)
c
−
2
Zt
e
− 2c (t−s)
c2
f (A(t, x, s)) + u(s, x + (t − s)) ds
4
e
− 2c (t−s)
c2
f (B(t, x, s)) + u(s, x − (t − s)) ds
4
e
− 2c (t−s)
fx (A(t, x, s)) + fp (A(t, x, s))ux (s, x + (t − s))
0
c
−
2
Zt
0
1
+
2
Zt
0
+
n
X
i=1
fri (A(t, x, s))(vi )x (s, x + (t − s)) +
c2
ux (s, x + (t − s)) ds
4
944
Lucjan Sapa
1
−
2
Zt
e
− 2c (t−s)
fx (B(t, x, s)) + fp (B(t, x, s))ux (s, x − (t − s))
0
n
X
c2
+
fri (B(t, x, s))(vi )x (s, x − (t − s)) + ux (s, x − (t − s)) ds
4
i=1
+ f (t, x, u(t, x), v(t, x)) +
+
c2
u(t, x)
4
x+t
Z
c c
u1 (y)dy − e− 2 t [u1 (x + t) + u1 (x − t)]
2
c2 − c t
e 2
8
x−t
1 c
+ e− 2 t [(u1 )x (x + t) − (u1 )x (x − t)]
2
x+t
Z
c3 − c t
c2 c
2
+ e
u0 (y)dy − e− 2 t [u0 (x + t) + u0 (x − t)]
16
8
x−t
c c
− e− 2 t [(u0 )x (x + t) − (u0 )x (x − t)]
4
1 c
+ e− 2 t [(u0 )xx (x + t) + (u0 )xx (x − t)] ,
2
1
uxx (t, x) =
2
Zt
c
e− 2 (t−s) fx (A(t, x, s)) + fp (A(t, x, s))ux (s, x + (t − s))
0
+
n
X
fri (A(t, x, s))(vi )x (s, x + (t − s)) +
i=1
1
−
2
Zt
e
− 2c (t−s)
c2
ux (s, x + (t − s)) ds
4
fx (B(t, x, s)) + fp (B(t, x, s))ux (s, x − (t − s))
0
n
X
c2
+
fri (B(t, x, s))(vi )x (s, x − (t − s)) + ux (s, x − (t − s)) ds
4
i=1
1 c
+ e− 2 t [(u1 )x (x + t) − (u1 )x (x − t)]
2
c c
+ e− 2 t [(u0 )x (x + t) − (u0 )x (x − t)]
4
1 c
+ e− 2 t [(u0 )xx (x + t) + (u0 )xx (x − t)] ,
2
vt (t, x) = g (t, x, u(t, x), v(t, x)) .
It is clear that (u, v) ∈ C 2 ([0, T ] × R, R) × C 1 ([0, T ] × R, Rn ) and fulfills (1.1), (1.2).
Existence, uniqueness and estimates of classical solutions. . .
945
4. THE MAIN RESULT
Theorem 4.1. If Assumptions H1 [u0 , v0 , u1 ], H2 [f, g], H3 [Q] are satisfied, then there
1,α
is τ ∈ (0, T ] such that problem (1.1), (1.2) has an unique solution (u, v) ∈ Cb,τ
(Q)
2
and u ∈ C ([0, τ ] × R, R).
1,α
Proof. Define the operator H = (F, G) on Cb,τ
(Q) by the formula
1
F [u, v](t, x) =
2
Zt
x+(t−s)
Z
e
− 2c (t−s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y) dyds
4
0 x−(t−s)
1 c
+ e− 2 t
2
x+t
x+t
Z
Z
c −ct
2
u1 (y)dy + e
u0 (y)dy
4
x−t
(4.1)
x−t
1 c
+ e− 2 t [u0 (x + t) + u0 (x − t)] ,
2
Zt
G[u, v](t, x) = v0 (x) +
g (s, x, u(s, x), v(s, x)) ds,
(4.2)
0
1,α
1,α
where (u, v) ∈ Cb,τ
(Q), (t, x) ∈ [0, τ ] × R. It is obvious that (u, v) ∈ Cb,τ
(Q) is a
solution of the integral system (3.1) and consequently, under additional assumptions,
a solution of the initial differential problem (1.1), (1.2) (see Lemma 3.1) if and only
if its a fix-point of H.
1,α
Note that by the Arzeli-Ascola lemma, the set Cb,τ
(Q) is closed in the Banach
1+n
space Cb [0, τ ] × R, R
, k · k0 . We will show that there is τ ∈ (0, T ] such that
1,α
H maps Cb,τ
(Q) into itself and it is a contraction in the complete metric space
1,α
(Cb,τ
(Q), d) with the metric d generated by the norm k · k0 . Then we will use the
Banach fixed-point theorem.
Put for simplicity
A(t, x, s) = (s, x + (t − s), u(s, x + (t − s)), v(s, x + (t − s))) ,
B(t, x, s) = (s, x − (t − s), u(s, x − (t − s)), v(s, x − (t − s))) ,
C(x, s) = (s, x, u(s, x), v(s, x)) .
946
Lucjan Sapa
Definitions (4.1), (4.2) and Assumptions H1 [u0 , v0 , u1 ], H2 [f, g] imply
Zt
c
Ft [u, v](t, x) = −
4
x+(t−s)
Z
c
c2
e− 2 (t−s) f (s, y, u(s, y), v(s, y)) + u(s, y) dyds
4
0 x−(t−s)
1
+
2
Zt
e
− 2c (t−s)
c2
f (A(t, x, s)) + u(s, x + (t − s)) ds
4
0
+
1
2
Zt
c
c2
e− 2 (t−s) f (B(t, x, s)) + u(s, x − (t − s)) ds
4
(4.3)
0
x+t
Z
1 c
u1 (y)dy + e− 2 t [u1 (x + t) + u1 (x − t)]
2
c c
− e− 2 t
4
x−t
c2 c
− e− 2 t
8
x+t
Z
1 c
u0 (y)dy + e− 2 t [(u0 )x (x + t) − (u0 )x (x − t)] ,
2
x−t
1
Fx [u, v](t, x) =
2
Zt
− 2c (t−s)
e
c2
f (A(t, x, s)) + u(s, x + (t − s)) ds
4
0
1
−
2
Zt
e
− 2c (t−s)
c2
f (B(t, x, s)) + u(s, x − (t − s)) ds
4
(4.4)
0
1 c
+ e− 2 t [u1 (x + t) − u1 (x − t)]
2
c −ct
1 c
+ e 2 [u0 (x + t) − u0 (x − t)] + e− 2 t [(u0 )x (x + t) + (u0 )x (x − t)] ,
4
2
Gt [u, v](t, x) = g (t, x, u(t, x), v(t, x)) ,
(4.5)
Zt
Gx [u, v](t, x) = (v0 )x (x) +
[gx (C(x, s)) + gp (C(x, s)) ux (s, x)
(4.6)
0
+
n
X
gri (C(x, s)) (vi )x (s, x)] ds,
i=1
1,α
where (u, v) ∈ Cb,τ
(Q), (t, x) ∈ [0, τ ] × R.
We prove firstly that for a sufficiently small τ ∈ (0, T ] the operator H
1,α
1,α
maps Cb,τ
(Q) into itself. Let (u, v) ∈ Cb,τ
(Q) be fixed. Obviously, H[u, v] ∈
1
1+n
Cb [0, τ ] × R, R
and for x ∈ R
F [u, v] (0, x) = u0 (x),
G[u, v] (0, x) = v0 (x),
Ft [u, v] (0, x) = u1 (x).
(4.7)
947
Existence, uniqueness and estimates of classical solutions. . .
It is enough to show that there exists τ ∈ (0, T ] such that for each (t, x) ∈ [0, τ ] × R
(1)
|Fx [u, v](t, x)| ≤ Q1 ,
(1)
kGx [u, v](t, x)k ≤ Q2 ,
|F [u, v](t, x)| ≤ Q1 ,
|Ft [u, v](t, x)| ≤ Q1 ,
kG[u, v](t, x)k ≤ Q2 ,
kGt [u, v](t, x)k ≤ Q2 ,
(1)
(4.8)
(1)
(4.9)
and for an arbitrarily fixed triangle ∆(X, τ ), X ∈ R and for each two points
(t, x), (t, x) ∈ ∆(X, τ )
Ft [u, v](t, x) − Ft [u, v](t, x) ≤ Q(2) [|t − t| + |x − x|]α ,
1
(4.10)
Fx [u, v](t, x) − Fx [u, v](t, x) ≤ Q(2) [|t − t| + |x − x|]α ,
1
Gt [u, v](t, x) − Gt [u, v](t, x) ≤ Q(2) [|t − t| + |x − x|]α ,
2
(4.11)
Gx [u, v](t, x) − Gx [u, v](t, x) ≤ Q(2) [|t − t| + |x − x|]α .
2
From Assumptions H1 [u0 , v0 , u1 ], H2 [f, g], the properties of integrals and definitions
(2.19), (2.20) we have
(1.t)
(1.x)
|F [u, v](t, x)| ≤ S1τ , |Ft [u, v](t, x)| ≤ S1τ , |Fx [u, v](t, x)| ≤ S1τ
(1.t)
,
(4.12)
(1.x)
(4.13)
kG[u, v](t, x)k ≤ S2τ , kGt [u, v](t, x)k ≤ S2τ , kGx [u, v](t, x)k ≤ S2τ
for (t, x) ∈ [0, τ ] × R. Let ∆(X, τ ) be fixed and (t, x), (t, x) ∈ ∆(X, τ ). Using the mean
value theorem for the functions:
x+(t−s)
Z
Zt
f1 (t, x) =
e
− 2c (t−s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y) dyds,
4
0 x−(t−s)
Zt
f2 (t, x) =
e
− 2c (t−s)
c2
f (A(t, x, s)) + u(s, x + (t − s)) ds,
4
e
− 2c (t−s)
c2
f (B(t, x, s)) + u(s, x − (t − s)) ds,
4
0
Zt
f3 (t, x) =
0
− 2c t
x+t
Z
u
e1 (t, x) = e
u1 (y)dy,
x−t
u
e0 (t, x) = e
− 2c t
x+t
Z
u0 (y)dy,
x−t
u0
948
Lucjan Sapa
and the relations: |t − t| = |t − t|1−α |t − t|α , |x − x| = |x − x|1−α |x − x|α , |t − t| ≤ τ ,
|x − x| ≤ 2τ we obtain
Ft [u, v](t, x) − Ft [u, v](t, x)
x+(t−s)
Z
Zt
c ≤ 4
e
− 2c (t−s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y)
4
0 x−(t−s)
Zt
x+(t−s)
Z
Z
−
e
0 x−(t−s)
− 2c (t−s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y) 4
Zt
c
1 c2
+ e− 2 (t−s) f (A(t, x, s)) + u(s, x + (t − s)) ds
2
4
0
Zt
−
e
− 2c
0
2
c
(t−s) f A t, x, s + u s, x + t − s
ds
4
Zt
1 c2
− 2c (t−s)
+ e
f (B(t, x, s)) + u(s, x − (t − s)) ds
2
4
0
Zt
2
c
− e (t−s) f B t, x, s + u s, x − t − s
ds
4
0
x+t
x+t
Z
Z
c
c −ct
−
t
u1 (y)dy + e 2
u1 (y)dy − e 2
4
x−t
x−t
c
c
c
1 c
+ e− 2 t u1 (x + t) − e− 2 t u1 (x + t) + e− 2 t u1 (x + t) − e− 2 t u1 (x + t)
2
c
c
c
1 c
+ e− 2 t u1 (x − t) − e− 2 t u1 (x − t) + e− 2 t u1 (x − t) − e− 2 t u1 (x − t)
2 x+t
x+t
Z
Z
2 c
c −ct
+
u0 (y)dy − e− 2 t
u0 (y)dy e 2
8 x−t
x−t
c
1 c
+ e− 2 t (u0 )x (x + t) − e− 2 t (u0 )x (x + t)
2
c
c
+ e− 2 t (u0 )x (x + t) − e− 2 t (u0 )x (x + t)
c
1 c
+ e− 2 t (u0 )x (x − t) − e− 2 t (u0 )x (x − t)
2
c
c
+ e− 2 t (u0 )x (x − t) − e− 2 t (u0 )x (x − t)
− 2c
(4.14)
Existence, uniqueness and estimates of classical solutions. . .
Zt1
c c
≤ −
4 2
x1 +(t
Z 1 −s)
e
− 2c (t1 −s)
c2
f (s, y, u(s, y), v(s, y)) + u(s, y) ds
4
0 x1 −(t1 −s)
Zt1
+
e
− 2c (t1 −s)
c2
f (A(t1 , x1 , s)) + u (s, x1 + (t1 − s)) ds
4
e
− 2c (t1 −s)
c2
f (B(t1 , x1 , s)) + u (s, x1 − (t1 − s)) ds|t − t|
4
0
Zt1
+
0
Zt1
c
c c2
+ e− 2 (t1 −s) f (A(t1 , x1 , s)) + u (s, x1 + (t1 − s)) ds
4
4
0
Zt1
−
e
0
− 2c (t1 −s)
c2
f (B(t1 , x1 , s)) + u (s, x1 − (t1 − s)) ds|x − x|
4
Zt2 c2
1 c − c (t2 −s)
f (A(t2 , x2 , s)) + u(s, x2 + (t2 − s))
+ − e 2
2
2
4
0
c
+e− 2 (t2 −s) fx (A(t2 , x2 , s)) + fp (A(t2 , x2 , s))ux (s, x2 + (t2 − s))
+
n
X
fri (A(t2 , x2 , s))(vi )x (s, x2 + (t2 − s))
i=1
2
c
ux (s, x2 + (t2 − s)) ds
4
c2
+ f (t2 , x2 , u(t2 , x2 ), v(t2 , x2 )) + u(t2 , x2 )|t − t|
4
+
Zt2
"
c
1 + e− 2 (t2 −s) fx (A(t2 , x2 , s)) + fp (A(t2 , x2 , s))ux (s, x2 + (t2 − s))
2
0
+
n
X
fri (A(t2 , x2 , s))(vi )x (s, x2 + (t2 − s))
i=1
# c2
+ ux (s, x2 + (t2 − s)) ds|x − x|
4
949
950
Lucjan Sapa
t
Z3 c c
1 c2
− e− 2 (t3 −s) f (B (t3 , x3 , s)) + u (s, x3 − (t3 − s))
+ 2
2
4
0
−e
− 2c (t3 −s)
[fx (B (t3 , x3 , s)) + fp (B (t3 , x3 , s)) ux (s, x3 − (t3 − s))
n
X
c2
+
fri (B (t3 , x3 , s)) (vi )x (s, x3 − (t3 − s)) + ux (s, x3 − (t3 − s))
4
i=1
##
ds
c2
+f (t3 , x3 , u (t3 , x3 ) , v (t3 , x3 )) + u (t3 , x3 ) |t − t|
4
t
Z3
c
1 + e− 2 (t3 −s) [fx (B (t3 , x3 , s)) + fp (B (t3 , x3 , s)) ux (s, x3 − (t3 − s))
2
0
# c2
fri (B (t3 , x3 , s)) (vi )x (s, x3 − (t3 − s)) + ux (s, x3 − (t3 − s)) ds
+
4
i=1
n
X
× |x − x|
"
x4Z+t4
c
c c − c t4
+
u1 (y)dy + e− 2 t4 [u1 (x4 + t4 ) + u1 (x4 − t4 )] |t − t|
− e 2
4
2
x4 −t4
#
+e
− 2c t4
|u1 (x4 + t4 ) − u1 (x4 − t4 )| |x − x|
1 c 1 c − c t5
− e 2 (t − t) |u1 (x + t)| + e− 2 t u1 (x + t) − u1 (x + t)
2
2
2
1 c c
1 c + − e− 2 t5 (t − t) |u1 (x − t)| + e− 2 t u1 (x − t) − u1 (x − t)
2
2
2
"
x6Z+t6
c
c2 c − c t6
+
u0 (y)dy + e− 2 t6 [u0 (x6 + t6 ) + u0 (x6 − t6 )] |t − t|
− e 2
8
2
+
x6 −t6
#
+e
− 2c t6
|u0 (x6 + t6 ) − u0 (x6 − t6 )| |x − x|
1 c − c t5
− e 2 (t − t) |(u0 )x (x + t)| +
2
2
1 c c
+ − e− 2 t5 (t − t) |(u0 )x (x − t)| +
2
2
+
1 − c t e 2 (u0 )x (x + t) − (u0 )x (x + t)
2
1 − c t e 2 (u0 )x (x − t) − (u0 )x (x − t)
2
951
Existence, uniqueness and estimates of classical solutions. . .
≤
c c
c2
c2
M1∗ + Q1 τ 2 τ 1−α + 2 M1∗ + Q1 τ τ 1−α
4 2
4
4
c2
1 c
c2
∗
1−α
∗
+2 M1 + Q1 τ (2τ )
M1 + Q1 τ τ 1−α
+2·
4
2 2
4
c2 (1)
c2
(1)∗
(1)
(1)
+ M1
1 + Q1 + nQ2 τ τ 1−α + Q1 τ τ 1−α + M1∗ + Q1 τ 1−α
4
4
c2 (1)
(1)∗
(1)
(1)
+M1
1 + Q1 + nQ2 τ (2τ )1−α + Q1 τ (2τ )1−α
4
i c
c h c (1)
(1)
(2)
(1)
(2)
Λ1 2τ + 2Λ1 τ 1−α + Λ1 (2τ )α (2τ )1−α + Λ1 τ 1−α + Λ1
+
4 2
2
i c
c2 h c
(1)
(1)
(2)
Λ1 2τ + 2Λ1 τ 1−α + Λ1 2τ (2τ )1−α + Λ1 τ 1−α + Λ1
+
8
2
2
× [|t − t| + |x − x|]α
(2.t)
= S1τ [|t − t| + |x − x|]α .
The points (ti , xi ) ∈ ∆(X, τ ), i = 1, . . . , 5, are intermediate ones. Making similar
calculations as above we get
Fx [u, v](t, x) − Fx [u, v](t, x)
nh c
i
(1)∗
(1)
(1)
≤
M1∗ + M1
1 + Q1 + nQ2
τ + M1∗ τ 1−α
2
i
c2 h c
(1)∗
(1)
(1)
(1)
+ M1
1 + Q1 + nQ2 (2τ )1−α τ +
Q1 + Q1 τ + Q1 τ 1−α
4
2
(4.15)
h
i
2
c (1)
c (1) 1−α c (1)
c
(2)
1−α
1−α
1−α
+ Q1 (2τ )
τ + Λ1 + Λ1 τ
+
Λ (3τ )
+ Λ1 τ
4
2
2 1
2
o
c
(2)
(1)
+Λ1 + Λ1 τ 1−α [|t − t| + |x − x|]α
2
(2.x)
= S1τ
[|t − t| + |x − x|]α ,
h
n
Gt [u, v](t, x) − Gt [u, v](t, x) ≤ H2∗ + M (1)∗ (2τ )1−α + Q(1) τ 1−α + (2τ )1−α
2
1
io
(1)
+nQ2 τ 1−α + (2τ )1−α
[|t − t| + |x − x|]α
(2.t)
= S2τ [|t − t| + |x − x|]α
(4.16)
for a fixed ∆(X, τ ) and (t, x), (t, x) ∈ ∆(X, τ ). Because of the non-sufficient regularity
of g we can not use the mean value theorem for g1 (t, x) = Gx [u, v](t, x). But using
the additivity of an integral we have
952
Gx [u, v](t, x) − Gx [u, v](t, x)
Lucjan Sapa
Zt
Zt
≤ k(v0 )x (x) − (v0 )x (x)k + gx (C(x, s)) ds − gx (C(x, s)) ds
0
0
Zt
Zt
+ gp (C(x, s))ux (s, x)ds − gp (C(x, s))ux (s, x)ds
0
+
0
n Zt
X
i=1
Zt
gri (C(x, s))(vi )x (s, x)ds −
0
gri (C(x, s))(vi )x (s, x)ds
0
Zt
Zt
Zt
(2)
α
≤ Λ2 |x − x| + gx (C(x, s)) ds − gx (C(x, s)) ds − gx (C(x, s)) ds
0
t
0
Zt
Zt
Zt
+ gp (C(x, s))ux (s, x)ds − gp (C(x, s))ux (s, x)ds − gp (C(x, s))ux (s, x)ds
0
t
0
Zt
n Zt
X
+
gri (C(x, s))(vi )x (s, x)ds − gri (C(x, s))(vi )x (s, x)ds
i=1
0
0
Zt
−
gri (C(x, s))(vi )x (s, x)ds
t
≤
(2)
Λ2 |x
α
Zt
− x| +
Zt
(1)∗ kgx (C(x, s)) − gx (C(x, s))k ds + M2 ds
0
t
Zt
kgp (C(x, s))ux (s, x) − gp (C(x, s))ux (s, x)k ds
+
0
Zt
+
Zt
(1)∗ (1) kgp (C(x, s))ux (s, x) − gp (C(x, s))ux (s, x)k ds + M2 Q1 ds
t
0
+
n Zt
X
kgri (C(x, s))(vi )x (s, x) − gri (C(x, s))(vi )x (s, x)k ds
i=1 0
+
n Z
X
i=1 0
t
Zt
(1)∗ (1) kgri (C(x, s))(vi )x (s, x) − gri (C(x, s))(vi )x (s, x)k ds + M2 Q2 ds
t
(4.17)
953
Existence, uniqueness and estimates of classical solutions. . .
≤
(2)
Λ2 |x
Zt
α
(1)∗
− x| +
H2
[|x − x| + |u(s, x) − u(s, x)|
0
(1)∗
α
+kv(s, x) − v(s, x)k] ds + M2
Zt
+
(1)∗ (2)
M2 Q1 |x
Zt
α
|t − t|
(1)∗
− x| ds +
H2
0
0
(1)
α
(1)∗
+kv(s, x) − v(s, x)k] Q1 ds + M2
+
[|x − x| + |u(s, x) − u(s, x)|
" Zt
n
X
i=1
(1)∗
M2
(1)
Q1 |t − t|
(2)
Q2 |x − x|α ds
0
Zt
(1)∗
H2
+
h
|x − x| + |u(s, x) − u(s, x)|
0
#
iα
(1)
(1)∗ (1)
+ kv(s, x) − v(s, x)k Q2 ds + M2 Q2 |t − t|
(1)∗
(2)
≤ Λ2 |x − x|α + H2
(1)∗
+ M2
(1)∗
h
iα
(1)
(1)
(1)∗
|x − x| + Q1 |x − x| + Q2 |x − x| τ + M2 |t − t|
(2)
(1)∗
Q1 |x − x|α τ + H2
h
iα
(1)
(1)
(1)
|x − x| + Q1 |x − x| + Q2 |x − x| Q1 τ
(1)∗
+ M2 Q1 |t − t|
h
iα
h
(1)
(1)∗
(1)
(1)
(1)∗ (2)
|x − x| + Q1 |x − x| + Q2 |x − x| Q2 τ
+n M2 Q2 |x − x|α τ + H2
i
(1)∗ (1)∗
+ M2 Q2 |t − t|
α
(2)
(1)∗
(1)
(1)
(1)∗
≤ Λ2 + H2
1 + Q1 + Q2
τ + M2 τ 1−α
(1)∗
+ M2
(2)
(1)∗
Q1 τ + H2
(1)
(1)
1 + Q1 + Q2
α
(1)
(1)∗
Q1 τ + M2
(1)
Q1 τ 1−α
i
α
h
(1)
(1)
(1)∗ (1)
(1)∗
(1)
(1)∗ (2)
1 + Q1 + Q2
Q2 τ + M2 Q2 τ 1−α
+ n M2 Q2 τ + H2
× [|t − t| + |x − x|]α
(2.x)
= S2τ
[|t − t| + |x − x|]α
for a fixed ∆(X, τ ) and (t, x), (t, x) ∈ ∆(X, τ ). The inequalities (4.12)–(4.17) and
Remark 2.4 imply the existence of τ ∈ (0, T ] such that (4.8)–(4.11) are true.
954
Lucjan Sapa
Now we prove that for a sufficiently small τ ∈ (0, T ] the operator H is a contrac1,α
(Q) and (t, x) ∈ [0, τ ] × R, then using H2 [f, g], the
tion. Indeed, if (u, v), (u, v) ∈ Cb,τ
mean value theorem for f and the theorem on the estimate of an increment of g we
have
1
|F [u, v](t, x) − F [u, v](t, x)| ≤
2
x+(t−s)
Z
Zt
c
e− 2 (t−s) |f (s, y, u(s, y), v(s, y))
0 x−(t−s)
−f (s, y, u(s, y), v(s, y))| dyds
c2
+
8
Zt
x+(t−s)
Z
c
e− 2 (t−s) u(s, y) − u(s, y)dyds
0 x−(t−s)
≤
1
2
"
Zt x+(t−s)
Z
|fp (P1 )| |u(s, y) − u(s, y)|
(4.18)
0 x−(t−s)
+
n
X
#
|fri (P1 )| |vi (s, y) − v i (s, y)| dyds
i=1
c2
+
8
Zt
x+(t−s)
Z
|u(s, y) − u(s, y)| dyds
0 x−(t−s)
1
c2 2
(1)∗
≤
(1 + n)M1 +
τ k(u − u, v − v)k0 ,
2
4
kG[u, v](t, x) − G[u, v](t, x)k
Zt
≤ kg (s, x, u(s, x), v(s, x)) − g (s, x, u(s, x), v(s, x))k ds
0
Zt "
≤
kgp (P2 )k |u(s, x) − u(s, x)| +
n
X
#
(4.19)
kgri (P2 )k |vi (s, x) − v i (s, x)| ds
i=1
0
(1)∗
≤ (1 + n)M2
τ k(u − u, v − v)k0 ,
n
where P1 , P2 ∈ [0, τ ]×R×[−Q1 , Q1 ]×[−Q2 , Q2 ] are intermediate points. Inequalities
(4.18), (4.19) imply
where
kH[u, v] − H(u, v)k0 ≤ Dτ k(u − u, v − v)k0 ,
(4.20)
c2 2
1
(1)∗
(1)∗
Dτ = max
(1 + n)M1 +
τ , (1 + n)M2 τ .
2
4
(4.21)
955
Existence, uniqueness and estimates of classical solutions. . .
It is obvious that there is τ ∈ (0, T ] such that
Dτ < 1.
(4.22)
The use of the Banach fixed-point theorem and Lemma 3.1 finishes the proof.
Remark 4.2. It follows from the proof of Theorem 4.1 that the set [0, τ ] × R of
the existence and uniqueness of solutions to (1.1), (1.2) can be found by solving the
(j.t)
(j)
(j.x)
(j)
inequalities Siτ ≤ Qi , Siτ ≤ Qi , Siτ ≤ Qi , i, j = 1, 2, (see Remark 2.4) and
inequality (4.22).
Remark 4.3. If additionally, in Theorem 4.1, u0 ∈ Cb2 (R, R) and u1 ∈ Cb1 (R, R),
then u ∈ Cb2 ([0, τ ] × R, R).
Remark 4.4. If the functions f , g generating the system (1.1) do not depend on
t, x and f is of C 1 class and g is of C 2 class on R1+n , then all the assumptions of
Theorem 4.1 are satisfied. It is true especially for the equations with any polynomial
right-hand sides with respect to p, r. Add that Theorem 4.1 works if we consider
the first equation in (1.1) only (the nonlinear wave equation for c = 0 and the nonlinear telegraph equation for c > 0). The examples of such equations are the forced
dissipative sine-Gordon equation with a variable coefficient
utt − uxx + cut + a(t, x) sin u = h(t, x)
(4.23)
and the superlinear telegraph equation
utt − uxx + cut + dum = h(t, x),
(4.24)
where d ∈ R, m ∈ N and a, ax , h, hx are continuous and bounded on [0, T ] × R
(see [12]).
Acknowledgements
The research of the author was partially supported by the Polish Ministry of Science
and Higher Education.
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Lucjan Sapa
[email protected]
AGH University of Science and Technology
Faculty of Applied Mathematics
al. Mickiewicza 30, 30-059 Krakow, Poland
Received: May 12, 2014.
Revised: February 12, 2015.
Accepted: February 16, 2015.