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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
2024, Т. 166, кн. 3 ISSN 2541-7746 (Print)
С. 377-394 ISSN 2500-2198 (Online)
ORIGINAL ARTICLE
UDC 517.956.35 doi: 10.26907/2541-7746.2024.3.377-394
Initial-Boundary Value Problem with Dirichlet and Wentzell Conditions for a Mildly Quasilinear Biwave Equation
V.l. Korzyuka'b, J. V. Rudzkob
a Belarusian State University, Minsk, 220000 Republic of Belarus
bInstitute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220000 Republic of Belarus
Abstract
For a nonstrictly hyperbolic mildly quasilinear biwave equation in the first quadrant, an initial-boundary value problem with the Cauchy conditions specified on the spatial half-line and the Dirichlet and Wentzell conditions applied on the time half-line was examined. The solution was constructed in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations was investigated using the parameter continuation method. For the problem under study, the uniqueness of the solution was proved, and the conditions under which its classical solution exists were established. In the case when the data were not smooth enough, a mild solution was constructed.
Keywords: method of characteristics, mildly quasilinear biwave equation, nonlinear equation, nonstrictly hyperbolic equation, initial-boundary value problem, matching conditions, classical solution, parameter continuation method, mild solution
Introduction
The classical linear biwave equation
(d2 - a2A)(d2 - b2A)u(t, x) = f (t, x) (1)
applies to mathematical models related to the mathematical theory of elasticity. For example, the Cauchy-Kovalevski-Somigliana solution of the elastodynamic wave equation can be obtained by solving the biwave equation [1]. The Cauchy problem for Eq. (1) was examined in [1,2] for the cases a = b and a = b, respectively.
The following equation is one of the simplest one-dimensional linear generalizations of Eq. (1)
(d2 - a2дX)(дX - b2dx)u(t,x) + m2d2tu(t,x) = f (t,x), (2)
on which the Timoshenko-Ehrenfest beam theory relies [3]. When the axial effect is considered, the equation becomes [4]
(d2 - a232x)(d2^ - b2dx)u(t,x)+ m2dt2u(t,x)+ N32xu(t,x) = f (t,x), (3)
taking the place of (2).
A large class of boundary value problems was investigated in [5-9] for the linear generalization of Eq. (1) expressed as
(d2 - a2A)(d2 - b2A)u(t, x) + ^ a(a)(x)Dau(t, x) = f (t, x).
|a|<3
In [10], it was proposed to describe various physical processes by nonlinear equations of the form
(d2 - A)'u(t, x) = F(u(t, x), Au(t, x)). (4)
For l = 1 and F(u, w) = F(u), Eq. (4) reduces to the standard nonlinear wave equation (dt2 — A)u(t, x) = F(u(t,x)), which describes a scalar, spinless, and uncharged particle in the quantum field theory [11]. The symmetry properties of Eq. (4) with l = 2 and F(u,w) = F(u) were studied in [11]. The solvability of boundary value problems for Eq. (4) was analyzed using the Leray-Schauder fixed point theorem in [12-15].
All of Eqs. (1)-(4), where l = 2, in the one-dimensional case, represent a special instance of the following equation:
(dt2 - a2d2)(5t2 - b2dx)u(t,x) = f (t, x, u(t, x), dtu(t, x), dxu(t, x),
d2u(t, x), dtdxu(t, x), dXu(t, x)), (5)
which is classified as (strictly) hyperbolic if a = b and as nonstrictly hyperbolic if a = b.
This article focuses on the nonstrictly hyperbolic case of Eq. (5). Section 1 contains a statement of the initial-boundary value problem. In Section 2, this problem is reduced to the solution of integro-differential equations, and their solvability, uniqueness, and well-posedness are established. In Section 3, the existence and uniqueness theorem for the initial-boundary value problem is formulated. Section 4 outlines a mild solution and proves its existence and uniqueness. The last section summarizes the findings of the study.
1. Statement of the Problem
In the domain Q = (0, oo) x (0, oo) of two independent variables (t,x) € Q £ M2, the following one-dimensional nonlinear equation is considered:
(d^ - a2dX)2u(t, x) = F[u](t, x) := f (t, x, u(t, x), dtu(t, x), dxu(t, x),
dt2u(t, x), dtdxu(t, x), d2u(t, x)), (6)
where a > 0 for definiteness, and f is a function defined on the set [0, to) x [0, to) x r6. Equation (6) is equipped with the initial conditions
u(0,x) = yo(x), dtu(0, x) = yi(x), dt2u(0, x)= y>2(x), d3u(0,x) = ^(x), x G [0, to),
(7)
and the boundary conditions
u(t, 0) = ^o(t), d2u(t, 0) = ^1(t), t G [0, to), (8)
where y0, yi, y2, y3, , and are the functions defined on the half-line [0, to) .
As noted above, equations of the form (6) are used for modeling Timoshenko beams [3] in the nonstrictly hyperbolic case, i.e., when the equality E = kg holds, where E is the elastic modulus of the beam material, G is the shear modulus of the beam material, and k is the Timoshenko shear coefficient. The homogeneous boundary conditions of the form u(t, 0) = ¿>Xu(t, 0) = 0 correspond to a simply supported
beam, the parameter a = \JEp 1, and the function / can be defined by the formula F [u](t,x) := (kAGJ -1m-1 + m-1^2 - EIJ -1m-1)q(t,x) - kAGJ-1d2u(t,x), where A is the cross-sectional area of the beam, I is the second moment of cross-sectional area, q(t,x) is a distributed load (force per unit length), m := pA, and J := pI.
2. Integro-Differential Equation
The domain Q is divided by the characteristic x — at = 0 into two subdomains Qti) = {(t,x) G Q: (-1Y (at - x) > 0}, j = 1, 2. In the closure QW of each of the subdomains Q(j), the integro-differential equations considered are
w(i)(t , = X+rat6a\i(z) + 2aHy2{z) + (a2t2 - (x - z)2) y3(z) ^ |
J 8 a3
x-at
(x — at) + ^o(x + at) t (^i(x — at) + ^i(x + at)) + 2 4 +
at (D^o(x — at) — D^o(x + at)) + 4 +
t x+a(t-t) , ,
f f (a2(t-r)2 ~{x-z)2)T\u^]{t,z) + dr / ^--- 8ffl3 ; -(t,x)eg(1), (9)
0 x-a(t-T )
u(2) ^ = Xjat6a2Vl(z) + 2a2tV2{z) + (a2t2 - (x - z)2) y3(z) ^ | ' 8a3
0
0
6aVi(>) + 2a2t<p2{z) + (a2t2 - (x + z)2) y>3(z) r]
+ J 8a3 +
at-x
x-at
x ¡' f z \ f x\ f0 (x + at) — fo (at — x) xf i(0)
+ 2 J ^[-a)dZ + ^[t-a)+-2---^ +
o
t(ifl(at-x)-ifl(x + at)) xDpo{0) xDl°{t~a, + 4 2a + 2a +
at (Dfo(at — x) — Dfo(x + at)) + 4 +
t x+a(t-T) , ,
/ ,T / (.'(«-rl'-fr-W»]^)^
8a3
(a2(t — t)2 -(x-z)2)
8 a3
((x + z)2 - a2{t-T)2)
8a3
t-x/a x-a(t-T ) t-x/a x+a(t-t )
+ J dr j ---o ' ' L"—J dz +
00 t-x/a a(t—t ) — x
+ J dr J ^ ' -J v''dz, (t,x)eg(2),
00
where D is the ordinary derivative operator.
On the closure Q of the domain Q, a function u is defined as the one coinciding with the solution u(j) of the integral equations (9) and (10)
u(t,x) =u<-j)(t,x), j = 1,2, (11)
on the closure of the domain .
Theorem 1. Let the conditions cp0 eC5([0, oo)), y>i GC4([0, oo)), cp2 GC3([0, oo)), (f3 G C2([0,oo)); no G C5([0,oo)); /xi G C3([0,oo)), / G C2(QxI6) hold. The function u belongs to the class C4(Q) and satisfies Eq. (6), the initial conditions (7), and the boundary conditions (8) if and only if, for each j = 1, 2, it is a solution of Eqs. (9) and (10) in the space C2(Q(j)), subject to the following matching conditions:
dmQ(0) = yi(0), Mi(0) = DVo(0), (12)
d2mQ(0) = ^2(0), DMi(0)= DVi(0), (13)
d3mQ(0) = ^S(0), D2Mi(0) = DV2(0), (14)
d4mQ(0) = 2a2D2^2(0) - a4DVo(0) +
+ f (0, 0, yo(0), ^i(0), d^q(0), ^2(0), D^i(0), dVq(0)), (15)
D5^q(0) = a2D2^3(0) + a2D3^i(0) - a4D4^i(0) + DV(0) x x dUxxf (0,0, ^q(0), ¥>i(0), D^q(0), ^2(0), D^i(0), uxx = DVq(0)) + + D^2(0)öutxf (0, 0, ^q(0), ^i(0), D^Q(0), ^2(0), utx = D^i(0), DVQ(0)) + + ^s(0)öuttf (0, 0yo(0), ^i(0), D^q(0), utt = ^2(0), D^i(0), DVQ(0)) + + D^i(0)öuxf (0,0, ^q(0), ^i(0), ux = D^Q(0), ^2(0), D^i(0), DVQ(0)) + + ^2(0)öutf(0, 0, yo(0), ut = ^i(0), D^q(0), ^2(0), D^i(0), DVQ(0)) + + ^i(0)öuf (0,0, u = yo(0), ^i(0), D^q(0), ^2(0), D^i(0), DVq(0)) + + dtf (t = 0,0, ^q(0), ^i(0), D^o(0), ^2(0), D^i(0), D2^q(0)). (16)
Proof. 1. Let the function u G C\Q) satisfy Eq. (6) in Q, the initial conditions (7), and the boundary conditions (8) everywhere. Under a linear nondegenerate change of the independent variables £ = x — at, n = x + at and with u(t, x) expressed as v(£, n), the differential equation is transformed into
Integrating it four times yields the equation
v(£, n) = fi(£) + nf2(£) + fs(n) + f4(n) +
« n
q i«I
Returning to the original variables t and x, we obtain
u(t,x) = fi(x — at) + (x + at)f2(x — at) + f3(x + at) + (x — at)f4(x + at) +
x —at x+at
1 f f (z — y z + y \
+ l6a4 dy (x-at-y)(x + at-z)J7[u]l^—,^—jdz. (17)
0 |x — at |
By introducing functions gi, g2, g3, and g4 we can rewrite (17) as
u(t, x) = gi(x — at) + tg2(x — at) + g3(x + at) + tg4(x + at) +
"2a ' 2
x —at x+at
Iff ( z_y z ++ y \
+ Tßä4 dy (x - at - y)(x + at - —, — dz. (18)
Note that the functions g1, g2, g3, and g4 in the representation (18) should be determined by the initial conditions (7) and the boundary conditions (8). Substituting (18) into (7), we obtain the following system
gi(x) + g3(x) = yo(x), x G [0, œ), (19)
g2(x) + g4(x) — aDgi(x) + aDg3(x) = ^i(x), x G [0, œ), (20)
x
f{X-yWX'y)dy- 2a£g2(x) + o
+ 2aDg4(x) + a2D2g1(x) + a2D2g3(x) = ^2(x), x G [0, œ), (21)
X
f (x — y)dxG (x,y) + 3G(x,y)
4a
■ ¿y +
+ a2 (aD3g3(x) - aD3gi(x) + 3D2g2(x) + 3D2g4(x)) = (x), x G [0, to),
(22)
where G(y, z) is denoted as
/-c ^ xr ifz~y z + y
From (19) and (20), we have
gi(x) = ¥>o(x) - g3(x), g2(x) = ^i(x) - g4(x)+ aDgi(x) - aDg3(x), x G [0, to). (23) Substituting (23) into (21) and (22), we get two ordinary differential equations
x
n3 , N f (x-y)dxg(y,x)+3g(y,x)
Dg3(x) = j--^-dy -
o
w3(x) 3D2w1(x) D3^0(x) r
X
o
\ aD2u0(x) (x) Dy>i (x) r .
— a_D g3(x) +-+ o » x G [0, to). (25)
4 4a 2
x—at
Let us integrate Eqs. (24) and (25)
93(X) = c1 + C2X + cw + \J[x- ^ J Jt-yWM + WM dy +
0 0 x x
+ XG[0,CX), (26)
x
g4(x) =C\- aDg3(x) + j-J <p2($ + aD^x\
0
x «
+ d£--^^-dy-\--—, x G [0, oo), (27)
00
where C1, C2, C3, and C4 are the integration constants. Substituting (23), (26), and (27) into (18) results in
= X+rat&a2Vi{z) + 2 a2tcp2(z) + (a2t2 - (x - z)2) cp3(z) ^ |
J 8 a3
x-at
(po(x — at) + ^o(x + at) t (^(x — at) + ^(x + at))
+ 2 4 +
at (Dlpq{x — at) — Dtpoix + at))
x-at x+at
1 /" /" , ,,/z — y z +
+ 16a4 dy (x-at - y)(x + at - —, — dz+
0 x-at
V' z G(y,z)(—3a2t2 + 2at(z — y) + 3(x — z)2)—(y — z)dz G(y,z)((x — z)2 —a2t2) + dz -^-dy +
00 x+at z
• (y — z)3zG(y, z)((x — z)2 — a2t2) + G(y, z)(3a2t2 + 2at(y — z) — 3(x — z)2) + I dz I -^-dy,
00
(t,x) G Q(1). (28)
To simplify the expression (28), we integrate by parts, i.e.,
x+at z
f , f(y-z)((x-z)2-a2t2)dzg(y,z)
J J 32a4 V
00
x+at z
f {{x-z)2-a2t2)dzj^-Z^^dy =
0
z
J ((x _ zf _ aH2) dz J 32fl4
0
x+at
s - Z)2 _ aH2) dz I (y-z)dMy,z) -g(y,z) + g(y,z) ^
x+at
2 2,2\ , f (y _ z)dzG(y,z) _G(y,z)
J ((x _ z)2 _ a2t2) dz J
dV =
UV
x+at
+ J ((x _ z)2 _ a2t2
0
U = (x _ z)2 _ a2t2,
(y ~ z)dzÇ(y,z) - G{y,z) 32 a4
x+at x+at
32a4
Q{y,z) 32 a4
dy +
dy =
dy
0
z=x+at z=0
dU = 2(z _ x) dz
z
dZj v= [{y-z)G^z)dy
32a4
Q{y,z) 32 a4
00
J VdU + J ((x _ z)2 _ a2t2) dz y
0 0 0 x+at
■dy+ J ((x - z)2 - a2t2) dz J ■
dy
at+x z x+at z
dzi{x-z){y:z)G^z)dy+ f ((x-z)2-a2t2)dz f dy. (29)
32a4
The resulting expression is
x+at z
dz
(y - z) ((x - z)2 - a2t2) + z) (3a2t2 + 2af (y - z) -3(x - z)2)
32a4
dy =
00
x+at z
/"(at _ x + y)(at + x _ z )G (y, z ) dz I -„ „ ,-ay.
16a4
00
Similarly, the following calculation is performed:
g(y,z)(-3a2t2 + 2at(z-?/) + 3(x- z)2)-(y - z)dzÇ(y,z)((x - z)2 - a2t2)
32 a4
dy =
00
. (at _ x + y)(at + x _ z)G(y, z) dz I -——;-ay.
16a4
x—at 0
Thus, there is an equation
x+at
u(t, x) =
6a2^i(z) + 2a2t^2(z) + (a2t2 _ (x _ z)2) <^(z)
8a3
dz +
^o(x _ at) + <£>o(x + at) t (yi(x _ at) + yi(x + at)) H--n---A--
at (D(po(x — at) — Dtpoix + at))
x —at x+at
1 f f , ,,/z _ yz + y\
+ Î6a4 y (x - at - y)(x + at - z)^N —, —
dz _
0 x—at
z
z
z
x—at z
0
x—at
x+at
16a4
J dz J(x — at — y)[x + at — z) J-"[tt] ^ ^ ^,—dy, (t,x)£Q(1\
(30)
Then, changing the variables t = (z — y)/(2a), £ = (z + y)/(2) in the double integral in the formula (30), we arrive at Eq. (9).
To define the functions g1 and g2 for the negative values of the argument, the boundary conditions (8) are used. Substituting the expression (18) yields the equations
gi( —at) + tg2(—at) + 53(at) + tg4(at) = ^o(t), t G [0, œ),
(-at - y)G(y, a t) 4 a4
dy + D2gi(—at) + tD2g2( —at) +
(31)
(32)
+ D2g3(at) + tD2g4(at) = ^1(t), t G [0, œ).
From (31), we have
(33)
-ag3(-z) + ap0 (--) + zg2(z) + zg4(-z)
9i(z) =--, 2 G (—oo,0]. (34)
a
Substituting (34) into (33) leads to a first-order ordinary differential equation
Dg2(z) = —
(z - y)G{y, -z)
8 a3
dy —
£>V o —
z
a/j 11--
2a
+ ■
+ Dg4(—z),
Integrating (35), we get
z G (—œ, 0]. (35)
z ç
g2(z) = 52(0)+ | d^y —
(e — y)G(y, —e)
8a3
— g4(—z) +
z
dy+^J Aii(-f) ^+ 54(0)-
2a2
2a2
z G (—œ, 0], (36)
where the values g2(0), g4(0), and g4(—z) can be calculated by the formulas (23) and (27). Then, using the representations (23), (26), (27), (34), and (36), we substitute the functions g1, g2, g3, and g4 into the formula (17) for (t,x) G Q(2), integrate by parts as in (29), and get Eq. (10).
The continuity conditions for the function u and its partial derivatives up to and including the fourth order, i.e.,
dtfcdpu(1)(t,x = at) = dtfcdxw(2)(t,x = at), 0 < k + p < 4,
(37)
are also satisfied, where k and p are nonnegative integers. It turns out that the equalities (37) entail the matching conditions (12)-(16), which can be verified directly using the algorithm outlined in [16]. Note that, in this case, the conditions (12)-(16) cannot be strictly justified by differentiating the initial and boundary conditions, as was done, for example, in [17].
z
1
2
2. Assume that the representations (9)—( 11) hold for the function u, which belongs to the classes C2(and C2(Q(2)) , and the conditions (12)-(16) are satisfied. Then, by virtue of the smoothness conditions G C5([0, o)), yi G C4([0, o)), ^>2 G C3([0, oo)), cf3 G C2([0, oo)), /t0 G C5([0, oo)), G C3([0,oo)), / G C2 (Q x
similar to [18], it follows that the function u belongs to the classes C4(Q(1)) and C4(Q(2)) . We substitute the representations (9)-(11) into Eq. (6) and verify that the function u satisfies this equation in Q(1) and Q(2). In this case, for the function u to belong to the class Ca(Q), it is sufficient that the values of the functions u^ and u^ and the values of their derivatives up to and including the fourth order coincide with each other on the characteristic x = at, i.e., that the equalities (37) hold. The latter is equivalent to the validity of the conditions (12)-(16), as can be easily derived by following the argument in the reverse order to that in item 1 of the proof, based on the representations (9)—(11). □
Theorem 2. Let the conditions cp0 GC3([0, oo)), y>i GC2([0, oo)), cp2 GC'dO, oo)), cf3 G C([0,oo)), /t0 G C3([0,oo)); /xi G C^oo)), / G C(Q x R6) hold and the function f satisfy the Lipschitz condition with L G C(Q) in the last six variables, i.e., there exists the function L G C(Q) such that
6
\f (t, x, U1,U2,U3,U4,U5,U6) - f (t, x, zi, Z2, Z3, Z4, Z5, Z6)| < L(t, x) ^ |u - Zj|.
i=i
Then, there exist unique solutions of Eqs. (9) and (10) in the spaces C2(Q(1)) and C2(Q(2)), respectively, and these solutions continuously depend on the initial data.
Proof. To be definite, consider Eq. (9) for the function u(1), which can be solved by the parameter continuation method [19,20]. Set
X+ai6aV(z) + 2a2t^2(z)+ (a2t2 - (x - z)2) ^j(z) V(t,x)= j -^-dz +
x-at
^o(x - at) + ^o(x + at) t (^(x - at) + ^(x + at)) + 2 4 +
at (D^o(x - at) - D^o(x + at)) + 4 '
t x+a(t-t )
«>]«,.)=/* /
0 x-a(t-T)
Rewrite Eq. (9) in the operator form
M(1)(t,x) = if[w(1)](t,x) + w(t,x), (t,x)GQ(1). (38)
Let us also introduce the following family of equations with the parameter e G [0,1]:
41}(t,x) -e(K[u^] -K[0]){t,x) =w{t,x), (t, x) G (39)
where w(t, x) = v(t,x) + K[0](t, x). It is clear that any solution Ug1)(t,x) of Eq. (39) with e = 1 is also a solution of Eq. (38), and vice versa. Hence, the task reduces to solving Eq. (39) when e = 1 .
Let us introduce the set = {(t,x) | (t, x) € Q(i) A x + at ^ m}, m € N. Due to the smoothness conditions y0 € C3 ([0, to)), y € C2([0, to)), y2 € C 1([0, to)), G C([0, to)) , /t0 G C3([0, to)) , /ti G C1([0, to)) , / G C (Q x R6), as in [18], we conclude that K[g] € C2(^m), assuming that, for example, g € C2(^m). It implies that the operator K maps from the space C2(^m) to the space C2(^m). Let us show that the operator K: C2(^m) ^ C2(Om) is Lipschitz-continuous. We have
|K[u1](t,x) - K[u2](t,x)| =
t x+a(t —t )
r r (q2(t-r)2-(x-z)2)F[Ml](r,z)
I J 8 a3 *
0 x — a (t—t )
t x+a(t—t )
a^it- (x - FIwqKt.z)
(a2(t - t)2 - (x - z)2) FM(t,z)
<
<
where
-Jdr J -—3-dz
0 x — a (t—t ) t x+a(t—t )
JdT J (a2(t - r)2 - (x - z)2) (FH(r, z) - FH(t, z)) ^
0 x — a (t—t )
t x+a(t—t )
< ajdT j |L(t,x)|(|ui - U21 + |dtui - dtU21 + |dxui - dxU21 +
0 x — a (t —t )
+ |dt2Mi - dt2M2| + |dtdx«i - dtdxU2| + ^ui - |)(t,x) dz <
< aaNLNc(Om)H«i - u2yc2(fim)t2 < am||L||C(fim)||ui - u2||c2(fim),
(t,x) € ui € C2(nm), «2 € C2(^m), (40)
!(t - t)2 - (x - z)2
a = max
8a3
Proceeding to (40), we arrive at the estimate
|dpdkK[ui](t,x) - dpdkK[u2](t,x)| < ap,fc|ui - U2|c*(nm), (t,x) €
0 < p + k < 2, «4 € C2(^m), u2 € C2(^m), (41)
where k and p are nonnegative integers, and ap k is a constant determined by the function L, the number a, and the set . It follows from (41) that
||K[ui] - KM||c2(nm) < £||ui - «2|c2(nm), ui € C2(^m), «2 € C2(^m), (42)
where 0 = a0,0 + aij0 + a0ji + a2,0 + ai;i + a0,2 . The inequality (42) implies that the operator K: C2(Om) ^ C2(^m) is Lipschitz-continuous. Consider the operator K£ defined by the formula
K£[u] = u - e(K[u] - K[0]).
Since the operator K: C2(Om) ^ C2(Om) is Lipschitz-continuous, the operator K: C2(^m) ^ C2(^m) retains this property.
Let us prove that the operator K£ : C2(^m) ^ C2(^m) is coercive. To achieve this, it suffices to derive an a priori estimate of the form
lu^l
£
C2(Qm) ^ C|w|C2(0m), (43)
<
for the solution u£i) of Eq. (39), where C is some constant that does not depend on the function u£i) and the number e. We have
|u£i)(t,x)| = |w(t,x) + e(J[u£i)](t,x) - K[0](t,x))| < ||w|c(nm)+
t x+a(t—t )
f f (a2(t - r)2 - (x - z)2) (r, z) - -F[0](r, z)) J + ejdr j -g-g-dz
0 x — a (t—t )
t x+a(t—t )
< |m|C(fim) + aj dT j U(t, z) dz, (t,x) € , (44)
0 x — a(t—t )
where
U (t,x) = |u(i)(t,x)| + |dtu(i)(t, x)| + |dxu (i)(t , x)| +
+ |dtu (i)(t , x)| + |dtdxu (i)(t , x)| + |d_2u (i)(t ,x)|. (45)
Similarly, we get
t x+a(t—t )
|dtu£i)(t,x)| < ||dtw||C(nm) + 7i,^y dT J U (t,z) dz, (t,x) € , (46)
0 x — a (t—t ) t x+a(t—t )
|dxu£i)(t,x)| < ||dxwyC(nm) + 70,^dT J U(T,z) dz, (t,x) € (47)
0 x — a (t—t )
|d?u£i)(t, x)| <
^ ydi2wyC(Qm) + A^ U(r,x - a(t - t)) dr + U(r, x + a(t - t)) dr +
0 0
t x+a(t-t )
+ 72,0 y dr J U(r,z) dz, (t,x) G , (48)
0 x-a(t-T )
|dtdxu£X) (t,x)| <
t t < ||dtdxw||c(nm) + A2 J U(r,x - a(t - r)) dr + fa J U(r, x + a(t - r)) dr +
00
t x+a(t-t )
+ 71,1 y dr J U(r, z) dz, (t, x) G , (49)
0 x-a(t-T )
|d2u£1)(t,x)| <
t t
2
< ||dXw|C(nm) + A^ y U(r,x - a(t - r)) dr + J U(r, x + a(t - r)) dr +
00
t x+a(t-t )
+ 70,2 y dr J U(r,z) dz, (t,x) G , (50)
where 71j0 , 70j1 , 72,0 , 71,1, 70,2 , Aj (i = 1, 2, 3), and -j (i = 1, 2, 3) are the constants, which depend on the function L, the number a, and the set ilm. Summation of the inequalities (44)-(50) yields
|U (t,x)| < |H|c 2(fim) + A J U (r, x - a(t - r )) dr + - J U (t,x + a(t - r )) dr +
0 0
t x+a(t-t )
+ yJ dr J U(r, z) dz, (t,x) G , (51)
where A = A1 + A2 + A3, - = -1 + -2 + -3, Y = a + 71,0 + 70,1 + 72,0 + 71,1 + 70,2.
Let us denote V(s) = U(s, x - a(t - s)) for fixed x G {x | 3t : (t, x) G . Then, we have
|V (t)| < ||w|C2(nm) + A j V (r) dr + -J U (r,x + a(t - r)) dr +
00
t x+a(t-t )
+ 7 J dr J U(r, z) dz, (t, x) G ,
0 x-a(t-T)
Applying the Gronwall lemma to the preceding inequality, we obtain
|V(t)| < ^|M|C2(fim) + - f U(r,x + a(t - r)) dr +
t x+a(t-t ) .
+ 7 j dr j U(r, z) dz j exp(At), (t, x) G .
0
0 x-a(t-T)
Using this technique iteratively, we get
t x+a(t-t )
(t x + a (t t ) *
INIc^^/dr J U(T,z)dz jexp(^exp0^,
0 x-a(t-T) '
(t, x) G .
Applying the multidimensional Groonwall lemma [21] to the preceding inequality, we derive the estimate
,TT, ...........f-m /Am\\ f 2 /-m /Am\\\
| t/(i, x) | < || HI c*(nra) exp exp j j exp (jai exp exp^-jjj,
(t, x) G ttm. (52)
The formulas (45) and (52) are actually the a priori estimates of the form (43). Therefore, we have proved that the operator Ke: C2(^m) ^ C2(0m) is coercive.
Note that the function B: [0,1] 9 e ^ Ae is continuous in the seminorm of the space of Lipschitz-continuous operators [19]. It is obvious that B(0) = A0 is continuously
t
invertible, as it corresponds to the identity operator. Considering this, we conclude that the conditions of [19, Theorem 4] hold for the operator-function B. Therefore, Eq. (39) has a unique solution in the space C2(^m) for any e G [0,1], and this solution continuously depends on the initial data. Thus, we have solved Eq. (38) in the space C2 (Om).
To construct the solution u of Eq. (38) in the space C2(Q(1)), consider the following limit
u = lim um, (53)
where um (m € N) is the solution of Eq. (38) in the space C2(^m). We also assume that the functions um € C2(Q(i)) are extended in some way outside the set .
Let us prove the existence of the limit (53). Consider the functions un and um, where n < m. In this case, um|^n = un. Otherwise, there would be a contradiction with the uniqueness of the solution of Eq. (38) in the class C2(fi„). Thus, for any e > 0, there exists an integer N(e) = m such that for any integer M > N(e) we have ||uN - uM ||c2(nm) < e. This indicates that the sequence (um) is fundamental in any
oo _
seminorm of the form ||-||c2(nk), where k € N. Since |J = Q(i), the topology
m= 1
of the Frechet space C2(Q(i)) can be induced by a countable family of seminorms |H|c2(nk). So, the sequence (um) converges in the space C2(Q(i)).
Next, we can prove that the limit (53) solves Eq. (38). Consider a point (t0,x0) €
Q(1). There exists a number m such that (to,xo) G ^m. We have u|^n = um, where m > n. Otherwise, there would be a contradiction with the uniqueness of the solution of Eq. (38) in the class C2(^„). Then,
u(to,xo) = Um(to,xo) = K [um](to,xo) + w(t,x). (54)
Let us pass to the limit as m ^ œ in (54) and obtain
u(to,xo)= lim (K[um](to,xo) + w(to,xo)) = K[ lim um] (to,xo) + w(to,xo) =
= K [u](to, xo) + w(to ,xo ).
Given the arbitrariness of the point (t0,x0) € Q(i) and the preceding equality, we conclude that the function u defined by the limit (53) is a solution of Eq. (38) in the class C2(Q).
Let us prove that the limit (53) is the unique solution of Eq. (38). Assume that Eq. (38) has two solutions u and m in the space C2(Q(i)). Then, the functions and u]nm are the solutions of Eq. (38) in the class C2(^m). Therefore, = u]nm .
Since y = Q(1), we arrive at the equality u = u. This proves that Eq. (38) has
m= 1
a unique solution in the class C2(Q(1)).
Therefore, we have constructed a unique solution of Eq. (9) in the class C2(Q(1)). The existence of a unique solution of Eq. (10) in the class C2(Q(2)), which continuously depends on the initial data, can be proved in a similar way. □
3. Classical Solution
The following theorem is a consequence of Theorems 1 and 2.
Theorem 3. Let the conditions cp0 eC5([0, oo)), y>i G C4([0, oo)), cp2 G C3([0, oo)), fa G C2([0,oo)), /j,0 G C5([0, oo)), m G C3([0,oo)), / G C2 (Q x R6) hold and the function f satisfy the Lipschitz condition with L G C(Q) in the last six variables, i.e., there exists the function L G C(Q) such that
6
\f (t, x, U1,U2,U3,U4,U5,U6) - f (t, x, z1, z2, z3, z4, z5, z6)\ < L(t, x) ^ |uj - z»|.
i=1
Th(n, thee initial-boundary value problem (6)-(8) has a unique solution u in the class C4(Q) if and only if the conditions (12)—(16) are satisfied. This solution is determined by the formulas (9)-(11).
4. Mild Solution
Consider the problem (6)-(8) for the case, where the functions ^>0, , ^>2 , ^>3, , yU4 , and f are not smooth enough.
Definition 1. We define the function u representable in the form (9)-( 11) as a mild solution of the problem (6)-(8).
Remark 1. Any classical solution of the problem (6)-(8) is also a mild solution of this problem.
Remark 2. If the additional smoothness conditions ^>0 G C5([0, to)), G C4([0, oo)), G C3([0, oo)), cf3 G C2([0, oo)), G C5([0,oo)), G C3([0,oo)), / G C2(Q x R6) and the matching conditions (12)—(16) hold, then the mild solution of problem (6)-(8) is classical.
Let Q = Q\{(t,x) | x = at}.
Theorem 4. Let the conditions ^0GC3([0, oo)), y>i GC2([0, oo)), G C1 ([0, oo)), cf3 G C([0,oo)), no G C3([0, oo)), m G C^^oo)), / G C(Q x R6) hold and the function f satisfy the Lipschitz condition with L G C(Q) in the last six variables, i.e., there exists the function L G C(Q) such that
6
\f (t, x, U1, U2, U3, U4, U5, U6) - f (t, x, z1, z2, z3, z4, z5, z6) \ ^ L(t, x) |uj - zj|.
i=1
Then, the initial-boundary value problem (6)-(8) has a mild solution u in the class
C2(Q).
Proof. The solvability of the integral equations (9) and (10) and the belonging of their solutions to the classes of C2(Q(1)) and C2(Q(2)), respectively, follows from Theorem 2. □
If the matching conditions (12)-(16) are partially met, the smoothness of the mild solution can be increased, i.e., the following theorem holds.
Theorem 5. Let the conditions (po G C3([0, oo)), ^ieC2([0,oo)), ^eC'fjOjOo)), (f3 G C([0,oo)), no G C3([0,oo)); m G C^oo)), / G C(Q x R6) hold and the function f satisfy the Lipschitz condition with L G C(Q) in the last six variables, i.e., there exists the function L G C(Q) such that
6
|f(t, x, ui, u2, u3, u4, u5, u6) - f (t, x, zi, z2, z3, z4, z5, z6) | ^ L(t, x) |u - zj|.
i=i
Then, the initial-boundary value problem (6)-(8) has a mild solution u in the class C\Q) n C(Q) if and only if 920(0) = /x0(0).
Proof. 1. Let us prove the necessity of the condition cpo(O) = jno(0). If uGC(Q), then u(0, 0) = lim u(t, 0) = lim u(0,x). The representations (9)—( 11) imply lim u(t, 0) =
t^0 x^0 t^0
lim ^0(t) = ^(0) and lim u(0,x)=lim ^0(x) = (0). Hence, <^>0 (0) = ^0(0).
t^0 x^0 x^0
2. Let us prove the sufficiency of the condition ^>0(0) = ^0(0). According to Theorem 4, there exist a unique mild solution u € C2(Q) of the problem (6)-(8). Using the formulas (9)-(11), we compute
[(u)+ - (u)—](t,x = at) = u(i)(t, at) - u(2)(t, at) = y>0(0) - ^0(0), (55)
where (w)±(t, x = at) = ^lim u(t, ai±5). Using (55), we conclude that u G C(<3) • d
Conclusions
Sufficient conditions for the existence of a unique classical solution of the initial-boundary value problem in the first quadrant for a nonstrictly hyperbolic mildly quasilinear biwave equation are established. The results obtained show that the failure to meet the matching conditions makes it impossible to construct a classical solution in the entire first quadrant. In the case when the initial data are insufficiently smooth, a mild solution of the initial-boundary value problem is constructed, and its uniqueness is proved.
Acknowledgments. This study was supported by the Ministry of Science and Higher Education of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics (agreement no. 075-15-2022-284).
Conflicts of Interest. The authors declare no conflicts of interest.
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Received August 18, 2024 Accepted August 27, 2024
Korzyuk Viktor Ivanovich, Academician, Doctor of Physics and Mathematics, Professor;
Leading Research Fellow
Belarusian State University
pr. Nezavisimosti, 4, Minsk, 220000 Republic of Belarus Institute of Mathematics of the National Academy of Sciences of Belarus
ul. Surganova, 11, Minsk, 220000 Republic of Belarus E-mail: [email protected]
Rudzko Jan Viaczaslavavicz, Junior Research Fellow
Institute of Mathematics of the National Academy of Sciences of Belarus
ul. Surganova, 11, Minsk, 220000 Republic of Belarus E-mail: janycz@yahoo. com
ОРИГИНАЛЬНАЯ СТАТЬЯ
УДК 517.956.35 10.26907/2541-7746.2024.3.377-394
Начально-граничная задача с условиями Дирихле и Вентцеля для слабо квазилинейного биволнового уравнения
В.И. Корзюк1,2, Я.В. Рудько2
1 Белорусский государственный университет, г. Минск, 220000, Республика Беларусь
2Институт математики НАН Беларуси, г. Минск, 220000, Республика Беларусь
Аннотация
Для нестрого гиперболического слабо квазилинейного биволнового уравнения, заданного в первом квадранте, рассматривается начально-граничная задача, в которой на пространственной полупрямой заданы условия Коши, а на временной полупрямой - условия Дирихле и Вентцеля. Решение строится в неявном аналитическом виде как решение некоторых интегро-дифференциальных уравнений. Методом продолжения по параметру исследуется разрешимость этих уравнений. Для рассматриваемой задачи доказывается единственность решения и установлены условия существования ее классического решения. Если данные задачи недостаточно гладкие, то строится слабое решение.
Ключевые слова: метод характеристик, слабо квазилинейное биволновое уравнение, нелинейное уравнение, нестрого гиперболическое уравнение, начально-краевая задача, условия согласования, классическое решение, метод продолжения по параметру, слабое решение.
Благодарности. Работа выполнена при финансовой поддержке Министерства науки и высшего образования Российской Федерации в рамках реализации программы Московского центра фундаментальной и прикладной математики (соглашение № 075-15-2022-284).
Конфликт интересов. Авторы заявляют об отсутствии конфликта интересов.
Поступила в редакцию 18.08.2024 Принята к публикации 27.08.2024
Корзюк Виктор Иванович, академик, доктор физико-математических наук, профессор; главный научный сотрудник
Белорусский государственный университет
пр. Независимости, д. 4, г. Минск, 220000, Республика Беларусь Институт математики НАН Беларуси
ул. Сурганова, д. 11, г. Минск, 220000, Республика Беларусь E-mail: [email protected] Рудько Ян Вячеславович, младший научный сотрудник Институт математики НАН Беларуси
ул. Сурганова, д. 11, г. Минск, 220000, Республика Беларусь E-mail: janycz@yahoo. com
For citation : Korzyuk V.I., Rudzko J.V. Initial-boundary value problem with Dirich-/ let and Wentzell conditions for a mildly quasilinear biwave equation. Uchenye Zapiski \ Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2024, vol. 166, no. 3, pp. 377-394. https://doi.org/10.26907/2541-7746.2024.3.377-394.
Для цитирования : Korzyuk V.I., Rudzko J.V. Initial-boundary value problem / with Dirichlet and Wentzell conditions for a mildly quasilinear biwave equation // \ Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. 2024. Т. 166, кн. 3. С. 377-394. https: //doi.org/10.26907/2541-7746.2024.3.377-394.
