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ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗ
INTEGRO-DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS
Онлайн-доступ к журналу: http://mathizv.isu.ru
Серия «Математика»
2021. Т. 35. С. 34—48
УДК 518.517 MSC 03C07, 03C60
DOI https://doi.org/10.26516/1997-7670.2021.35.34
An Initial Problem for a Class
of Weakly Degenerate Semilinear Equations
with Lower Order Fractional Derivatives *
G. D. Baybulatova1, M. V. Plekhanova1'2
1 Chelyabinsk State University, Chelyabinsk, Russian Federation,
2 South Ural State University (National Research University), Chelyabinsk, Russian Federation
Abstract. An initial value problem is studied for a class of evolutionary equations with a weak degeneration, which are nonlinear with respect to lower order fractional Gerasimov -Caputo derivatives. The linear part of the equations contains a respectively bounded pair of operators. Unique local solvability is proved in the case of a nonlinear operator depending on elements of the degeneration space only. Examples of an equation and a system of partial differential equations are given, the initial-boundary value problems for which are reduced to the initial problem for an equation in a Banach space of the studied class.
Keywords: fractional Gerasimov - Caputo derivative, fractional order differential equation, degenerate evolution equation, semilinear equation.
* The reported study was funded by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011; and by the Russian Foundation for Basic Research, project numbers 20-31-90015 and 21-51-54003.
THE INITIAL PROBLEM FOR A CLASS OF WEAKLY DEGENERATE... 35 1. Introduction
Among the equations of mathematical physics, a special place is occupied by equations and systems of equations that are not solvable with respect to the highest time derivative, called degenerate evolutionary equations. In this paper, we investigate the solvability of a class of degenerate evolution equations with fractional derivatives.
In Banach spaces X, y, a continuous linear operator L is given (briefly, L £ C(X, y)), M is a linear closed operator with domain Dm dense in A', (M £ Cl(X, y)) and N : X ^ y is a nonlinear operator, where X is an open set in R x Xn. Consider the equation
D?Lx(t) = Mx(t) + N (t, D?1 x(t),D?2 x(t),..., x(t)), (1.1)
where 0 < a,\ < a2 < ••• < an < a, m — 1 < a < m £ N, Df is the Gerasimov — Caputo derivative.
Note the studies of the solvability of degenerate evolution equations of form (1.1) of integer [3; 4; 20-22] and fractional [2; 5; 6; 8; 9; 15; 16] orders. For various equations resolved with respect to the fractional derivative, as well as for the corresponding integral equations, results on the existence of a unique solution were obtained by such authors as J. Priiss [19], E.G. Bajlekova [1], A.V. Glushak [10], M. Kostic [12], V.E. Fedorov [7].
Solvability conditions for the Cauchy problem to an equation of form (1.1) with ^ = y, L = I, M £ C(X, y) are examined in [18]. Solvability of initial problems for a degenerate (ker L = {0}) equation with a relatively bounded pair of operators L and M the authors of the article investigated for two types of constraints on the nonlinear operator [17; 18]: if the image of the nonlinear operator belongs to a subspace without degeneration, or if the nonlinear operator depends on the elements of such subspace only. In this paper, on the contrary, we use the condition that the nonlinear operator depends on the elements of the degeneration subspace only. Using the results of the theory of degenerate evolution equations (see [22]), we investigate the equation (1.1) equipped with the initial conditions
x(k)(to) = Xk, k = 0,1,...,r — 1, (Px)(l)(to) = Xi, I = r,...,m — 1, (1.2)
where P is the projector onto the space without degeneration, the number r is determined by the value of an (see further). Such a problem is reduced to the Cauchy problem for a system consisting of a linear equation solved with respect to the highest fractional derivative on a subspace without degeneration and a semilinear equation of a lower fractional order on the subspace of degeneration obtained using the implicit function theorem. To illustrate the abstract results obtained, examples of initial-boundary value problems for an equation and a system of partial differential equations that are nonlinear with respect to the lowest fractional time derivatives are given.
2. Equations solvable with respect to the highest fractional derivative
Introduce the notations gs(t) := r(5)-1tS-1, gs(t) := r(^)-1(i - to)S-1, t
jf h(t) := / gs(t -s)h(s)ds for 5 > 0, t > 0. Let D™ be the usual derivative
to
of the order m £ N, Jj0 be the identity operator. The fractional Gerasi-mov — Caputo [1, p. 11] derivative of the function h is defined as
(m-1 \
h(t) - £ h(k)(to)~gk+1(t)\ , t>to.
k=o J
Let Z be a Banach space, A £ C(Z). Consider the Cauchy problem
k) (to) = Zk, k = 0,1,...,m - 1, (2.1)
for the inhomogeneous linear equation
D?z(t) = Az(t) + f(t), t £ [to,T), (2.2)
where T £ (t0, +rc>]. A function z £ Cm-1([t0,T); Z) is called a solution of
/ m-1 \
problem (2.1), (2.2), if J^-« ( z - £ k)(io)5k+1 £ Cm([to,T); Z) and
V k=0 /
equalities (2.1), (2.2) are valid.
Theorem 1. [15]. Let A £ C(Z), f £ C([to,T); Z). Then for all z0,z1,..., zm-1 £ Z there exists a unique solution of problem (2.1), (2.2).
Let n £ N, Z be an open set in R x Zn, B : Z ^ Z be a nonlinear operator, zk £ Z, k = 0,1,... ,m - 1, 0 < a1 < a2 < ■ ■ ■ < an < a, m - 1 < a < m £ N/ Consider the semilinear equation
Dfz(t) = Az(t) + B(t, Df1 z(t),D.T z(t),..., Dfn z(t)). (2.3)
By a solution to problem (2.1), (2.3) on an interval [i0,i1] we mean a function 2 £ Cm-1([t0, i1]; Z), for which the condition
j m
Jt
m— 1 \
z - ^ z(k)(to)gk+i) £ Cm([to, iij; Z)
k=1 J
is satisfied, and for any t £ [io,ii] (tz(t),D?2z(t),...,^"nz(t)) £ Z, equalities (2.1) and (2.3) are true for all t £ [i0,i1].
Further, the line above the symbol will denote a set of n elements with indices from 1 to n, for example, x = (x1,x2,... ,xn). Let Ss(x) = {y £ Zn : \\yk - xk\\z < 5, к = 1, 2,...,n}. A mapping^ : Z ^ Z will be
(
called locally Lipschitzian in z if for each (t, x) £ Z there are 5 > 0 and I > 0 for which [t0 — 5, t0 + 5] x Ss (S) C Z and for any (s, y), (s ,v) £ [io — 5, to + x Ss (x)
n
\\B (s, y) — B (s, v)\\z < lY. II y* — Wz.
k=1
Using the initial data z0, z1,..., zm-\ we define
* = ¿o + 1!(i — io) + |f(i — to )2 + • • • + (^—^(i — to)m-\
¿1 = D^1 \t=t0 z(t), *2 = D\r\t=t0 z(t), ..., zn = Dp\t=t0 z(t).
Theorem 2. [18]. Let A £ C(Z), a set Z be open in R x Zn, a mapping B £ C(Z;Z) be locally Lipschitzian in z, zk £ Z, k = 1,2,... ,m — 1 be such that (to, *1, *2,..., zn) £ Z. Then problem (2.1), (2.3) has a unique ssolution on a segment [io, i1] for some t1 > to.
3. Semilinear equation with a weak degeneration
Let X, y are Banach spaces, L £ C(X; y), kerL = {0}, M £ CI(X; y), Dm be the domain of M operator equipped with the graph norm \\ • \ \oM :=
II-II* + \\M J*.
We define an L-resolvent set pL(M) := {p £ C : (pL — M)-1 £ C(y; X)} of operator M and denote R%(M) := (pL — M)-1L, := L(pL — M)-1. Operator M is called (L, a)-bounded, if
3a > 0 Vp £ C (\p\ >a) ^ (p £ pL(M)).
Under the condition that the operator M is (L, a)-bounded, we define the projectors
P :=2tiJ R»(M) ^ £ C(X), Q :=2tiJ LL^(M) ^ £ C(y), (3.1)
i i
where 7 = {p £ C : \p\ = r > a} (see [22, p. 89, 90]). Let Xo := kerP, X1 := imP, yo := kerQ, y1 := imQ. Let us denote by Lk (Mk) the constriction of the operator L (M) on *k (DMk := Dm nXk), k = 0,1.
Theorem 3. [22, p. 90, 91]. Let an operator M be (L, a)-bounded. Then
(i) M1 £t{X1; y 1e, Mo £Cl {Xo;yoe, Lk £ C{Xk; yk e, k = 0,1;
(ii) there exist operators Mf1 £ C(yo; Xoe, L-1 £ £(y1; X^.
We denote G := M-1L0. For p e N0 := {0} U N an operator M is called (L,p)-bounded, if it is (L, a)-bounded, Gp = 0, Gp+1 = 0. Consider the problem
x(k) (to ) = xk, k = 0,1,...,r — 1, (Px)(l) (to ) = Xi, l = r,...,m — 1, (3.2)
for the equation
DÇLx(t) = Mx(t) + N (t, D?1 x(t),D?2 x(t),..., D?" x(t)), (3.3)
where 0 < a1 <a2 < • • • < an < a, m — 1 < a < m e N, r — 1 <an < r e N, X is an open set in R x Xn, N : X ^ y is a nonlinear operator.
Since kerL = {0}, equation (3.3) is degenerate. In the case of an (L, 0)-bounded operator M, we have ker P = ker L [22], therefore, the degeneration subspace X° in this case is minimal and the corresponding class of equations (3.3) is called weakly degenerate.
By a solution of problem (3.2), (3.3) on a segment [t0,t1] we mean a function x e C([t0,t1]; DM) n Cr-1([t0,t1]; X), such that
(m-1 \
Lx — ^ (Lx)(k)(to)gk+A e Cm([to,t1]; y), k=o J
for all t e [to,t1] (t,^?1 x(t),D?2x(t),...,D?nx(t)) e X and equalities (3.2), (3.3) hold.
By [(/ — Q)N]'Xn (t,z1,z2,... ,zn) we denote the Frechet derivative of the operator (I — Q)N at the point (t, z1, z2,..., zn) e X by the last argument xn. For brevity, we denote the projector along A'1 on X° as P0 := I — P. We denote W = X n (R x (X0)n),
X = X0 + (t — t0 ) + (t — t0)2 + • • • + T^^r (t — t0)m-1, 1! 2! (m — 1)!
for xk, k = 0,1,... ,m — 1, from conditions (3.2).
Theorem 4. Let 0 < a1 < a2 < • • • <an < a, m—1 < a < m e N, r—1 <
< r e N, an operator M be (L, 0)-bounded, a set X be open in R x N e C (X ; y ), for all (t,z1,..., zn) e X, such that (t, P0z1,..., P0zn) e X, N (t, z1,...,zn) = N1 (t, P0z1,..., P0 zn) at some N1 e C (W ; y), such that (I — Q)N1 e C ; x1, x2 ... xr-1 e X, xr,xr+1,..., xm-1 e X1, the mapping M0"1[(/ — Q)N1]'Xn (t, z1,..., zn) : X0 ^ X0 be a bijection for all elements (t,z1,z2,... ,zn) of the point (t0,D(?111=t0 x,...,D?" 11=t0 %) e W neighborhood, herewith
P0X0 + M-1(I — Q)N (t0,Dt?11 t=to P0X,...,D?" | t=to m) = 0. (3.4)
Then there exists such t1 > t0, that problem (3.2), (3.3) has a unique solution on the segment [to,^].
Proof. Let us act on equation (3.3) by a continuous operator Mo"1(/ — Q), the existence of which follows from Theorem 3. We get the equation
0 = w(t) + Mf1(I — Q)N1 (t, D^^1w(t),D^^2w(t),..., D?"w(t)),
where w(t) := Pox(t). By the implicit function theorem, since there exists the inverse operator
(M0-1[( I — Q)N1]'^ (t, D^1w(t),D^2w(t),..., D^w(t))e- £ C(Xo),
this equation can be solved with respect to D""w at t from some interval (to — 5,to + 5). Hence, we have the equation
D?"w(t) = R(t, D'^1w(t),^^2w(t),..., D"n-1w(t)) (3.5)
with a continuously differentiable mapping R. Theorem 2 implies the existence of a unique solution to the Cauchy problem w(fc)(io) = Po%k, k = 0,1,... ,r — 1, for equation (3.5) on some segment [t o, *1]. Moreover, under the conditions of this theorem Lo = 0, therefore, Lw = 0.
Consider the Cauchy problem for the second equation obtained after the action by the operator L-1Q on equation (3.3),
D?v (t) = S1 v (t) + L-1QN (t, ^t1w(t), D*2w(t),..., D?nw(t)),
v(k)(to) = Pxk, k = 0,1,... ,m — 1,
where S1 = L-1M1 £ C(X1). The unique solvability of this problem on [io, i1] at some t1 £ (to,i1] follows from Theorem 2. It completes the proof of the theorem. □
Remark 1. Note that at r = m (3.2) is the Cauchy problem
4. Initial-boundary value problem for a nonlinear integro-differential equation
In the region (0,1) x [io, to), to £ R, consider the initial-boundary value problem
dlw
(s, to) = vi(s), 1 = 0,1,...,r — 1, s £ (0,1), (4.1)
dk Aw
-^r- (s, to) = Avk (s), k = r,r + 1,...,m — 1, s£ (0,1), (4.2)
dw dw
w(0, t) = w(1, t), — (0, t) = — (1, t), t> to, (4.3)
for a semilinear fractional-order equation
P 1
D?Aw +
i
J D^wis, t)ds 0
J D?2w(s, t)ds = 0, s E (0,1), t > to. (4.4)
Here m — 1 < a < m, 0 < ai < a2 < a, r — 1 < a2 < r, ft > 0. Let us denote by (■, ■) the inner product in the space L2(0,1). Let
X = {v e H2(0,1) : v(0) = v(1), v'(0) = v'(1)}, y = L2(0,1), « = + 1(t — io) + 22(f — ^0)2 + ■ ■ ■ + ^ — io )m_i, for vk, k = 0,1,... ,m — 1, from conditions (4.1), (4.2).
Theorem 5. Suppose m — 1 < a < m, 0 < ai <a2 < a, r — 1 <a2 < r, P > 0, Vi e X, 1 = 0,1,... ,m — 1, (vk, 1) = 0, k = r,r + 1,...,m — 1, D?1\1=t0 (v, 1) = 0, D?2\t=t0 (v, 1) = 0. Then for some ti > t0 problem (4.1)-(4.4) has a unique solution on the set (0,1) x [i0,iij.
Proof. Let's take L = A, Mx = (x, 1) at x e X, then for p = 0, x e X, y e y we have
(pL — M)x = pAx — (x, 1) = —(x, 1) + ^ 27rkp(x(s), e27Tkis)e2lTkis,
kez\{0}
(pL — M)~ly = —(y, 1) + £ (y(s), e2Ms)e2lTkis,
kez\{0} 77 p
1 + 472 k2
\\(pL — M)-ly\\2H2{0A) = 1)|2 + £ \(y^Hs)? <
kez\{0} p
<C 2\\y\\l2io,iy Therefore, the operator M is (L, a)-bounded. Wherein
R^(M) = L^(M)= ^ 1 (■, e27kis)e27kis, fcez\{o} ^
consequently, P = Q = ^ (■, e2l7k%s)e2l7kts, X1 is the closure of the
kez\{o}
linear span of the set {e±2ms, e±477ls,... } in the space X, У1 is the closure of the same set in L2(0,1), and the subspaces X0 = У0 = span{1} coincide and are one-dimensional. Insofar as kerL = kerP, then operator M is ( L, 0)-bounded.
The nonlinear operator in the considered equation will have the form N(x,y,z) = —{x, 1) — Hy, 1)1?(z, 1), it is defined on X = R x X3, so W = R x (A'0)3. Let's show the action N : X ^ y, for x,y,z e X we have
W(x,y,z)h2(o,1) <l(x, 1)| + l(y, 1)1?I(z, 1)|.
It's obvious that N e C 1(X;L2(tt)), N(x,y,z) = N(P0x,P0y,P0z), since, say, Pox = (x, 1), (Pox, 1) = {{x, 1), 1) = (x, 1).
Note that conditions (4.2) are of the form (Lx)(k(0) = Lxk and therefore, in the case of an (L, a)-bounded operator, by virtue of Theorem 3, they are equivalent to the conditions (Px)(k (0) = xk, k = r, r+1,..., m—1.
The equalities (vk, 1) =0, k = r,r + 1,... ,m — 1, entail the conditions Vk(•) = Xk e X1, k = r,r + 1,... ,m — 1, of Theorem 4.
The operator M-1 is identical, for all y e y (I — Q)y = (y, 1) e y0, Pov(to, •) = Povo = (vo, 1),
M-1 (I — Q)N(to, Pov(to, •), D?11t=toPov, D?21t=toPo*) =
= (/ — Q)(—(vo, 1) — ID?11 t=to (v, 1)1? D?21 t=to (v, 1)) = = —(vo, 1) — ID?11t=to (V, 1)1?D?21t=to (V, 1), hence condition (3.4) from Theorem 4 has a form
ID?11t=to («, 1)1?D?21t=to («, 1) = 0.
It is true in this case, since D?211=to (v, 1) = 0. The Frechet derivative has the form
[(/ — Q)N]'Z(x, y, z)h = l(y, 1)l?I(h, 1)| = l(y, 1)l?h
at h e X0. Since by the hypothesis of this theorem D?111=to(v, 1) = 0, then the operator M0-1[(I — Q)N]'z(x, y, z) is a multiplication by a nonzero number l(y, 1)|? for all (x,y,z) from the neighborhood of the point
(vo,D'?11 t=t o (v, 1),D?21 t=t o (v, 1))
in the spase X0, therefore all conditions of Theorem4 are satisfied. □
Remark 2. For example conditions D?111=to (v, 1) = 0, D?21t=to (v, 1) = 0 are met, if a1 = r < a2 e N, (vr, 1) = 0.
5. Initial-boundary value problem for a nonlinear system
Let Q C be a bounded domain with a smooth boundary dQ. Consider the initial boundary value problem
d k x'
z-(s,to) = xik (s), k = 0,1,...,r - 1, s e Q, i = 1,2,3, (5.1)
dtk
dlxi
(s, to) = xu (s), I = r,r + 1,...,т — 1, s e Q, (5.2)
dtl
Xi(s, t) = 0, s e дQ, t > to, г = 1,2,3, (5.3)
D?Axi = Axi + hi (s, t, D?1 x2, D?1 x3,x2, x3), 0 = Ax2 + h2 (s, t, D?1 x2, D?1 x3,..., x2, x3), 0 = Ажэ + h3 (s, t, D?1 X2, D?1 x3,..., x2, x3),
s e Q, t > t0,
(5.4)
where m — 1 < a < m e N,0 < ai < a2 < ■ ■ ■ < an < a, r — 1 < an < r e N, functions hi are defined on R2n+2, i = 1,2,3.
Let A be the Laplace operator with the domain H2(Q) = {z e H2(Q) : z(s) = 0, s e 9Q} С L2(Q), {(pk} be an orthonormal in L2(Q) system of its eigenfunctions corresponding to the eigenvalues of the operator A, numbered in the non-increasing order, taking into account their multiplicity.
We reduce problem (5.1)-(5.4) to abstract problem (3.2), (3.3), by choosing the spaces
* = (#02+2j(Q))3, у = (tf2j(Q))3, (5.5)
where j > f — 1, #02+2i(Q) = {z e H2+2j(Q) : z(s) = 0, s e dQ}, and the operators
A 0 0 A 0 0
L = | 0 0 0 I eC(X; У), M = I 0 A 0 | e C(X; У). (5.6) 0 0 0/ \ 0 0 A
Lemma 1. Let spaces (5.5) and operators (5.6) be given. Then the operator M is (L, 0)-bounded and the projectors have the form
1 0 0 1 0 0 P = 1 000 | , Q = | 000 1 . (5.7)
0 0 0 0 0 0
Proof. For ^ = 1 we have the operator
)-V
(p-L — M)-1 = > y,<fikЫ I 0 —X-1 0 I eC(y; *),
~ (¡л. — 1)-1Л-1 0 0
Ы | 0 —\-1 k=i V 0 0 -A-1
so M (L, a)-bounded,
(p - l)-1 0 0
RLß(M) = lL(M) = £(-,Vk)<pk I 0 0 0
tf(M) = lL(
k=i V 0 0 0
From these equalities, using formulas (3.1) and the residue theorem, we obtain the form of projectors (5.7). Since kerP = kerL, the operator M is (L, 0)-bounded. □
It follows from this lemma that
X1 = H20+2j(Q) x {0} x {0}, X° = {0} x H2+2j(Q) x H°2+2i (Q),
y1 = H2j(Q) x {0} x {0}, = {0} x H2j(Q) x H2j(Q).
From the form of the projector P it follows that the initial conditions (5.1), (5.2) for system (5.3), (5.4) can be written as (3.2). Let's construct according to the initial data elements
¿1 = «1° + ^(t - t°) + ^ (t - to)2 + ■ ■ ■ + (t - t°)m-1
Xi = Xl° + Xy(t - to) + X2 (t - to)2 + ■ ■ ■ + (t - t°)r-1, i = 2,3.
Note that the functions h = h(s, t, z1, z2, z3, . . . , z2n) depend on the 2n phase variables z1, z2,..., z2n. Let us introduce the notation
( jJ^i (S A Z^..^ Z2n) it (S ^ Z2n) \
' ^, ^ Z1,..., ^ ^ (*, t, ^1,..., ) J^ (*, f, ,1,..., Z2n)) .
Theorem 6. Let m - 1 < a < m e N, 0 < a1 < a2 < ■ ■ ■ < an < a, r- 1 <a,n <r e N, hi e C~(R2n+2; R), j > 4 - 1, xik,xu e H°2+2i(Q), i = 1,2,3, k = 0,1,... ,r - 1, I = r,r + 1,... ,m - 1, for some c > 0 for all s e Q
| det J(s, t°, D?11t=t0X2, D*11 = 0Xs,..., D?- |=0Xs) | > c > 0, (5.8)
AXi° + hi (■, t °, D?11=0X2, D?11 t=t 0X3,..., D?" 11=toXa) = 0, i = 2, 3. (5.9)
Then there exists such t1 > t° that problem (5.1)—(5.4) has a unique solution on the set Q x [i°, i1j.
Proof. For the proof, we check the conditions of Theorem 4. First of all, note that Hi(t) e C~((H2+2j(Q))2n; H2+2j(Q)), where
Hi(t)(z 1, Z2,..., Z2n) := hit,t, Z1^), %(■),..., ¿2n(0), i = 1, 2, 3,
for a fixed t by virtue of Proposition 1 [11, p. 197], since 2 + 2j > d/2.
From the form of the projectors obtained in the previous lemma it follows that the nonlinear part of the equation depends only on the elements of the subspace X0, and the mapping (I — Q)N is defined by two functions h2,h3. The bijectivity condition for the operators of the Frechet derivative follows from condition (5.8) of this theorem. Condition (3.4) for this problem is (5.9). By Lemma 1 the operator M is (L, 0)-bounded and by Theorem 4 we obtain the required. □
6. Conclusion
A new class of initial value problems for degenerate evolution equations that are nonlinear with respect to the lowest fractional derivatives is investigated. In what follows, we will study the solvability of optimal control problems for systems whose state is described by equations of this class.
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Guzel Baybulatova, Chelyabinsk State University, 129, Br. Kashirins st., Chelyabinsk, 454021, Russian Federation, tel.: +7(3517)997235, email:
baybulatova_g_d@mail.ru,
ORCID iD https://orcid.org/0000-0002-1629-7959
Marina Plekhanova, Doctor of Sciences (Physics and Mathematics), South Ural State University (National Research University), 76, Lenin av., Chelyabinsk, 454080, Russian Federation; Chelyabinsk State University, 129, Br. Kashirins st., Chelyabinsk, 454021, Russian Federation, tel.: +7(3517)997235, email: mariner79@mail.ru, ORCID iD https://orcid.org/0000-0002-0904-518X
Received 31.01.2021
Начальная задача для одного класса слабо вырожденных полулинейных уравнений с младшими дробными производными
Г. Д. Байбулатова1, М. В. Плеханова1'2
1 Челябинский государственный университет, Челябинск, Российская Федерация,
2 Южно-Уральский государственный университет (НИУ), Челябинск, Российская Федерация
Аннотация. Исследована разрешимость одной начальной задачи для класса эволюционных уравнений со слабым вырождением, нелинейных относительно младших дробных производных Герасимова - Капуто. Линейная часть уравнения содержит относительно ограниченную пару операторов. Доказана однозначная локальная разрешимость в случае нелинейного оператора, зависящего только от элементов подпространства вырождения. Приведены примеры уравнения и системы уравнений в частных производных, начально-краевые задачи для которых сведены к начальной задаче для уравнений в банаховом пространстве изученного класса.
Ключевые слова: дробная производная Герасимова - Капуто, дифференциальное уравнение дробного порядка, вырожденное эволюционное уравнение, полулинейное уравнение.
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Гузель Дамировна Байбулатова, Челябинский государственный университет, Российская Федерация, 454001, г. Челябинск, ул. Братьев Кашириных, 129, тел.: +7(3517)997235, email: baybulatova_g_d@mail.ru, ORCID iD https://orcid.org/0000-0002-1629-7959
Марина Васильевна Плеханова, доктор физико-математических наук, доцент, Южно-Уральский государственный университет (НИУ), Российская Федерация, 454080, г. Челябинск, проспект Ленина, 76; Челябинский государственный университет, Российская Федерация, 454001, г. Челябинск, ул. Братьев Кашириных, 129, тел.: +7(3517)997235, email: mariner79@mail.ru, ORCID iD https://orcid.org/0000-0002-0904-518X
Поступила в 'редакцию 31.01.2021
