**5. A Modified Sobolev System**

Consider another problem

$$\frac{\partial^k \upsilon}{\partial t^k}(\mathbf{x}, 0) = \mathbf{z}\_k(\mathbf{x}), \ \mathbf{x} \in \Omega, \ k = 0, 1, \ldots, m - 1,\tag{17}$$

$$w(\mathbf{x}, t) = h(\mathbf{x}, t), \quad \mathbf{x} \in \Omega, \ t \in [-r, 0], \tag{18}$$

$$v\_{\boldsymbol{\pi}}(\mathbf{x},t) := \sum\_{i=1}^{3} v\_i(\mathbf{x},t)n\_i(\mathbf{x}) = 0, \ (\mathbf{x},t) \in \partial\Omega \times [0,T],\tag{19}$$

$$D\_t^a v(\mathbf{x}, t) = [v(\mathbf{x}, t), \overline{\boldsymbol{\omega}}] - r(\mathbf{x}, t) + \int\_{-r}^0 (K\_1(s)v(t + s) + K\_2(s)r(t + s))ds, \quad (\mathbf{x}, t) \in \Omega \times [0, T], \tag{20}$$

$$\nabla \cdot \boldsymbol{v}(\mathbf{x}, t) = 0, \ (\mathbf{x}, t) \in \Omega \times [0, T], \tag{21}$$

where <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup>3, is a bounded region with a smooth boundary *<sup>∂</sup>*Ω, *<sup>ω</sup>* <sup>∈</sup> <sup>R</sup>3.

Such a system without delay and at *α* = 1 describes the dynamics of small internal movements of a stratified fluid in an equilibrium state [25].

Following the approach of S.L. Sobolev [25], we use the generalized statement of the problem (17)–(21), replacing incompressibility Equation (21) and boundary condition (19) with the equation

$$
\Pi \boldsymbol{w}(\cdot, t) = 0, \quad t \in [0, T]. \tag{22}
$$

where <sup>Π</sup> is the same orthoprojector as in the previous section. Indeed, the set {∇*<sup>ϕ</sup>* : *<sup>ϕ</sup>* <sup>∈</sup> *<sup>C</sup>*∞(Ω)} is dense in the subspace H*<sup>π</sup>* and the integral identity

$$\int\_{\Omega} \langle v, \nabla \boldsymbol{q} \rangle\_{\mathbb{R}^3} = \int\_{\partial \Omega} v\_n \boldsymbol{q} d\boldsymbol{s} - \int\_{\Omega} (\nabla \cdot \boldsymbol{v}) \boldsymbol{q} d\boldsymbol{x}$$

is true for all *<sup>ϕ</sup>* <sup>∈</sup> *<sup>C</sup>*∞(Ω), *<sup>v</sup>* <sup>∈</sup> <sup>H</sup>1, hence, for every *<sup>v</sup>* <sup>∈</sup> <sup>H</sup><sup>1</sup> the satisfaction of conditions (19), (21) is equivalent to the inclusion *<sup>v</sup>* <sup>∈</sup> <sup>H</sup>*σ*. Rejecting the restriction <sup>H</sup><sup>1</sup> we obtain condition (22).

Define by *Bw* = [*w*, *<sup>ω</sup>*] at a fixed *<sup>ω</sup>* <sup>∈</sup> <sup>R</sup><sup>3</sup> the linear operator *<sup>B</sup>* ∈ L(L2;L2). Put <sup>X</sup> <sup>=</sup> <sup>Y</sup> <sup>=</sup> <sup>L</sup><sup>2</sup> <sup>=</sup> <sup>H</sup>*<sup>σ</sup>* <sup>×</sup> <sup>H</sup>*π*,

$$L = \begin{pmatrix} I & O \\ \mathbb{O} & \mathbb{O} \end{pmatrix} \in \mathcal{L}(\mathcal{X}; \mathcal{Y}), \quad M = \begin{pmatrix} \Sigma B & O \\ \Pi B & -I \end{pmatrix} \in \mathcal{L}(\mathcal{X}; \mathcal{Y}).$$

Then it can be shown directly (see [26]), that the operator *M* is (*L*, 0)-bounded and the projectors have the form

$$P = \left(\begin{array}{ccc} I & \mathbb{O} \\ & \Pi B & \mathbb{O} \end{array}\right), Q = \left(\begin{array}{ccc} \mathbb{O} & \mathbb{O} \\ & \mathbb{O} & \mathbb{O} \end{array}\right)$$

.

Therefore, <sup>X</sup> <sup>0</sup> <sup>=</sup> {0} × <sup>H</sup>*π*, <sup>X</sup> <sup>1</sup> <sup>=</sup> {(*w*1, *<sup>w</sup>*2) <sup>∈</sup> <sup>H</sup><sup>2</sup> *<sup>σ</sup>* <sup>×</sup> <sup>H</sup>*<sup>π</sup>* : *<sup>w</sup>*<sup>2</sup> <sup>=</sup> <sup>Π</sup>*Bw*1}, <sup>Y</sup><sup>0</sup> <sup>=</sup> {0} × <sup>H</sup>*π*, <sup>Y</sup><sup>1</sup> <sup>=</sup> <sup>H</sup>*<sup>σ</sup>* × {0}. As in the previous section Theorem <sup>3</sup> at *<sup>p</sup>* <sup>=</sup> 0, *<sup>g</sup>* <sup>≡</sup> 0 implies the next result.

**Theorem 5.** *Let <sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*([−*r*, 0]; <sup>H</sup>*σ*)*, zk* <sup>∈</sup> <sup>H</sup>*σ, <sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>m</sup>* <sup>−</sup> <sup>1</sup>*, <sup>h</sup>*(·, 0) = *<sup>z</sup>*0(·)*,* <sup>K</sup>*<sup>i</sup>* <sup>∈</sup> *<sup>C</sup>m*([−*r*, 0]; <sup>R</sup>)*,* K(*n*) *<sup>i</sup>* (−*r*) = <sup>K</sup>(*n*) *<sup>i</sup>* (0) = 0 *at n* = 0, 1, ... , *m* − 1*, i* = 1, 2*. Then there exists T* > 0*, such that problems* (17)*,* (18)*,* (20)*,* (22) *have a unique solution.*

#### **6. Conclusions**

We studied the local unique solvability of the problem with the generalized Showalter–Sidorov conditions, which is associated by a given background for degenerate fractional evolution equations in Banach spaces with delay, including the Gerasimov–Caputo derivative and a relatively bounded pair of linear operators. The complexity of the studied problem is the simultaneous presence of a fractional derivative, a degenerate operator at it, and a delay argument in the equation. The obtained result shows that by the methods of the theory of resolving families of operators for degenerate evolution equations, this complex problem can be solved. Abstract results can be used for investigating problems for partial differential equations, demonstrated on a problem for Scott–Blair and modified Sobolev systems of equations with delays.

**Author Contributions:** Conceptualization, A.D.; methodology, V.E.F.; validation, A.D.; formal analysis, V.E.F.; investigation, V.E.F.; writing—original draft preparation, A.D., V.E.F.; writing—review and editing, A.D., V.E.F.; supervision, A.D.; project administration, A.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of the second author is done in the framework of the Ural Mathematical Center, it is also supported by Russian Foundation of Basic Research, project 19-41-450001, by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


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