*3.2. Second Order Abstract Evolution Equations*

**Corollary 1** (Logarithmic representation of infinitesimal generator)**.** *Let operators ∂t*U(*t*,*s*) *and <sup>∂</sup>t*V(*t*,*s*) = *<sup>∂</sup>*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*) *satisfy the sectorial property. If either Equation* (12) *or Equation* (13) *is well-defined, then their square roots being represented by either*

$$\pm \mathcal{A}(t)^{1/2} = \pm \left\{ (I - \kappa e^{-\mathfrak{A}(t,s)})^{-1} \, \partial\_l \widehat{a}(t,s) (I - \kappa e^{-a(t,s)})^{-1} \partial\_l a(t,s) \right\}^{1/2} \tag{14}$$

*or*

$$\pm \mathcal{A}(t)^{1/2} = \pm \left\{ (I - \kappa e^{-\mathfrak{a}(t,s)})^{-1} \partial\_t \mathfrak{a}(t,s) \left(I - \kappa e^{-\mathfrak{A}(t,s)}\right)^{-1} \partial\_t \mathfrak{A}(t,s) \right\}^{1/2} \tag{15}$$

*are the infinitesimal generators of Equation* (8) *in the sense that the solution of <sup>d</sup>*2*u*(*t*)/*dt*<sup>2</sup> <sup>=</sup> <sup>A</sup>(*t*)*u*(*t*) *is represented by*

$$u(t) = \mathcal{U}(t, s)u\_0 = \exp\left[+\mathcal{A}(t)^{1/2}\right]u\_+ + \exp\left[-\mathcal{A}(t)^{1/2}\right]u\_-,\tag{16}$$

*where u*<sup>+</sup> *and u*<sup>−</sup> *are the elements of* X *. The representation* (14) *is valid if* (12) *is true, and the representation* (15) *is valid if* (13) *is true.*

**Proof.** Under the commutation relation, the autonomous Equation (8) is also formally factorized as

$$\left(\partial\_t \mathcal{U}(t,s)^{1/2} + \mathcal{A}^{1/2} \mathcal{U}(t,s)^{1/2}\right) \left(\partial\_t \mathcal{U}(t,s)^{1/2} - \mathcal{A}^{1/2} \mathcal{U}(t,s)^{1/2}\right) u = 0$$

for any *u* ∈ X . It leads to the decomposition such that

$$\begin{cases} \partial\_t \mathcal{U}(t,s)^{1/2} \mathfrak{u}\_+ + \mathcal{A}^{1/2} \mathcal{U}(t,s)^{1/2} \mathfrak{u}\_+ = 0, \\\partial\_t \mathcal{U}(t,s)^{1/2} \mathfrak{u}\_- - \mathcal{A}^{1/2} \mathcal{U}(t,s)^{1/2} \mathfrak{u}\_- = 0. \end{cases}$$

The representation shown in Equation (16) is understood.

It is necessary to confirm the possibility of defining the fractional power of A. The possibility of defining square root of operator is justified if it is possible to define the exponential of

$$\mathbb{E}\log\left[\mathcal{A}(t)^{1/2}\right] := \frac{1}{2}\left\{\mathbb{L}\log\left[\partial\_t\mathcal{L}\log\mathcal{U}(t,s)\right] + \mathbb{L}\log\left[\partial\_t\mathcal{L}\log\mathcal{V}(t,s)\right]\right\},$$

where note that the logarithms of the right hand side are the formal form, and it does not matter whether they are well defined or not. According to the commutation assumption between *∂t*U(*t*,*s*) and *<sup>∂</sup>t*V(*t*,*s*) = *<sup>∂</sup>*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*), the exponential of each logarithm of the right hand side are independently well-defined. According to the commutation relation between U(*t*,*s*), *∂t*U(*t*,*s*) and *∂t*V(*t*,*s*) = *∂*2 *<sup>t</sup>* U(*t*,*s*), the exponential function of right hand side is equal to

$$\left[\partial\_t \mathrm{Log} \mathcal{U}(t,s)\right]^{1/2} \left[\partial\_t \mathrm{Log} \mathcal{V}(t,s)\right]^{1/2} = \left[\partial\_t \mathrm{Log} \mathcal{V}(t,s)\right]^{1/2} \left[\partial\_t \mathrm{Log} \mathcal{U}(t,s)\right]^{1/2},$$

and each square root is well-defined by the sectorial assumption (for the sectorial property, see Section 2.10 of Chapter 5 in [11]), where the logarithms of U(*t*,*s*) and V(*t*,*s*) leading to the definition of fractional powers of U(*t*,*s*) and V(*t*,*s*) are valid as seen in Equations (10) and (11). Consequently, the logarithmic representations of infinitesimal generators ±A(*t*)1/2 are true.

## **4. Conclusions**

In this article, the Miura transform is generalized in the following sense:


where, in terms of applying to theory of higher order abstract evolution equations, the variable is taken as *t* in this article. For the preceding work dealing with the general choice of the evolution direction, see [16,18]. Consequently, the generalized Miura transform is obtained as the product of two logarithmic representations of operators in a general Banach space framework and they are applied to clarify the structure of the general solutions of second order abstract evolution equations defined in finite and infinite dimensional Banach spaces. Since the linear operator A is a generalized concept of matrices, the presented result potentially includes any matrix situations.

**Funding:** This work was partially supported by JSPS KAKENHI Grant No. 17K05440.

**Acknowledgments:** The author is grateful to Emeritus Hiroki Tanabe for valuable comments. The referee's comment on the Reference [6] is acknowledged for giving an idea for initiating this research.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
