The Maximal Regularity of Nonlinear Second-Order Hyperbolic Boundary Differential Equations

Abstract

In this paper, we show the maximal regularity of nonlinear second-order hyperbolic boundary differential equations. We aim to show if the given second-order partial differential operator satisfies the specific ellipticity condition; additionally, if solutions of the function, which are related to the first-order time derivative, possess no poles nor algebraic branch points, then the maximal regularity of nonlinear second-order hyperbolic boundary differential equations exists. This study explores the use of taking the positive definite second-order operator as the generator of an analytic semi-group. We impose specific boundary conditions to make this positive definite second-order operator self-adjoint. As a linear operator, the self-adjoint operator satisfies the linearity property. This, in turn, facilitates the application of semi-group theory and linear operator theory.

1. Introduction

The maximal regularity of evolution equations within the Lebesgue space scale $L^p(\mathbb{T},X)$ has been a focal point of attention, and the maximal regularity of parabolic equations has been widely investigated in recent decades [1,2,3,4,5,6,7]. However, the maximal regularity of hyperbolic equations has never been studied because the eigenvalues of the differential operator could be complex. In this paper, we will fill in the research gap in this area, applying the approach of studying maximal regularity of parabolic equations to hyperbolic equations.

We study the initial value problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\partial }^{2}u}{\partial t^2}+Lu=f,in{U}_T\hfill \\ u=0,on\partial U\times [0,T]\hfill \\ u=g,onU\times \{t=0\},\hfill \end{array}\right.\end{array}$$

(1)

where $U_T=U\times (0,T]$, where $T>0$ and $U\subset {\mathbb{R}}^n$ is an open, bounded set, $f:U_T\to \mathbb{R}$ is some nonlinear function, $g:U\to \mathbb{R}$, and $u:\overline{U_T}\to \mathbb{R}$ is the unknown, $u=u(x,t)$. The symbol L represents a second-order partial differential operator at each time t, having either the divergence form

$$\begin{array}{c}\hfill Lu=-\sum _{i,j=1}^n{\left(a^{ij}(x,t)u_{x_i}\right)}_{x_j}+\sum _{i=1}^n{b}^i(x,t)u_{x_i}+c(x,t)u\end{array}$$

(2)

or else the nondivergence form

$$\begin{array}{c}\hfill Lu=-\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j}+\sum _{i=1}^n{b}^i(x,t)u_{x_i}+c(x,t)u\end{array}$$

(3)

for given smooth coefficients $a^{ij},b^i,c$$(i,j=1,\cdots ,n),$ with the coefficients $a^{ij}=a^{ji}$ satisfying the strong ellipticity condition:

$$\begin{array}{ccc}\hfill & K^{-1}{\left|\xi \right|}^2\le {\sum }_{i,j=1}^N{a}^{ij}(x,t){\xi }_i{\xi }_j\le 1·{\left|\xi \right|}^2,& \mathrm{for}\xi \in {\mathbb{R}}^N\mathrm{and}(x,t)\in U_T.\hfill \end{array}$$

(4)

where K are some positive constants.

Remark 1.

Note (4) indicates that for each point $(x,t)$, the symmetric $n\times n$ matrix $\mathbf{A}(x,t)=\left(\left(a^{ij}(x,t)\right)\right)$ is positive definite, possessing the smallest eigenvalue greater or equal to $K^{-1}$, and with the greatest eigenvalue less or equal to 1.

Remark 2.

The divergence form (2) is nonlinear since the term $a^{ij}(x,t)u_{x_i{x}_j}$ involves a product of $a^{ij}(x,t)$ and $u_{x_i}$ before calculating the derivative with respect to $x_j$. The nondivergence form (3) is linear. In this paper, we take $Lu$ to be in the nondivergence form (3) since we need $Lu$ to be linear to apply linear operator theory and semigroup theory in the following sections. Since the coefficients of the first and the second derivatives of u do not depend on u itself, $Lu$ is a linear operator in the nondivergence form.

Since u depends on time, and consequently $Lu$ also depends on time, $Lu$ is classified as a non-autonomous operator. In Equation (1), f represents some nonlinear functions of u whose outputs belong to $L^p(J;X)$, where $J:=(0,T)$. Additionally, $▿u$ is assumed to belong to $L^p(J;Y)$ for some space Y that is compatible with X. Given these conditions, Equation (1) is, therefore, nonlinear. Referenced from [8], we have:

Remark 3.

The partial differential operator $\frac{\partial }{\partial t}+L$ is uniformly parabolic and the partial differential operator $\frac{{\partial }^2}{\partial t^2}+L$ is uniformly hyperbolic if there exists a constant $\theta >0$ such that

$$\begin{array}{c}\hfill \sum _{i,j=1}^n{a}^{ij}(x,t){\xi }_i{\xi }_j\ge \theta {\left|\xi \right|}^2\end{array}$$

(5)

for all $(x,t)\in U_T,\xi \in {\mathbb{R}}^n$.

To construct the self-adjoint operator corresponding to the expression ${\sum }_{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j}$ using its weak form, we multiply both sides of the equation by a test function v and then integrate the result over the domain:

$$\begin{array}{c}\hfill {\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j})vdx={\int }_U{f}_{\ast }(x,t)vdx.\end{array}$$

(6)

Since $a^{ij}(x,t)$ are smooth and symmetric $(a^{ij}=a^{ji}),$ based on the boundary conditions in (8), we continue computing (6):

$$\begin{array}{ccc}& & {\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j})vdx\hfill \\ & =& a^{ij}(x,t)u_{x_i}{|}_{\partial U}-{\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i}{v}_{x_j})dx\hfill \\ & =& {\int }_{\partial U}{a}^{ij}(x,t)u_{x_i}v{n}_{j}ds-{\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i}{v}_{x_j})dx\hfill \\ & =& -{\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i}{v}_{x_j})dx\hfill \\ & =& {\int }_U{f}_{\ast }(x,t)vdx\hfill \end{array}$$

the boundary term is zero due to the boundary conditions.

Define the operator $A\left(t\right):H^1\left(U\right)\to H^1{\left(U\right)}^{\prime }$ by

$$\begin{array}{ccc}\hfill & & \left(A\right(t)w,v)\hfill \\ & =& {\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)w_{x_i}{v}_{x_j})dx\hfill \\ & =& {\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)v_{x_i}{w}_{x_j})dx\hfill \\ & =& \left(A\right(t)v,w)\hfill \end{array}$$

(7)

for $\forall w,v\in H^1\left(U\right)$, then $A\left(t\right)$ is a self-adjoint operator.

Lemma 1.

The operators $A\left(t\right)$ defined in (7) are closed and densely defined in some Banach space X.

Proof. 

We consider X to be a Sobolev space, such as $H^1\left(U\right)$, which is the space of functions on U with square-integrable first derivatives. This space is a Banach space with the norm:

$$\begin{array}{c}\hfill {\parallel w\parallel }_{H^1\left(U\right)}=({\int }_U(w^2{+|▿w|}^2{\left)dx\right)}^{\frac{1}{2}}\end{array}$$

The operator $A\left(t\right)$ is densely defined if its domain is dense in X. The domain of $A\left(t\right)$ would be $H_0^1\left(U\right)$, the closure of $C_c^{\infty }\left(U\right)$ in $H^1\left(U\right)$. Since $C_c^{\infty }\left(U\right)$ is dense in $H^1\left(U\right)$, $A\left(t\right)$ is densely defined.

To show that $A\left(t\right)$ is closed, we need to verify that the graph of $A\left(t\right)$ is closed in $X\times X$. The graph of $A\left(t\right)$ is the set:

$$\begin{array}{c}\hfill \Gamma \left(A\right(t\left)\right)=\left\{\right(w,A\left(t\right)w):w\in \mathrm{Dom}(A\left(t\right)\left)\right\}.\end{array}$$

For $A\left(t\right)$ to be closed, if $(w_n,A\left(t\right)w_n)\to (w,v)$ in $X\times X$ as $n\to \infty$, then w must be in the domain of $A\left(t\right)$ and $v=A\left(t\right)w$.

Since $A\left(t\right)$ is symmetric and linear, $A\left(t\right)$ is a continuous operator from $H_0^1\left(U\right)$ to $H^{-1}\left(U\right)$ by the Riesz Representation Theorem.

Under such assumptions, if $w_n\to w$ in $H_0^1\left(U\right)$ and $A\left(t\right)w_n\to v$ in $H_0^1\left(U\right)$, then by continuity of $A\left(t\right)$:

$$\begin{array}{c}\hfill A\left(t\right)w=\underset{n\to \infty }{lim}A\left(t\right)w_n=A\left(t\right)v\end{array}$$

thus, $(w,A(t\left)w\right)=(w,v)$ is in the graph of $A\left(t\right)$, showing that $A\left(t\right)$ is closed. □

We rewrite (1) as

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\frac{{\partial }^{2}u}{\partial t^2}-\sum _{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j}=f-\sum _{i=1}^n{b}^i(x,t)u_{x_i}-cu\hfill & & in{U}_T,\hfill \\ \sum _{i,j=1}^n{a}^{ij}{u}_{x_i}u{n}_j=0,on\partial U\times [0,T]\hfill \\ u=g,onU\times \{t=0\},\hfill \end{array}\right.\end{array}$$

(8)

the reason for rewriting (1) in the form of (8) is that we only left the linear self-adjoint positive definite operator, that is, the term ${\sum }_{i,j=1}^n{a}^{ij}(x,t)u_{x_i{x}_j}$, in the left side of the equation. The linear self-adjoint positive definite operator possesses positive eigenvalues, which facilitates the application of linear operator theory and semigroup theory in the further process.

Subsequently, Equation (8) can be expressed in an abstract manner

$$\begin{array}{c}\hfill \frac{{\partial }^{2}u}{\partial t^2}-Au=f-▿u-cu,u\left(0\right)=g\end{array}$$

(9)

In the following sections, we will transition our analysis from Equations (1)–(9).

2. Methodology

The methodology employed in this study is based on the approach outlined in [9]. Notable achievements in the abstract theory were attained during the 1990s and the first decade of the 21st century, notably by Amann (refer to [10,11]) and Prüss (refer to [12]). This method has been adapted and tailored to the specific requirements of our research question and context.

The core principle of maximal regularity aims to address nonlinear partial differential equations by employing a linearization technique. Consider our nonlinear partial differential Equation (9); since $f-▿u-cu$ are some nonlinear functions depending on u, we define

$$\begin{array}{c}\hfill G\left(u\right)=f-▿u-cu\end{array}$$

so (9) can be written in

$$\begin{array}{c}\hfill \frac{{\partial }^{2}u}{\partial t^2}-A\left(t\right)u=G\left(u\right),u\left(0\right)=g\end{array}$$

(10)

For the purpose of linearizing (10), we determine a particular function u and then search for a solution to the Cauchy problem

$$\begin{array}{c}\hfill \frac{{\partial }^{2}y}{\partial t^2}-A\left(t\right)y=G\left(u\right),u\left(0\right)=g\end{array}$$

(11)

It is worth noting that (11) is a linear equation in terms of y, making it amenable to methods drawn from linear operator theory and semigroup theory. Since (11) is a non-autonomous problem, as u and therefore also $G\left(u\right)$ still depend on time, the original equation can be transformed into the non-autonomous Cauchy problem. Hence, we formulate our primary finding within an abstract framework

$$\begin{array}{c}\hfill \frac{{\partial }^{2}y}{\partial t^2}-A\left(t\right)y=f\left(t\right),u\left(0\right)=g\end{array}$$

(12)

Since all operators $A\left(t\right)$ are closed and densely defined operators in some Banach space X as shown in (1) and have the same domain $D_A$, we possess a norm ${\parallel ·\parallel }_A$ on $D\left(A\right)$ that, for all $t\in (0,T)$, is equivalent to the graph norm of $A\left(t\right)$, defined as ${\parallel ·\parallel }_X+{\parallel A\left(t\right)·\parallel }_X.$ This allows us to equate the unbounded operator $A\left(t\right):X\subset D_A\to X$ with the bounded operator $A\left(t\right)\in L(D_A,X)$.

Assume $a^{ij}(x,t)$ are some measurable functions on $U\times (0,T)$, and are bounded in absolute value by some constant M, $|a^{ij}(x,t)|\le M$. We would verify that for each $t\in (0,T)$, $A\left(t\right)$ is a bounded linear operator from $D_A$ to X. Let $w\in D_A$, we need to bound ${\parallel A\left(t\right)w\parallel }_X$ by ${C\parallel w\parallel }_{D_A}$ for some constant C independent of t.

By the Cauchy–Schwarz inequality and the boundedness of $a^{ij}$:

$$\begin{array}{cc}\hfill \left|\right(A\left(t\right)w,v\left)\right|& \le \left|{\int }_U(\sum _{i,j=1}^n{a}^{ij}(x,t)w_{x_i}{v}_{x_j})dx\right|\hfill \\ \hfill & \le {\int }_U(\sum _{i,j=1}^n|a^{ij}(x,t)\left|\right|w_{x_i}\left|\right|v_{x_j}\left|\right)dx\hfill \\ \hfill & \le M{\int }_U(\sum _{i,j=1}^n|w_{x_i}\left|\right|v_{x_j}\left|\right)dx\hfill \\ \hfill & \le M{n}^2{\int }_U\left(\right|w_x\left|\right|v_x\left|\right)dx\hfill \\ \hfill & \le M{n}^2({\int }_U|w_x{|}^2{dx)}^{\frac{1}{2}}({\int }_U|v_x{{|}^{2}dx)}^{\frac{1}{2}}\hfill \\ \hfill & =M{n}^2\parallel w_x{\parallel }_2{\parallel v_x\parallel }_2\hfill \end{array}$$

(13)

since ${\parallel ·\parallel }_2$ and ${\parallel ·\parallel }_X$ are equivalent norms on $D_A$, subsequently, we obtain the estimates:

$$\begin{array}{c}\hfill \left|(A\left(t\right)w,v)\right|\le M{n}^2{C}_1{C}_2{\parallel w\parallel }_X{\parallel v\parallel }_X\end{array}$$

According to the Riesz Representation Theorem, there exists $C=M{n}^2{C}_1{C}_2$ such that

$$\begin{array}{c}\hfill {\parallel A\left(t\right)w\parallel }_X\le C{\parallel w\parallel }_X\end{array}$$

Since $M{n}^2{C}_1{C}_2$ are independent of t, the bound C is uniform over $(0,T)$. Therefore, $A\left(t\right)$ is a bounded linear operator from $D_A$ to X for each $t\in (0,T)$, and the norm of $A\left(t\right)$ as a linear operator is uniformly bounded over $(0,T)$. Hence, $A\in L^{\infty }((0,T);L(D_A,X))$.

We regard $\mathbb{F}:=L^p(J;X)$ as the basic space for the right-hand side, where J denotes the interval $(0,T)$

$$\begin{array}{c}\hfill \mathbb{E}:=W_p^2(J;X)\cap L^p(J;D_A)\end{array}$$

(14)

We equate $A:(0,T)\to L(D_A,X)$ with a function on $\mathbb{E}$ through the definition of

$$\begin{array}{c}\hfill \left(Ay\right)\left(t\right):=A\left(t\right)y\left(t\right),(t\in (0,T),y\in \mathbb{E})\end{array}$$

For the initial value $u\left(0\right)=g$, we define the trace space referenced from [9]:

Definition 1.

The trace space ${\gamma }_t\mathbb{E}$ is defined by ${\gamma }_t\mathbb{E}:=\{{\gamma }_{t}u:u\in \mathbb{E}\}$, where ${\gamma }_t{u:=u|}_{t=0}$ stands for the time trace of the function u at time $t=0$.

In case of maximal regularity, we expect a unique solution of (12) and a continuous solution operator $S_u$ (depending on $A\left(t\right)$ and therefore on u)

$$\begin{array}{c}\hfill S_u:\mathbb{F}\times {\gamma }_t\mathbb{E}\to \mathbb{E},(f,g)\to y\end{array}$$

of the linear Equation (12). Then the nonlinear Cauchy problem has a unique solution if, and only if, the fixed point equation

$$\begin{array}{c}\hfill y=S_u(G\left(u\right),g)\end{array}$$

has a unique solution $y\in \mathbb{E}$.

When $n=1$ in (12), if the coefficients $a^{ij}$ are time-independent and $a^{ij}\in W^{1,\infty }\left(\Omega \right)$, it is commonly known that the solution possesses the maximal parabolic regularity [13,14,15]:

$$\begin{array}{c}\hfill \parallel {\partial }_t{y\parallel }_{L^p(0,T;L^q)}+{\parallel Ay\parallel }_{L^p(0,T;L^q)}\le {\parallel f\parallel }_{L^p(0,T;L^q)},\mathrm{if}u\left(0\right)=0.\end{array}$$

(15)

for $1<p,q<\infty$ and some specific boundary conditions. The findings have been expanded to cover cases with time-dependent coefficients as well [16,17,18,19,20,21,22,23,24].

By combining the general results presented in [9] with our model (12), we obtain results on the existence and uniqueness of the boundary value problem, as well as $L^p$ maximal regularity estimates.

The primary finding of this paper is presented in the following theorem:

Theorem 1.

For the initial value problem (1) in the abstract setting (12) where $f\in L^p(\mathbb{T},X)$, the $L^p$ maximal regularity

$$\begin{array}{c}\hfill \parallel {\partial }_{tt}{y\parallel }_{L^p(0,T;X)}+{\parallel Ay\parallel }_{L^p(0,T;X)}\le {\parallel f\parallel }_{L^p(0,T;X)}\end{array}$$

(16)

holds only if every solution $y\left(t\right)$ to (16) yields a solution to $\frac{\partial y}{\partial t}=f\left(y\left(t\right)\right)$ without poles or algebraic branch points, given $u\left(0\right)=0$.

3. Preliminaries

We establish the Lemma, which will be utilized in the proof of the theorem presented in Section 4.

Lemma 2.

If A is a positive definite linear self-adjoint operator, $\sqrt[n]{A}$ is also a positive definite linear operator.

Proof. 

First, we want to prove the existence of the operator $\sqrt[n]{A}$. Since A is a positive definite self-adjoint operator, according to spectrum theorem, there is a unique positive definite, self-adjoint operator, denoted as $\sqrt[n]{A}$, that satisfies the condition

$$\begin{array}{c}\hfill {\left(\sqrt[n]{A}\right)}^n=A.\end{array}$$

Next, we prove the linearity of the operator $\sqrt[n]{A}$; a self-adjoint operator on a finite-dimensional complex vector space is diagonalizable with real eigenvalues. Since A is a self-adjoint operator, it possesses an orthonormal basis of eigenvectors $\{e_1,e_2,\cdots ,e_n\}$ with associated eigenvalues $\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\}$. Therefore, A can take the form of:

$$\begin{array}{c}\hfill A=\sum _i^n{\lambda }_i{e}_i{e}_i^{*}\end{array}$$

where $e_i^{*}$ denotes the conjugate transpose (or dual) of $e_i$.

We introduce the operator $\sqrt[n]{A}$ defined as

$$\begin{array}{c}\hfill \sqrt[n]{A}=\sum _i^n\sqrt[n]{{\lambda }_i}{e}_i{e}_i^{*}\end{array}$$

since ${\lambda }_i$ are real and non-negative as A is positive definite, $\sqrt[n]{{\lambda }_i}$ is well defined.

To show that $\sqrt[n]{A}$ is a linear operator, we need to verify that it satisfies the fundamental linearity conditions. Specifically, we will check the following:

• Additivity: For any vectors u and v

  $$\begin{array}{c}\hfill \sqrt[n]{A}(u+v)=\sqrt[n]{A}\left(u\right)+\sqrt[n]{A}\left(v\right)\end{array}$$

  (17)

  given that u and v can be represented as linear combinations of the eigenvectors $\left\{e_i\right\}$, we can use the linearity of A to show that (17) is satisfied.
• Homogeneity: For any scalar c and vector u

  $$\begin{array}{c}\hfill \sqrt[n]{A}\left(cu\right)=c\sqrt[n]{A}\left(u\right).\end{array}$$

  (18)
  The proof method is the same as above.

□

4. Main Results

We consider the main problem in (12):

$$\begin{array}{c}\hfill y^{\left(n\right)}\left(t\right)-A\left(t\right)y=f\left(t\right),y\left(0\right)=g,\end{array}$$

(19)

for every fixed $t_0\in [0,T]$, $A\left(t_0\right)$ belongs to a family of analytic generators on the closed right half plane, and there exists a maximal regularity estimate for $n=1$ with a constant C that does not depend on t within the interval $[0,T]$. Moreover, we suppose that the domain $D\left(A\right(t\left)\right)$ is isomorphic to a fixed Banach space D which is a subset of X, for all $t\in [0,T]$. Additionally, the mapping $t\in [0,T]\to A\left(t\right)\in B(D,X)$ is continuous.

When $f=0$, the characteristic equation for (19) is:

$$\begin{array}{c}\hfill {\lambda }^n-A\left(t\right)=0,y\left(0\right)=g,\end{array}$$

(20)

Here, $\lambda$ represents the eigenvalues (or characteristic roots) of the differential operator. When $n=1$ in (19), the problem becomes one of parabolic equations, we would use the maximal regularity of parabolic equations to generalize the result on hyperbolic equations for $n=2$ in (19).

4.1. When $n=1$: The Maximal Regularity of Parabolic Equations

When $n=1$, solving (20) for $\lambda$, we have $\lambda =A\left(t\right)$. $A\left(t\right)$ generates the analytic semigroup $\left\{e^{sA\left(t_0\right)}\right|s\in \mathbb{R}\}$ for every fixed $t_0\in [0,T]$.

To verify $\left\{e^{sA\left(t_0\right)}\right|s\in \mathbb{R}\}$ is a semi-group under multiplication, we check:

• Closure:

  $$\begin{array}{c}\hfill e^{s_{1}A\left(t_0\right)}·e^{s_{2}A\left(t_0\right)}=e^{(s_1+s_2)A\left(t_0\right)}\end{array}$$
• Associativity:

  $$\begin{array}{c}\hfill (e^{s_{1}A\left(t_0\right)}·e^{s_{2}A\left(t_0\right)})·e^{s_{3}A\left(t_0\right)}=e^{s_{1}A\left(t_0\right)}·(e^{s_{2}A\left(t_0\right)}·e^{s_{3}A\left(t_0\right)})=e^{(s_1+s_2+s_3)A\left(t_0\right)}\end{array}$$

$\left\{e^{sA\left(t_0\right)}\right|s\in \mathbb{R}\}$ is an analytic semi-group if $e^{sA\left(t_0\right)}$ is an analytic function of s for all s in some interval containing 0.

For a linear operator $A\left(t_0\right)$ on a Banach space X, the exponential function $e^{sA\left(t_0\right)}$ is defined as:

$$\begin{array}{c}\hfill e^{sA\left(t_0\right)}=\sum _{n=0}^{\infty }\frac{{\left(sA\left(t_0\right)\right)}^n}{n!}=\sum _{n=0}^{\infty }\frac{s^n{\left(A\left(t_0\right)\right)}^n}{n!}\end{array}$$

(21)

which converges for all $s\in \mathbb{R}$ since $sA\left(t_0\right)\in \mathbb{R}$; therefore, $\left\{e^{sA\left(t_0\right)}\right|s\in \mathbb{R}\}$ is an analytic semi-group.

For each $t_0$ in the interval $[0,T]$, $A\left(t_0\right)$ belongs to a family of analytic generators that fulfills

$$\begin{array}{c}\hfill \parallel R(\lambda ,A\left(t_0\right)){\parallel \le C(1-|\lambda \left|\right)}^{-1}\end{array}$$

(22)

and the maximal regularity estimate with a constant C that does not vary with t within the interval $t\in [0,T]$. To satisfy the R-boundedness (22), we check in (21):

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{\left(A\left(t_0\right)\right)}^n=\frac{1}{(1-A\left(t_0\right))}<\frac{C}{1-\left|\lambda \right|}\end{array}$$

only if $A\left(t_0\right)<1$.

In this scenario, we require that the solution $y\left(s\right)=e^{sA\left(t_0\right)}g$ of Equation (19) with $f=0$ belongs to the space $W_p^1([0,T],X)\cap L_p([0,T],D)$. The set of all $g\in X$ satisfying this condition defines an intermediate space between X and D, which, notably, does not vary with t in our case. We denote this intermediate space as $(Y,\parallel ·{\parallel }_Y)$ and suggest consulting the notes for additional information. For elements $g\in X$, we possess an estimate

$$\begin{array}{c}\hfill \parallel s\to e^{sA\left(t_0\right)}{g\parallel }_{W_p^1([0,T],X)}+\parallel s\to e^{sA\left(t_0\right)}{g\parallel }_{L_p([0,T],D)}\le C^n{\parallel g\parallel }_Y\end{array}$$

(23)

for a constant $C^n$ uniformly in $t\in [0,T]$.

In proving the existence of a solution to (19) when $n=1$ for $g\in Y$, we write

$$\begin{array}{c}\hfill {\partial }_{t}y=A\left(0\right)y\left(t\right)+g_y\left(t\right),g_y\left(t\right)=[A\left(t\right)-A\left(0\right)]y\left(t\right)+f\left(t\right).\end{array}$$

Denote by $L\left(y\right)$ the solution of

$$\begin{array}{c}\hfill {\partial }_{t}u=A\left(0\right)u\left(t\right)+g_y\left(t\right),u\left(0\right)=g\end{array}$$

(24)

Hence, at least formally, the solutions to Equation (24) can be regarded as fixed points of the operator L. We will now utilize maximal regularity to demonstrate that L behaves as a contraction in the space $L_p([0,a],D)$ for $1<p<\infty$, provided that $a\le T$ is sufficiently small. Specifically, for any $y_1,y_2\in L_p([0,a],D)$, the difference $L\left(y_1\right)-L\left(y_2\right)$ corresponds to the solution u of

$$\begin{array}{c}\hfill \frac{\partial u}{\partial t}=A\left(0\right)u\left(t\right)+g_{y_1}\left(t\right)-g_{y_2}\left(t\right),u\left(0\right)=0.\end{array}$$

(25)

Solving (25), we have

$$\begin{array}{c}\hfill u\left(t\right)=e^{A\left(0\right)t}{\int }_0^t[A\left(\tau \right)-A\left(0\right)][y_1\left(\tau \right)-y_2\left(\tau \right)]e^{-A\left(0\right)\tau }d\tau \end{array}$$

(26)

Since D is isomorphic to $D\left(A\right(0\left)\right)$ when equipped with the norm defined by $x\to \parallel A\left(0\right)x\parallel$, we obtain with the maximal regularity inequality

$$\begin{array}{c}\hfill \parallel y^{\left(1\right)}{\parallel }_{L_p([0,T),X)}+{\parallel Ay\parallel }_{L_p([0,T),X)}\le C_p{\parallel f\parallel }_{L_p([0,T),X)}\end{array}$$

we deduce

$$\begin{array}{ccc}& & \parallel L\left(y_1\right)-L\left(y_2\right){\parallel }_{L_p([0,a],D)}\hfill \\ & =& {\parallel u\left(t\right)\parallel }_{L_p([0,a],D)}\hfill \\ & \le & {C\parallel A\left(0\right)u\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel g_{y_1}\left(t\right)-g_{y_2}{\left(t\right)\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel [A(·)-A\left(0\right)](y_1-y_2){\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\underset{t\in [0,a]}{sup}{\parallel A\left(t\right)-A\left(0\right)\parallel }_{B(D,X)}{\parallel y_1-y_2\parallel }_{L_p([0,a],D)}\hfill \end{array}$$

Provided a is small enough, the inequality $C·sup\{\parallel A(t)-A(0)\parallel :t\le a\}<1$ holds, enabling us to invoke Banach’s fixed point theorem for the operator L mapping $L_p([0,a],D)$ to itself. This gives us a solution to the differential equation $y^{\left(1\right)}\left(t\right)-A\left(t\right)y=f\left(t\right)$ with initial condition $y\left(0\right)=g$ on $[0,a]$. Given that our constants are independent of t and $A\left(t\right)$ is uniformly continuous on $[0,T]$, we can extend this solution by repeatedly applying the same argument with updated initial values $g=y\left(a\right)$. After repeating this process multiple times, we obtain a unique solution on the entire interval $[0,T]$.

Remark 4.

The application of maximal regularity on parabolic equation above implies the rate of change of $A\left(t\right)$, $y_1\left(t\right)$ and $y_2\left(t\right)$ are sufficiently small so that there exists a sufficiently large number C to bound (26), ${\int }_0^t[A\left(\tau \right)-A\left(0\right)][y_1\left(\tau \right)-y_2\left(\tau \right)]e^{-A\left(0\right)\tau }\le C{e}^{-A\left(0\right)\tau }$ for some appropriate C.

4.2. When $n=2$: The Maximal Regularity of Parabolic Equations

When $n=2$, solving (20) for $\lambda$, we have $\lambda =\pm \sqrt{A\left(t\right)}$. Since $A\left(t\right)$ is positive definite according to (4), the eigenvalues corresponding to $A\left(t\right)$ are positive real numbers. Therefore, we only take the positive real $\lambda$, that is

$$\begin{array}{c}\hfill \lambda =\sqrt{A\left(t\right)}\end{array}$$

(27)

To satisfy the condition to generate the analytic semigroup, $\sqrt{A\left(t\right)}$ must be a linear self-adjoint operator; from Lemma (2), we have

$$\begin{array}{c}\hfill (A^{\frac{1}{2}}v,A^{\frac{1}{2}}v)=(A^{2}v,v)\end{array}$$

(28)

and

$$\begin{array}{c}\hfill A^{\frac{1}{2}}(\alpha v+\beta w)=\alpha A^{\frac{1}{2}}v+\beta A^{\frac{1}{2}}w\end{array}$$

(29)

which means $A^{\frac{1}{2}}$ is a linear operator. So, $A^{\frac{1}{2}}$ generates the analytic semigroup $\left\{e^{s\sqrt{A\left(t_0\right)}}\right|s\in \mathbb{R}\}$ for every fixed $t_0\in [0,T]$.

To verify $\left\{e^{sA\left(t_0\right)}\right|s\in \mathbb{R}\}$ is a semi-group under multiplication, we check:

• Closure:

  $$\begin{array}{c}\hfill e^{s_1\sqrt{A\left(t_0\right)}}·e^{s_2\sqrt{A\left(t_0\right)}}=e^{(s_1+s_2)\sqrt{A\left(t_0\right)}}\end{array}$$
• Associativity:

  $$\begin{array}{cc}\hfill (e^{s_1\sqrt{A\left(t_0\right)}}·e^{s_2\sqrt{A\left(t_0\right)}})·e^{s_{3}A\sqrt{\left(t_0\right)}}& =e^{s_1\sqrt{A\left(t_0\right)}}·(e^{s_2\sqrt{A\left(t_0\right)}}·e^{s_3\sqrt{A\left(t_0\right)}})\hfill \\ \hfill & =e^{(s_1+s_2+s_3)\sqrt{A\left(t_0\right)}}\hfill \end{array}$$

$\left\{e^{s\sqrt{\left(t_0\right)}}\right|s\in \mathbb{R}\}$ is an analytic semi-group if $e^{s\sqrt{A\left(t_0\right)}}$ is an analytic function of s for all s in some interval containing 0.

For a linear operator $\sqrt{A(t_0})$ on a Banach space X, the exponential function $e^{s\sqrt{A\left(t_0\right)}}$ is defined as:

$$\begin{array}{c}\hfill e^{s\sqrt{A\left(t_0\right)}}=\sum _{n=0}^{\infty }\frac{\left(s\sqrt{A(t_0}\right){)}^n}{n!}=\sum _{n=0}^{\infty }\frac{s^n\left(\sqrt{A(t_0}\right){)}^n}{n!}\end{array}$$

(30)

which converges for all $s\in \mathbb{R}$ since $s\sqrt{A\left(t_0\right)}\in \mathbb{R}$; therefore, $\left\{e^{s\sqrt{A\left(t_0\right)}}\right|s\in \mathbb{R}\}$ is an analytic semi-group.

For each $t_0$ in the interval $[0,T]$, $A\left(t_0\right)$ belongs to a family of analytic generators that fulfills

$$\begin{array}{c}\hfill \parallel R(\lambda ,A\left(t_0\right)){\parallel \le C(1-|\lambda \left|\right)}^{-1}\end{array}$$

(31)

To satisfy the R-boundedness (31), we check in (30):

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\left(\sqrt{A(t_0}\right){)}^n=\frac{1}{(1-\sqrt{A\left(t_0\right))}}<\frac{C}{1-\left|\lambda \right|}\end{array}$$

only if $\sqrt{A(t_0})<1$ so that $A\left(t_0\right)<1$.

For the solution $y\left(s\right)=e^{s\sqrt{A\left(t_0\right)}}g$ of (19) with $f=0$ to be in the space $W_p^2([0,T],X)\cap L_p([0,T],D)$ is our requirement here. The collection of all such g in X forms an intermediate space between X and D, which, importantly, is independent of t in our context. We designate this intermediate space as $(Y,\parallel ·{\parallel }_Y)$ and refer to the notes for more details. For g belonging to Y, we have an estimation

$$\begin{array}{c}\hfill \parallel s\to e^{s\sqrt{A\left(t_0\right)}}{g\parallel }_{W_p^2([0,T],X)}+\parallel s\to e^{s\sqrt{A\left(t_0\right)}}{g\parallel }_{L_p([0,T],D)}\le C^n{\parallel g\parallel }_Y\end{array}$$

(32)

for a constant $C^n$ uniformly in $t\in [0,T]$.

In proving the existence of a solution to (19) when $n=2$ for $g\in Y$, we write

$$\begin{array}{c}\hfill y^{\left(2\right)}\left(t\right)=A\left(0\right)y\left(t\right)+g_y\left(t\right),g_y\left(t\right)=[A\left(t\right)-A\left(0\right)]y\left(t\right)+f\left(t\right).\end{array}$$

Let $L\left(y\right)$ represent the solution of

$$\begin{array}{c}\hfill {\partial }_{tt}u=A\left(0\right)u\left(t\right)+g_y\left(t\right),u\left(0\right)=g\end{array}$$

(33)

then, if the solution of (33) are fixed points of L, the solution is unique. Indeed, we prove the solution exists and is unique by contradiction. Suppose $y_1,y_2\in L_p([0,a],D)$, then $L\left(y_1\right)-L\left(y_2\right)$ corresponds to the solution u of

$$\begin{array}{ccc}\hfill \frac{{\partial }^{2}u}{\partial t^2}& =& A\left(0\right)u\left(t\right)+g_{y_1}\left(t\right)-g_{y_2}\left(t\right),\hfill \\ & =& A\left(0\right)u\left(t\right)+[A\left(t\right)-A\left(0\right)][y_1\left(t\right)-y_2\left(t\right)],u\left(0\right)=0.\hfill \end{array}$$

(34)

Integrating and transforming (34), we have

$$\begin{array}{cc}\hfill \frac{\partial u}{\partial t}& =\int \frac{{\partial }^{2}u}{\partial t}-C\hfill \\ & =\int A\left(0\right)u\left(t\right)dt+\int (g_{y_1}\left(t\right)-g_{y_2}\left(t\right))dt-C\hfill \\ & =\int A\left(0\right)u\left(t\right)dt+\int ([A\left(t\right)-A\left(0\right)][y_1\left(t\right)-y_2\left(t\right)])dt-C\hfill \end{array}$$

(35)

Solving (35), we define a new function $v\left(t\right)$ which is the integral of $u\left(t\right)$:

$$\begin{array}{c}\hfill v(t)=\int u(t)dt\end{array}$$

then

$$\begin{array}{c}\hfill \frac{\partial v}{\partial t}=u\left(t\right)\end{array}$$

here, we implicitly assume that the constant term of the integral is zero, as we can absorb this constant term by adjusting the initial condition of $v\left(t\right)$. Denote the second integral

$$\begin{array}{c}\hfill I\left(t\right)=\int ([A\left(t\right)-A\left(0\right)][y_1\left(t\right)-y_2\left(t\right)])dt\end{array}$$

Thus, we have

$$\begin{array}{c}\hfill \frac{\partial u}{\partial t}=A\left(0\right)v\left(t\right)+I\left(t\right)\end{array}$$

(36)

we can rewrite (36) as

$$\begin{array}{c}\hfill \frac{{\partial }^{2}v}{\partial t^2}=A\left(0\right)v\left(t\right)+I\left(t\right),v\left(0\right)=0,v^{\prime }\left(0\right)=0\end{array}$$

(37)

the general (homogeneous) solutions to (37) are

$$\begin{array}{c}\hfill v_{hom}\left(t\right)=c_1{e}^{\sqrt{A\left(0\right)}t}+c_2{e}^{-\sqrt{A\left(0\right)}t}\end{array}$$

now, we want to find the particular solution to (37), since we have

$$\begin{array}{c}\hfill {\int }_0^t[A\left(\tau \right)-A\left(0\right)][y_1\left(\tau \right)-y_2\left(\tau \right)]e^{-A\left(0\right)\tau }\le C{e}^{-A\left(0\right)\tau }\end{array}$$

for some appropriate C from the application of maximal regularity of the parabolic equation in the last section, for $I\left(t\right)\sim e^{-A\left(0\right)t}$, by Duhamel’s integral formula,

$$\begin{array}{c}\hfill v_p=\frac{1}{2}sinh\left(A\left(0\right)t\right)-\frac{1}{2}t{e}^{A\left(0\right)t}\end{array}$$

(38)

So, the solution to (37) is

$$\begin{array}{c}\hfill v\left(t\right)=v_{hom}\left(t\right)+v_p\left(t\right)\end{array}$$

(39)

substituting (39) back to (36), notice the equation

$$\begin{array}{c}\hfill \frac{\partial u}{\partial t}=A\left(0\right)v\left(t\right)+I\left(t\right)=f\left(y\left(t\right)\right)\end{array}$$

(40)

established since $f\left(y\right(t\left)\right)=A\left(0\right)v\left(t\right)+I\left(t\right)$ has no singularity and $u\left(t\right)$ can be expressed in the form of $C{e}^{Ct}$; if $f\left(y\right(t\left)\right)$ has singularity, $u\left(t\right)$ can not be expressed in the form of $C{e}^{Ct}$ as $C{e}^{Ct}$ is an entire function.

One can check [25], page 3, the movable singularities of all solutions to the equation $u^{\prime }=I(z,u)$, with I being irrational in u, are either algebraic branch points or logarithmic singularities.

One can check [25], Painlevé’s theorem in page 3, the movable singularities of all solutions to the equation of the form $u^{\prime }=R(z,u)$, where R is rational in u with coefficients that are analytic in z on some common open set, are either poles or algebraic branch points.

In condition $u^{\prime }=I\left(u\left(t\right)\right)$, where I is non-algebraic in u, all movable singularities of all complex solutions of $u^{\prime }=I\left(u\left(t\right)\right)$ are at most algebraic branch points; this is supported by the following theorems in [25]:

Theorem 2.

All movable singularities of all complex solutions of an autonomous first-order ordinary differential equation of the form $y^{\prime }=I\left(y\left(t\right)\right)$, where I is non-algebraic in y, are at most algebraic branch points.

Remark 5.

The properties of the integral $u\left(y\right)=\int f\left(y\right(t\left)\right)dt$ depend on the specific form of the function $f\left(y\right(t\left)\right)$. The integral of a non-algebraic function $f\left(y\right(t\left)\right)$ can be either algebraic or non-algebraic; for example, $f\left(y\left(t\right)\right)=\frac{1}{1+t^2}$ is non-algebraic but $\int f\left(y\right(t\left)\right)dt=arctan\left(t\right)+C$ is algebraic over its domain (though not rational) in the sense that it can be expressed as a finite combination of algebraic functions and operations, $f\left(y\left(t\right)\right)=e^t$ is non-algebraic and $\int f\left(y\left(t\right)\right)dt=C{e}^t$ is also non-algebraic. On the other hand, the integral of an algebraic function $f\left(y\right(t\left)\right)$ can be either non-algebraic or algebraic; for example, the function $\frac{1}{t}$ is algebraic at $t\ne 0$, its integral $\int \frac{1}{t}dt=ln\left|t\right|+C$ is non-algebraic in the sense that it cannot be expressed as a finite combination of algebraic functions and operations. This example also shows that the integral of a rational function $f\left(y\right(t\left)\right)$ can be an irrational function. $f\left(y\left(t\right)\right)=t^n$ is algebraic and $\int f\left(y\left(t\right)\right)dt=\frac{t^{n+1}}{n+1}+C$ is also algebraic. Finally, the integral of an irrational function $f\left(y\right(t\left)\right)$ can be a rational function; for example, $f\left(y\left(t\right)\right)=\sqrt{t}$ is irrational for non-square t, but $\int f\left(y\left(t\right)\right)dt=t^{\frac{3}{2}}+C$ is rational.

In general, the solution y to the differential equation involving a positive definite self-adjoint operator $A\left(t\right)$ in (12) will not be algebraic; it will typically involve transcendental functions. Based on these theorems, we have to restrict all solutions of $\frac{\partial y}{\partial t}=f\left(y\left(t\right)\right)$ for (16) have no poles nor algebraic branch points; this is the necessary condition in our proof.

Therefore,

$$\begin{array}{c}\hfill u=C{e}^{CA\left(0\right)t}\end{array}$$

so that

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_X\le C{\parallel e^{CA\left(0\right)t}\parallel }_X\end{array}$$

(41)

Combining with (34), we have

$$\begin{array}{ccc}\hfill & & \parallel \frac{{\partial }^{2}u}{\partial t^2}{\parallel }_{L_p([0,a],X)}\hfill \\ & =& C\parallel A^2\left(0\right)e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & {C\parallel A\left(0\right)u\left(t\right)\parallel }_{L_p([0,a],X)}+C{\parallel g_{y_1}\left(t\right)-g_{y_2}\left(t\right)\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel A\left(0\right)e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}+C{\parallel [A(·)-A\left(0\right)](y_1-y_2)\parallel }_{L_p([0,a],X)}\hfill \end{array}$$

(42)

By calculation,

$$\begin{array}{cc}\hfill \parallel A\left(0\right)e^{CA\left(0\right)t}{\parallel }_{W_p^1([0,a],X)}& =({\int }_0^a\parallel A\left(0\right)e^{CA\left(0\right)t}{\parallel }_X^{p}dt+{\int }_0^a\parallel C{A}^2\left(0\right)e^{CA\left(0\right)t}{{\parallel }_X^{p}dt)}^{\frac{1}{p}}\hfill \\ \hfill & =\left|A\left(0\right)\right|(1+{\left|C\right|}^p{)}^{\frac{1}{p}}({\int }_0^a\parallel e^{CA\left(0\right)t}{{\parallel }_X^{p}dt)}^{\frac{1}{p}}\hfill \\ \hfill & \le C({\int }_0^a\parallel e^{CA\left(0\right)t}{{\parallel }_X^{p}dt)}^{\frac{1}{p}}\hfill \\ \hfill & =C\parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\hfill \end{array}$$

Combining with (42), we have

$$\begin{array}{ccc}\hfill & & \parallel A^2\left(0\right)e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}-{\parallel A\left(0\right)e^{CA\left(0\right)t}\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel A\left(0\right)e^{CA\left(0\right)t}{\parallel }_{W_p^1([0,a],X)}-{\parallel A\left(0\right)e^{CA\left(0\right)t}\parallel }_{L_p([0,a],X)}\hfill \\ & \le & \left(\right|A\left(0\right)\left|\right(1+{\left|C\right|}^p{)}^{\frac{1}{p}}-1)\parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\left|A\left(0\right)\right|\parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel [A(·)-A\left(0\right)](y_1-y_2){\parallel }_{L_p([0,a],X)}\hfill \end{array}$$

(43)

Let ${\lambda }_{max}$ be the largest eigenvalue of $CA\left(0\right)$, then

$$\begin{array}{c}\hfill \parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}^p={\int }_0^a{\parallel e^{CA\left(0\right)t}\parallel }_X^{p}dt\end{array}$$

Remark 6.

The spectral radius $\rho \left(A\right)$ of a matrix A is the maximum absolute value of its eigenvalues; for any matrix norm ${\parallel A\parallel }_X$, it is known that ${\parallel A\parallel }_X\ge \rho \left(A\right)$. Therefore, $\parallel e^{CA\left(0\right)t}{\parallel }_X^p\ge e^{C\rho A\left(0\right)t}=e^{C{\lambda }_{max}t}$.

We have

$$\begin{array}{c}\hfill \parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}^p={\int }_0^a{\parallel e^{CA\left(0\right)t}\parallel }_X^{p}dt\ge {\int }_0^a{e}^{p{\lambda }_{max}t}dt,\end{array}$$

solving the integral:

$$\begin{array}{c}\hfill {\int }_0^a{e}^{p{\lambda }_{max}t}dt={[\frac{1}{p{\lambda }_{max}}{e}^{p{\lambda }_{max}t}]}_0^a=\frac{1}{p{\lambda }_{max}}(e^{p{\lambda }_{max}a}-1).\end{array}$$

Taking the p-th root to get back to the $L_p$ norm:

$$\begin{array}{c}\hfill \parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\ge {(\frac{1}{p{\lambda }_{max}}(e^{p{\lambda }_{max}a}-1))}^{\frac{1}{p}},\end{array}$$

the largest eigenvalues of $CA\left(0\right)$ are less than C, and we deduce

$$\begin{array}{ccc}& & \parallel L(y_1)-L(y_2){\parallel }_{L_p([0,a],D)}\hfill \\ & =& {\parallel u(t)\parallel }_{L_p([0,a],D)}\hfill \\ & \le & {C\parallel A(0)u\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel \frac{{\partial }^{2}u}{\partial t^2}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & (|A(0)|(1+{\left|C\right|}^p{)}^{\frac{1}{p}}-1)\parallel e^{CA(0)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\left|A(0)\right|\parallel e^{CA(0)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\parallel e^{CA(0)t}{\parallel }_{L_p([0,a],X)}\hfill \\ & \ge & C{(\frac{1}{p{\lambda }_{max}}(e^{p{\lambda }_{max}a}-1))}^{\frac{1}{p}}\hfill \\ & =& C{(\frac{1}{p}(e^{pa}-1))}^{\frac{1}{p}}\hfill \\ & \le & C\parallel [A(·)-A(0)](y_1-y_2){\parallel }_{L_p([0,a],X)}\hfill \\ & \le & C\underset{t\in [0,a]}{sup}{\parallel A(t)-A(0)\parallel }_{B(D,X)}{\parallel y_1-y_2\parallel }_{L_p([0,a],D)}\hfill \end{array}$$

Remark 7.

For a symmetric matrix A, the spectral norm is simply the absolute value of the largest eigenvalue of A. Therefore, in the fifth line from the bottom, we have $\left|A\left(0\right)\right|={max}_{1\le i\le n}\left|{\lambda }_i\right|=1$, $C\left|A\left(0\right)\right|\parallel e^{CA\left(0\right)t}{\parallel }_{L_p([0,a],X)}\le C{\parallel e^{CA\left(0\right)t}\parallel }_{L_p([0,a],X)}$.

Remark 8.

${(\frac{1}{p}(e^{pa}-1))}^{\frac{1}{p}}$ is used to check the lower bound of $C\parallel [A(·)-A\left(0\right)](y_1-y_2){\parallel }_{L_p([0,a],X)}$, since if the lower bound of $C\parallel [A(·)-A\left(0\right)](y_1-y_2){\parallel }_{L_p([0,a],X)}$ is greater than or equal to 1, L could not define a contraction map.

If a is small enough,

$$\begin{array}{c}\hfill {(\frac{1}{p}(e^{pa}-1))}^{\frac{1}{p}}\to 0\end{array}$$

(44)

and

$$\begin{array}{c}\hfill C·sup\{\parallel [A\left(t\right)-A\left(0\right)]\parallel :t\le a\}<1.\end{array}$$

(45)

By applying Banach’s fixed point to the mapping L: $L_p([0,a],D)\to L_p([0,a],D)$, we can obtain a solution for $y^{\left(2\right)}\left(t\right)-A\left(t\right)y=f\left(t\right),y\left(0\right)=g$ on $[0,a]$. Since our constants do not vary with t and $A\left(t\right)$ is uniformly continuous on $[0,T]$, we repeat the argument with the initial value $g=y\left(a\right)$, and after many repetitions, we obtain a solution on $[0,T]$, and this solution is unique. Given this unique solution, we can derive the maximal regularity inequality:

$$\begin{array}{c}\hfill \parallel y^{\left(2\right)}{\parallel }_{L_p([0,T),X)}+{\parallel Ay\parallel }_{L_p([0,T),X)}\le C_p{\parallel f\parallel }_{L_p([0,T),X)}\end{array}$$

5. Discussion

Nonlinear hyperbolic partial differential equations are of paramount importance in various fields, including hypoelastic solids (as referenced in [26]), astrophysics (as referenced in [21]), electromagnetic theory (as referenced in [27]), the propagation of heat waves (as referenced in [28]), and numerous other disciplines. In this article, we adopt special boundary conditions to ensure that the second-order differential operator in our second-order nonlinear hyperbolic equations becomes linear, positive definite, and self-adjoint. A positive definite operator implies that the eigenvalues corresponding to the operator are all positive, the square root of the positive eigenvalue is real, and only an operator with real eigenvalues can be made self-adjoint by imposing specific boundary conditions. These conditions are prerequisites for the application of semigroup theory and linear operator theory. Even so, the maximal regularity of the second-order nonlinear hyperbolic equation depends on whether the equation governing the first-order time derivative of the solution possesses algebraic branch points and poles, and we also have to restrict the spectrum norm of the second-order differential operator to be less than 1 to satisfy the R-boundedness condition. This paper provides a methodology for scholars to explore the maximal regularity of other equations.