Existence and Maximal Regularity of Solutions for a Class of Third-Order Differential Equations with an Unbounded Coefficient

Abstract

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space $L_2\left({\mathbb{R}}^2\right)$. The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in mathematical physics and wave theory, as well as by the limited development of the corresponding theory in the presence of unbounded coefficients. The main focus is on the existence, uniqueness, and maximal regularity of solutions. Within a functional-analytic framework, the well-posedness of the problem is established in natural function spaces under minimal assumptions on the coefficients. In particular, a priori estimates ensuring maximal regularity are derived.

1. Introduction

In recent years, third-order partial differential equations have attracted increasing attention due to their fundamental role in the mathematical modeling of complex processes in physics and engineering. Classical nonlinear models such as the Korteweg–de Vries equation, the Gardner equation, and the Burgers equation are widely used to describe wave propagation, nonlinear diffusion, and dispersive phenomena in various media.

Third-order equations also play a significant role in the construction of macroscopic models for semiconductors that incorporate quantum effects, including models of electric transmission lines and quantum hydrodynamic systems. These applications highlight both the practical importance and the interdisciplinary nature of third-order differential equations.

A substantial body of literature [1,2,3,4,5,6,7,8,9,10,11] is devoted to the study of well-posedness, regularity, and qualitative properties of solutions to third-order differential equations with constant or bounded coefficients, considered in various functional settings and under different boundary conditions. However, many modern applications naturally lead to differential operators defined in unbounded domains and involving unbounded coefficients.

The theory of third-order partial differential equations and the corresponding operator-differential equations has been further developed in the context of nonstandard conditions such as discontinuous coefficients, nonlocal constraints, and unbounded domains. In particular, Aliev and Muradova [12] studied third-order operator-differential equations with discontinuous coefficients and operator terms in boundary conditions, establishing well-posedness under minimal smoothness assumptions.

Nonlocal problems and equations with multiple characteristics were investigated by Khashimov and Smetanova [13], who derived solvability conditions and analyzed the influence of nonlocal terms on the structure of solutions. A related class of boundary value problems for third-order equations with multiple characteristics was considered by Apakov and Rutkauskas [14], where the authors obtained existence and uniqueness results and emphasized the analytical difficulties caused by degeneracy of characteristics.

Problems posed in unbounded domains form a particularly challenging direction. In this setting, Balkizov [15] analyzed a boundary value problem for a third-order parabolic–hyperbolic equation and derived a priori estimates that depend essentially on the behavior of coefficients at infinity. More recently, qualitative properties of third-order differential operators have also been studied; for example, Chaima [16] investigated the dynamical and chaotic behavior of solutions generated by such operators, revealing complex spectral and asymptotic features.

It should be emphasized that classical methods developed for elliptic, hyperbolic, and parabolic equations with unbounded coefficients cannot, in general, be directly applied to third-order differential equations of mixed or singular type. This significantly complicates the analysis and gives rise to unresolved theoretical problems.

At the same time, recent years have seen significant progress in the analysis of partial differential equation (PDE) systems and their applications in control theory, continuum mechanics, and dynamical systems. Modern studies emphasize the interplay between PDE models and advanced analytical methods, particularly in the presence of complex structures such as delays, coupling effects, and unbounded coefficients.

Recent developments in the analysis and control of distributed parameter systems, including PDE–ODE couplings, stabilization, and observer design (see, e.g., [17,18,19,20,21]), demonstrate the growing importance of operator-theoretic methods in complex systems with spatially varying coefficients. Although these works address different applied problems, they highlight modern analytical frameworks that motivate further study of differential operators with non-uniform and unbounded coefficients.

In the context of PDE–ODE coupled systems, Xu and Li [17,18] investigated stabilization and control problems for distributed parameter systems. Analytical approaches to complex transport phenomena have also been actively developed. For example, Bai et al. [15] derived an exact solution based on a three-energy equation model for gaseous transpiration cooling in porous media, illustrating how higher-order PDEs with nonstandard and possibly unbounded coefficients naturally arise in applications.

Taken together, these studies highlight the growing importance of developing a consistent functional-analytic framework for partial differential equations with unbounded coefficients. In particular, while many existing works are motivated by control-theoretic or applied considerations, the underlying analytical questions—especially those related to well-posedness and maximal regularity—remain insufficiently explored for third-order operators with unbounded coefficients.

Motivated by these considerations, the present paper is devoted to the study of existence, uniqueness, and maximal regularity of solutions to a class of third-order partial differential equations with an unbounded lower-order coefficient in the space $L_2\left({\mathbb{R}}^2\right)$. The analysis is carried out within a functional-analytic framework.

From a broader perspective, the linear operator considered in this work can be viewed as the principal part of a class of nonlinear third-order evolution equations. In this sense, the study of the corresponding linear problem constitutes a fundamental step in the analysis of nonlinear models, since it provides the basic coercive estimates and maximal regularity properties required for the application of perturbation techniques and fixed-point methods.

In particular, the obtained results create a rigorous analytical basis for treating nonlinear terms as perturbations in appropriate function spaces, which is essential in the study of nonlinear dispersive equations of Korteweg–de Vries or Gardner type. At the same time, it should be emphasized that the extension of these results to fully nonlinear settings involves additional difficulties related to dispersive interactions and remains a subject for future research.

2. Results

Consider the differential operator

$$(A+\mu I)u={\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}}+q_0\left(x\right)u+\mu u$$

(1)

initially defined in $C_0^{\infty }\left({\mathbb{R}}^2\right)$, where $C_0^{\infty }\left({\mathbb{R}}^2\right)$ is an infinitely differentiable function with compact support set.

Assume that the coefficient $q_0\left(x\right)$ satisfies the following conditions:

(i) $q_0\left(x\right)\ge {\delta }_0>0$ is a continuous function on $\mathbb{R}$, $\mathbb{R}=(-\infty ,\infty )$;

(ii) ${\mu }_0=\underset{|x-t|\le 1}{\sup }{\displaystyle \frac{q_0\left(x\right)}{q_0\left(t\right)}}<\infty$.

Here, $q_0\left(x\right)$ may be an unbounded function.

Under assumptions (i)–(ii), the operator $A+\mu I$ admits a closure, which will also be denoted by $A+\mu I$.

The operator A is closable in $L_2\left({\mathbb{R}}^2\right)$, and we denote by A its closure. The domain $D\left(A\right)$ of the closed operator A is given by

$$D\left(A\right)=\left\{u\in L_2\left({\mathbb{R}}^2\right)|{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}},{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}},{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}},q_0\left(x\right)u\in L_2\left({\mathbb{R}}^2\right)\right\},$$

where all derivatives are understood in the sense of distributions.

In particular, $D\left(A\right)$ is a Sobolev-type space adapted to the structure of the operator, and $C_0^{\infty }\left({\mathbb{R}}^2\right)$ is dense in $D\left(A\right)$ with respect to the norm

$${\parallel u\parallel }_{D\left(A\right)}={\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}+{\parallel Au\parallel }_{L_2\left({\mathbb{R}}^2\right)}.$$

Thus, A is a closed densely defined operator in $L_2\left({\mathbb{R}}^2\right)$.

We consider the equation

$$(A+\mu I)u=f(t,x)\in L_2\left({\mathbb{R}}^2\right).$$

(2)

Definition 1. 

A function $u\in L_2\left({\mathbb{R}}^2\right)$ is called a solution of Equation (2) if there exists a sequence $\left\{u_n\right\}\subset C_0^{\infty }\left({\mathbb{R}}^2\right)$ such that

$${∥u_n-u∥}_{L_2\left({\mathbb{R}}^2\right)}\to 0,{∥(A+\lambda I)u_n-f∥}_{L_2\left({\mathbb{R}}^2\right)}\to 0,asn\to \infty .$$

Definition 2 

([22]). A solution $u(t,x)\in L_2\left({\mathbb{R}}^2\right)$ of Equation (2) is said to be maximally regular if the estimate

$${∥{\displaystyle \frac{\mathsf{\partial }u}{dt}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{\mathsf{\partial }u}{dx}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{d{x}^3}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥q_0\left(x\right)u∥}_{L_2\left({\mathbb{R}}^2\right)}\le c{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)},$$

(3)

holds, where $c>0$ is a constant.

Theorem 1. 

Let the coefficient $q_0\left(x\right)$ satisfy conditions (i)–(ii). Then, for any $f(t,x)\in L_2\left({\mathbb{R}}^2\right)$, there exists a unique maximally regular solution $u(t,x)$ of Equation (2), and estimate (3) holds.

2.1. Auxiliary Results

To prove Theorem 1, we establish several auxiliary estimates.

Lemma 1. 

Assume that condition (i) is satisfied and $\mu \ge 0$. Then, the operator $A+\mu I$ satisfies the a priori estimate

$${∥(A+\mu I)u∥}_{L_2\left({\mathbb{R}}^2\right)}\ge ({\delta }_0+\mu ){∥u∥}_{L_2\left({\mathbb{R}}^2\right)},$$

(4)

for all $u\in D\left(A\right)$.

Proof. 

The proof follows from the properties of the functional $<(A+\mu I)u,u>$, $u\in D\left(A\right)$. Since the operator A is not symmetric, we consider the real part of the inner product:

$$\mathrm{Re}\langle (A+\mu I)u,u\rangle .$$

By the definition of A, we have

$$\langle (A+\mu I)u,u\rangle =〈{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}},u〉+〈{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}},u〉+〈{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}},u〉+\langle q_0\left(x\right)u,u\rangle +\mu {\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}^2.$$

We analyze each term. For sufficiently smooth $u\in D\left(A\right)$ (by density of $C_0^{\infty }\left({\mathbb{R}}^2\right)$), integration by parts yields

$$\mathrm{Re}〈{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}},u〉=0,\mathrm{Re}〈{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}},u〉=0,\mathrm{Re}〈{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}},u〉=0,$$

since boundary terms vanish due to compact support (or approximation argument).

Therefore,

$$\mathrm{Re}\langle (A+\mu I)u,u\rangle ={\int }_{{\mathbb{R}}^2}{q}_0{\left(x\right)\left|u\right|}^{2}dxdt+\mu {\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}^2.$$

Using Condition (i), $q_0\left(x\right)\ge {\delta }_0>0$, we obtain

$$\mathrm{Re}\langle (A+\mu I)u,u\rangle \ge ({\delta }_0+\mu ){\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}^2.$$

Finally, by the Cauchy–Schwarz inequality,

$$\mathrm{Re}\langle (A+\mu I)u,u\rangle \le {\parallel (A+\mu I)u\parallel }_{L_2\left({\mathbb{R}}^2\right)}{\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}.$$

Combining the inequalities, we arrive at

$${\parallel (A+\mu I)u\parallel }_{L_2\left({\mathbb{R}}^2\right)}\ge ({\delta }_0+\mu ){\parallel u\parallel }_{L_2\left({\mathbb{R}}^2\right)}.$$

□

Using the Fourier transform, the operator $A+\mu I$ is reduced to the study of an associated third-order differential operator

$$(A_{\tau }+\mu I)u\left(x\right)=u^{‴}\left(x\right)+u^{\prime }\left(x\right)+(i\tau +q_0\left(x\right))u+\mu I,u\in C_0^{\infty }\left(\mathbb{R}\right),-\infty <\tau <\infty .$$

The closure of this operator $A_{\tau }+\mu I$ in $L_2\left(\mathbb{R}\right)$ is again denoted by $A_{\tau }+\mu I$.

Lemma 2. 

Assume that condition (i) is satisfied and $\mu \ge 0$. Then, the operator $A_{\tau }+\mu I$ satisfies the estimate

$${∥(A_{\tau }+\mu I)u∥}_{L_2\left(\mathbb{R}\right)}\ge (\delta +\mu ){∥u∥}_{L_2\left(\mathbb{R}\right)}.$$

Proof. 

Let $u\in D\left(A_{\tau }\right)$. We consider the inner product

$${\langle (A_{\tau }+\mu I)u,u\rangle }_{L_2\left(\mathbb{R}\right)}.$$

By the definition of $A_{\tau }$, we have

$$(A_{\tau }+\mu I)u=u^{‴}+u^{\prime }+(i\tau +q_0\left(x\right)+\mu )u.$$

Therefore,

$$\langle (A_{\tau }+\mu I)u,u\rangle =\langle u^{‴},u\rangle +\langle u^{\prime },u\rangle +{i\tau \parallel u\parallel }_{L_2\left(\mathbb{R}\right)}^2+{\int }_{\mathbb{R}}(q_0\left(x\right)+\mu ){\left|u\left(x\right)\right|}^{2}dx.$$

We now analyze each term. Using the Fourier transform and the Plancherel theorem, we obtain

$$\langle u^{‴},u\rangle ={\int }_{\mathbb{R}}(-i{\xi }^3)\tilde{u}\left(\xi \right)\overline{\tilde{u}\left(\xi \right)}d\xi ,$$

$$\langle u^{\prime },u\rangle ={\int }_{\mathbb{R}}\left(i\xi \right)\tilde{u}\left(\xi \right)\overline{\tilde{u}\left(\xi \right)}d\xi .$$

Hence, both terms are purely imaginary, and therefore

$$\mathrm{Re}\langle u^{‴},u\rangle =0,\mathrm{Re}\langle u^{\prime },u\rangle =0.$$

Also,

$$\mathrm{Re}\left({i\tau \parallel u\parallel }_{L_2\left(\mathbb{R}\right)}^2\right)=0.$$

Thus, taking the real part, we obtain

$$\mathrm{Re}\langle (A_{\tau }+\mu I)u,u\rangle ={\int }_{\mathbb{R}}(q_0\left(x\right)+\mu ){\left|u\left(x\right)\right|}^{2}dx.$$

Using Condition (i), $q_0\left(x\right)\ge {\delta }_0>0$, we get

$$\mathrm{Re}\langle (A_{\tau }+\mu I)u,u\rangle \ge ({\delta }_0+\mu ){\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}^2.$$

On the other hand, by the Cauchy–Schwarz inequality,

$$|\langle (A_{\tau }+\mu I)u,u\rangle |\le \parallel (A_{\tau }+\mu I){u\parallel }_{L_2\left(\mathbb{R}\right)}{\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Combining the last two inequalities, we obtain

$$\parallel (A_{\tau }+\mu I){u\parallel }_{L_2\left(\mathbb{R}\right)}{\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}\ge ({\delta }_0+\mu ){\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}^2.$$

If $u\ne 0$, dividing by ${\parallel u\parallel }_2$ yields

$$\parallel (A_{\tau }+\mu I){u\parallel }_{L_2\left(\mathbb{R}\right)}\ge ({\delta }_0+\mu ){\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

□

Lemma 3. 

Assume that conditions (i)–(ii) are satisfied. Then there exists ${\mu }_0>0$ such that the operator $A_{\tau }+\mu I$ is continuously invertible in $L_2\left(\mathbb{R}\right)$ for all $\mu >{\mu }_0>0$ and

$${(A_{\tau }+\mu I)}^{-1}f={({\tilde{A}}_{\tau }+\mu I)}^{-1}{(I+A_{\mu })}^{-1}f,f\left(x\right)\in L_2\left(\mathbb{R}\right),$$

(5)

where $({\tilde{A}}_{\tau }+\mu I)u=u^{‴}\left(x\right)+(i\tau +q_0\left(x\right)+\mu )u,u\in D\left({\tilde{A}}_{\tau }\right),A_{\mu }=D_x{({\tilde{A}}_{\tau }+\mu I)}^{-1},D_x={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}},{\parallel A\mu \parallel }_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}<1$.

To prove Lemma 3, we need the following lemmas.

Lemma 4. 

Assume that conditions (i)–(ii) are satisfied and $\mu \ge 0$. Then, there exists the continuous inverse operator ${({\tilde{A}}_{\tau }+\mu I)}^{-1}$ in $L_2\left(\mathbb{R}\right)$ and the estimate

$${∥u^{‴}∥}_{L_2\left(\mathbb{R}\right)}+{∥(i\tau +q_0\left(x\right))u∥}_{L_2\left(\mathbb{R}\right)}\le c_0{∥({\tilde{A}}_{\tau }+\mu I)u∥}_{L_2\left(\mathbb{R}\right)},$$

holds for all $u\in D\left({\tilde{A}}_{\tau }\right)$, where $c_0>0$ is a constant.

Proof. 

For brevity, set

$$B_{\tau }:={\tilde{A}}_{\tau }+\mu I.$$

We divide the proof into several steps.

Step 1. Local model with frozen coefficient. For each $j\in \mathbb{Z}$, choose a point $x_j\in [j,j+1]$ and put

$$q_j:=q_0\left(x_j\right).$$

Consider the constant-coefficient operator

$$B_{\tau ,j}:={\displaystyle \frac{d^3}{d{x}^3}}+(i\tau +q_j+\mu )I.$$

Its Fourier symbol is

$$p_j(\xi ,\tau )={\left(i\xi \right)}^3+i\tau +q_j+\mu =-i{\xi }^3+i\tau +q_j+\mu .$$

Since $\Re p_j(\xi ,\tau )=q_j+\mu \ge {\delta }_0+\mu >0$, it follows that $p_j(\xi ,\tau )\ne 0$ for all $\xi ,\tau \in \mathbb{R}$. Hence, $B_{\tau ,j}$ is invertible in $L_2\left(\mathbb{R}\right)$.

Moreover, by Plancherel’s theorem,

$$\parallel v^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}^2+\parallel (i\tau +q_j){v\parallel }_{L_2\left(\mathbb{R}\right)}^2\le C{\parallel B_{\tau ,j}v\parallel }_{L_2\left(\mathbb{R}\right)}^2,$$

with a constant $C>0$ independent of j and $\tau$. Indeed, in the Fourier variables,

$$\widehat{B_{\tau ,j}v}\left(\xi \right)=\left(-i{\xi }^3+i\tau +q_j+\mu \right)\widehat{v}\left(\xi \right),$$

and therefore

$${\left|\xi \right|}^6|\widehat{v}{\left(\xi \right)|}^2+{\left(\left|\tau \right|+q_j\right)}^2|\widehat{v}{\left(\xi \right)|}^2\le C|p_j{(\xi ,\tau )|}^2{\left|\widehat{v}\left(\xi \right)\right|}^2.$$

After integration in $\xi$, the required estimate follows.

Step 2. Partition of unity and comparison of $q_0\left(x\right)$ with $q_j$. Let ${\left\{{\phi }_j\right\}}_{j\in \mathbb{Z}}\subset C_0^{\infty }\left(\mathbb{R}\right)$ be a standard partition of unity such that

$$0\le {\phi }_j\le 1,\sum _{j\in \mathbb{Z}}{\phi }_j^2\left(x\right)=1,$$

$$\mathrm{supp}{\phi }_j\subset (j-1,j+2),\underset{j}{\sup }{\parallel {\phi }_j^{\left(k\right)}\parallel }_{\infty }<\infty ,k=1,2,3,$$

and the multiplicity of the covering is finite.

For $x\in \mathrm{supp}{\phi }_j$, we have $|x-x_j|\le 2$. By continuity of $q_0$ and Assumption (ii), the values of $q_0$ are locally comparable on bounded intervals. Hence, there exists a constant $c_1\ge 1$, independent of j, such that

$$c_1^{-1}{q}_j\le q_0\left(x\right)\le c_1{q}_j,x\in \mathrm{supp}{\phi }_j.$$

Consequently,

$$\parallel (i\tau +q_0\left(x\right)){\phi }_j{u{\displaystyle \parallel }}_2\asymp {\parallel (i\tau +q_j){\phi }_{j}u\parallel }_2,$$

with constants independent of j and $\tau$.

Step 3. Localization of the operator. Applying the estimate from Step 1 to $v={\phi }_{j}u$, we obtain

$$\parallel {\left({\phi }_{j}u\right)}^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel (i\tau +q_j){\phi }_j{u\parallel }_{L_2\left(\mathbb{R}\right)}\le c{\parallel B_{\tau ,j}\left({\phi }_{j}u\right)\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Now

$$B_{\tau ,j}\left({\phi }_{j}u\right)={\phi }_j{B}_{\tau }u+\left[\frac{d^3}{d{x}^3},{\phi }_j\right]u+(q_j-q_0\left(x\right)){\phi }_{j}u,$$

where

$$\left[\frac{d^3}{d{x}^3},{\phi }_j\right]u=3{\phi }_j^{\prime }{u}^{″}+3{\phi }_j^{″}{u}^{\prime }+{\phi }_j^{‴}u.$$

Hence,

$$\parallel {\left({\phi }_{j}u\right)}^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+{\parallel (i\tau +q_j){\phi }_{j}u\parallel }_{L_2\left(\mathbb{R}\right)}\le$$

$$c\left(\parallel {\phi }_j{B}_{\tau }{u\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel {\chi }_j{u}^{″}{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel {\chi }_j{u}^{\prime }{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel {\chi }_j{u\parallel }_{L_2\left(\mathbb{R}\right)}+{\parallel (q_j-q_0\left(x\right)){\phi }_{j}u\parallel }_{L_2\left(\mathbb{R}\right)}\right),$$

where ${\chi }_j$ is the characteristic function of a slightly larger interval containing $\mathrm{supp}{\phi }_j$.

Using the local comparability of $q_0\left(x\right)$ and $q_j$ again, we get

$$\parallel (q_j-q_0\left(x\right)){\phi }_j{u\parallel }_{L_2\left(\mathbb{R}\right)}\le c{\parallel q_0\left(x\right){\phi }_{j}u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Step 4. Estimates of intermediate derivatives. By the standard interpolation inequalities in $W_2^3\left(\mathbb{R}\right)$, for every $\epsilon >0$,

$$\parallel u^{\prime }{\parallel }_{L_2\left(\mathbb{R}\right)}\le \epsilon \parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+{c\left(\epsilon \right)\parallel u\parallel }_{L_2\left(\mathbb{R}\right)},\parallel u^{″}{\parallel }_{L_2\left(\mathbb{R}\right)}\le \epsilon \parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+c\left(\epsilon \right){\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Since $q_0\left(x\right)\ge {\delta }_0>0$, we also have

$${\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}\le {\delta }_0^{-1}\parallel q_0{\left(x\right)u\parallel }_{L_2\left(\mathbb{R}\right)}\le {\delta }_0^{-1}{\parallel (i\tau +q_0\left(x\right))u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Therefore,

$$\parallel u^{\prime }{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel u^{″}{\parallel }_{L_2\left(\mathbb{R}\right)}\le \epsilon \parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+c\left(\epsilon \right){\parallel (i\tau +q_0\left(x\right))u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Substituting these inequalities into the localized estimate obtained in Step 3, summing with respect to j, and using the finite overlap of the partition of unity, we arrive at

$$\parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel (i\tau +q_0\left(x\right)){u\parallel }_{L_2\left(\mathbb{R}\right)}\le c\parallel B_{\tau }{u\parallel }_{L_2\left(\mathbb{R}\right)}+\epsilon c\parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+c\left(\epsilon \right){\parallel (i\tau +q_0\left(x\right))u\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Choosing $\epsilon >0$ sufficiently small and absorbing the lower-order terms into the left-hand side, we obtain

$$\parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel (i\tau +q_0\left(x\right)){u\parallel }_{L_2\left(\mathbb{R}\right)}\le c_0{\parallel B_{\tau }u\parallel }_{L_2\left(\mathbb{R}\right)},$$

which is the desired coercive estimate.

Step 5. Existence of the inverse operator. The previous estimate implies, in particular,

$${\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}\le {\delta }_0^{-1}\parallel (i\tau +q_0\left(x\right)){u\parallel }_{L_2\left(\mathbb{R}\right)}\le c{\parallel B_{\tau }u\parallel }_{L_2\left(\mathbb{R}\right)},u\in D\left(B_{\tau }\right).$$

Hence, $B_{\tau }$ is injective and has closed range in $L_2\left(\mathbb{R}\right)$.

Now, consider the adjoint operator

$$B_{\tau }^{\ast }=-{\displaystyle \frac{d^3}{d{x}^3}}+(-i\tau +q_0\left(x\right)+\mu )I,$$

defined on the same type of domain. Repeating the above argument for $B_{\tau }^{\ast }$, we obtain

$${\parallel v\parallel }_{L_2\left(\mathbb{R}\right)}\le c{\parallel B_{\tau }^{\ast }v\parallel }_{L_2\left(\mathbb{R}\right)},v\in D\left(B_{\tau }^{\ast }\right),$$

so $\ker B_{\tau }^{\ast }=\left\{0\right\}$. Since

$$\overline{\mathrm{Ran}\left(B_{\tau }\right)}={\left(\ker B_{\tau }^{\ast }\right)}^{\perp },$$

it follows that $\mathrm{Ran}\left(B_{\tau }\right)$ is dense in $L_2\left(\mathbb{R}\right)$. Because the range is also closed, we conclude that

$$\mathrm{Ran}\left(B_{\tau }\right)=L_2\left(\mathbb{R}\right).$$

Thus, $B_{\tau }$ is bijective, and by the Banach inverse theorem, its inverse $B_{\tau }^{-1}={({\tilde{A}}_{\tau }+\mu I)}^{-1}$ is bounded in $L_2\left(\mathbb{R}\right)$.

□

Lemma 5. 

Assume that the conditions of Lemma 4 are satisfied. Then, the derivatives with respect to the variable x satisfy the estimate

$${∥D_x^{\alpha }{({\tilde{A}}_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le {\displaystyle \frac{c_0}{{\mu }^{1-\frac{\alpha }{3}}}},$$

(6)

where $c_0>0$ is a constant, $D_x^{\alpha }={\displaystyle \frac{{\mathsf{\partial }}^{\alpha }}{\mathsf{\partial }{x}^{\alpha }}}$, $\alpha =0,1,2,3.$

Assume that the conditions of Lemma 4 are satisfied. Then, the following estimate holds:

$${∥D_x^{\alpha }{({\tilde{A}}_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le {\displaystyle \frac{C}{{\mu }^{1-\frac{\alpha }{3}}}},\alpha =0,1,2,3,$$

where $C>0$ is independent of $\mu$ and $\tau$.

Proof. 

Let $u={({\tilde{A}}_{\tau }+\mu I)}^{-1}f$. Then,

$$({\tilde{A}}_{\tau }+\mu I)u=f.$$

By Lemma 4, we have the coercive estimate

$$\parallel u^{‴}{\parallel }_{L_2\left(\mathbb{R}\right)}+\parallel (i\tau +q_0\left(x\right)){u\parallel }_{L_2\left(\mathbb{R}\right)}\le C{\parallel f\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Since $q_0\left(x\right)\ge {\delta }_0>0$, it follows that

$${\parallel u\parallel }_{L_2\left(\mathbb{R}\right)}\le C{\parallel f\parallel }_{L_2\left(\mathbb{R}\right)}.$$

Thus,

$$\parallel u^{‴}{\parallel }_{L_2}+{\mu \parallel u\parallel }_{L_2}\le C{\parallel f\parallel }_{L_2}.$$

Now we apply interpolation inequalities in Sobolev spaces $W_2^3\left(\mathbb{R}\right)$:

$$\parallel D_x^{\alpha }{u\parallel }_{L_2}\le C\parallel u^{‴}{\parallel }_{L_2}^{{\displaystyle \frac{\alpha }{3}}}{\parallel u\parallel }_{L_2}^{1-{\displaystyle \frac{\alpha }{3}}},\alpha =0,1,2,3.$$

Using the previous estimates, we obtain

$$\parallel D_x^{\alpha }{u\parallel }_{L_2}\le {C(\parallel f\parallel }_{L_2}{)}^{{\displaystyle \frac{\alpha }{3}}}{\left({\displaystyle \frac{{\parallel f\parallel }_{L_2}}{\mu }}\right)}^{1-{\displaystyle \frac{\alpha }{3}}}=C{\displaystyle \frac{{\parallel f\parallel }_{L_2}}{{\mu }^{1-\frac{\alpha }{3}}}}.$$

Therefore,

$$\parallel D_x^{\alpha }{({\tilde{A}}_{\tau }+\mu I)}^{-1}{f\parallel }_{L_2}\le {\displaystyle \frac{C}{{\mu }^{1-\frac{\alpha }{3}}}}{\parallel f\parallel }_{L_2}.$$

Taking the supremum over $f\ne 0$, we obtain the desired operator norm estimate. □

Proof of Lemma 3. 

Consider the following equation in $L_2\left(\mathbb{R}\right)$

$$(A_{\tau }+\mu I)u=u^{‴}\left(x\right)+u^{\prime }\left(x\right)+(i\tau +q_0\left(x\right)+\mu )u\left(x\right)=f\left(x\right)\in L_2\left(\mathbb{R}\right).$$

(7)

We rewrite Equation (7) in the form

$$\vartheta +A_{\mu }\vartheta =f\left(x\right),$$

(8)

where $({\tilde{A}}_{\tau }+\mu I)u=u^{‴}\left(x\right)+(i\tau +q_0\left(x\right)+\mu )u=\vartheta ,A_{\mu }\vartheta =D_x{({\tilde{A}}_{\tau }+\mu I)}^{-1}\vartheta$. According to Estimate (6), there exists a number ${\mu }_0>0$ such that

$${∥A_{\mu }∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}={∥D_x{({\tilde{A}}_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le {\displaystyle \frac{c_0}{{\mu }^{1-\frac{\alpha }{3}}}}<1.$$

(9)

for $\mu \ge {\mu }_0$.

From Inequality (9) it follows that Equation (8) has a unique solution

$$\vartheta ={(I+A_{\mu })}^{-1}f\left(x\right).$$

(10)

Hence, and taking into account $({\tilde{A}}_{\tau }+\mu I)u=\vartheta$ and by virtue of Lemma 4, we find

$$u={({\tilde{A}}_{\tau }+\mu I)}^{-1}\vartheta .$$

From the last equality and using Equality (10) we obtain

$$u={(A_{\tau }+\mu I)}^{-1}f={({\tilde{A}}_{\tau }+\mu I)}^{-1}{(I+A_{\mu })}^{-1}f\left(x\right).$$

□

Lemma 6. 

Assume that Conditions (i)–(ii) are satisfied and $\mu \ge 0$. Then, there exists ${\mu }_0>0$ such that

$${∥u^{‴}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥u^{\prime }∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥(i\tau +q_0\left(x\right))u∥}_{L_2\left({\mathbb{R}}^2\right)}\le c_0{∥({\tilde{A}}_{\tau }+\mu I)u∥}_{L_2\left({\mathbb{R}}^2\right)},$$

holds for $\mu \ge {\mu }_0>0$, where $c_0>0$ is a constant.

Proof. 

Using estimate (6) and Lemma 3, we obtain the proof of Lemma 6. □

2.2. Proof of the Main Theorem

2.2.1. Existence and Uniqueness

Let us consider the equation

$$(A+\mu I)u={\displaystyle \frac{\mathsf{\partial }u}{dt}}+{\displaystyle \frac{\mathsf{\partial }u}{dx}}+{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{d{x}^3}}+(q_0\left(x\right)+\mu )u=f(t,x).$$

Applying the Fourier transform with respect to the variable t, we obtain

$$(A_{\tau }+\mu I)\tilde{u}={\tilde{u}}^{‴}\left(x\right)+{\tilde{u}}^{\prime }\left(x\right)+(i\tau +q_0\left(x\right)+\mu )\tilde{u}=\tilde{f}(\tau ,x),$$

where $\tilde{u},\tilde{f}$ are the Fourier transform with respect to t of functions $u(t,x)$ and $f(t,x)$.

In the following, we denote the Fourier transform by $F_{t\to \tau }$, and the Fourier inversion formula by $F_{\tau \to t}^{-1}$. From Lemma 3 and using the properties of the Fourier transform, we have

$$u(t,x)={(A+\mu I)}^{-1}f=F_{\tau \to t}^{-1}{(A_{\tau }+\mu I)}^{-1}\tilde{f}.$$

(11)

The continuity of the Fourier transform and the operator ${(A_{\tau }+\mu I)}^{-1}$ the last equality holds for any $f(t,x)\in L_2\left({\mathbb{R}}^2\right)$. The uniqueness follows from Lemma 1. The existence of a solution to the equation $(A+\mu I)u=f$ is proven.

2.2.2. Maximal Regularity

Lemma 7. 

Assume that the conditions of Lemma 6 are satisfied. Then, the following estimates hold:

$${∥q_0\left(x\right){(A_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le c_0<\infty ,$$

(12)

$${∥{\displaystyle \frac{d}{dx}}{(A_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le c_1<\infty ,$$

(13)

$${∥{\displaystyle \frac{d^3}{d{x}^3}}{(A_{\tau }+\mu I)}^{-1}∥}_{L_2\left(\mathbb{R}\right)\to L_2\left(\mathbb{R}\right)}\le c_2<\infty ,$$

(14)

where $c_0>0,c_1>0,c_2>0$ are constants.

Proof. 

The proof of Lemma 7 follows from Lemma 6. □

Now, we find the norm ${∥q_0\left(x\right)u(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}$. Using Equality (11) and the property of the Fourier transform, we obtain

$${∥q_0\left(x\right)u(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}^2=\underset{-\infty }{\overset{\infty }{\int }}{∥q_0\left(x\right){(A_{\tau }+\mu I)}^{-1}\tilde{f}(\tau ,x)∥}_{L_2\left({\mathbb{R}}^2\right)}^{2}d\tau \le$$

$$\le \underset{\tau \in R}{\sup }{∥q_0\left(x\right){(A_{\tau }+\mu I)}^{-1}∥}_{L_2\left({\mathbb{R}}^2\right)\to L_2\left({\mathbb{R}}^2\right)}^2·$$

$$·\underset{-\infty }{\overset{\infty }{\int }}(\underset{-\infty }{\overset{\infty }{\int }}|\tilde{f}(\tau ,x){|}^{2}dx)d\tau \le c_0^2·{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}^2,$$

i.e.,

$${∥q_0\left(x\right)u(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}\le c_0{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)},$$

(15)

In the same way, using Lemma 7 and repeating the calculations that were used in the proof of (15), we obtain the following estimates

$${∥D_{x}u∥}_{L_2\left({\mathbb{R}}^2\right)}={∥{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}∥}_{L_2\left({\mathbb{R}}^2\right)}\le c_1{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)},$$

(16)

$${∥D_x^{3}u∥}_{L_2\left({\mathbb{R}}^2\right)}={∥{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}}∥}_{L_2\left({\mathbb{R}}^2\right)}\le c_2{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}.$$

(17)

Using Inequalities (15)–(17) and Estimate (4), we obtain the following inequality

$${∥{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}∥}_2={∥(A+\mu I)-[{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}}+{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+(q_0\left(x\right)+\mu )u]∥}_{L_2\left({\mathbb{R}}^2\right)}\le$$

$$\le {∥(A+\mu I)u∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥q_0\left(x\right)u∥}_{L_2\left({\mathbb{R}}^2\right)}+\mu {∥u∥}_{L_2\left({\mathbb{R}}^2\right)}\le$$

$$\le {∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}+c_2{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}+c_1{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)}+c_0{∥f(t,x)∥}_2+{∥f(t,x)∥}_2\le$$

$$\le c{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)},$$

(18)

where $c=max\{1,c_2,c_1,c_0\}$.

From Inequalities (15)–(18) follows the estimate

$${∥{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥{\displaystyle \frac{{\mathsf{\partial }}^{3}u}{\mathsf{\partial }{x}^3}}∥}_{L_2\left({\mathbb{R}}^2\right)}+{∥q_0\left(x\right)u∥}_{L_2\left({\mathbb{R}}^2\right)}\le c{∥f(t,x)∥}_{L_2\left({\mathbb{R}}^2\right)},$$

where $c>0$ is a constant.

Thus, Theorem 1 is completely proved.

We provide example of coefficient $q_0\left(x\right)$ satisfying Conditions (i)–(ii).

Example 1. 

Consider the function

$$q_0\left(x\right)={(2+3x^2)}^{\alpha },\alpha >2.$$

Then,

$$q_0\left(x\right)\ge 2^{\alpha }>0,$$

so Condition (i) holds with ${\delta }_0=2^{\alpha }$.

Moreover, if $|x-t|\le 1$, then $\left|x\right|\le \left|t\right|+1$, and hence

$$x^2\le {\left(\right|t|+1)}^2=t^2+2\left|t\right|+1.$$

Therefore,

$$2+3x^2\le 5+3t^2+6\left|t\right|\le 8+6t^2\le 4(2+3t^2),$$

which implies

$${\displaystyle \frac{q_0\left(x\right)}{q_0\left(t\right)}}={\left({\displaystyle \frac{2+3x^2}{2+3t^2}}\right)}^{\alpha }\le 4^{\alpha }.$$

Thus,

$$\underset{|x-t|\le 1}{\sup }{\displaystyle \frac{q_0\left(x\right)}{q_0\left(t\right)}}<\infty ,$$

and condition (ii) is satisfied.

Hence, $q_0\left(x\right)={(2+3x^2)}^{\alpha }$ is an admissible unbounded coefficient.

3. Discussion

In this paper, we obtain results aimed at investigating the existence, uniqueness, and maximum regularity of solutions to a third-order partial differential equation with an unbounded lowest-order term in the space $L_2\left({\mathbb{R}}^2\right)$. Within the framework of the developed approach, we establish the correct solvability of the corresponding problem in natural function spaces, enabling us to rigorously justify the existence and uniqueness of weak solutions under minimal assumptions on the equation’s coefficients.

Particular attention is paid to analyzing the influence of the unbounded lowest-order term on the properties of solutions. It is shown that, under certain conditions on the structure of the coefficients, it is possible to obtain a priori estimates that ensure maximum regularity of solutions in the space $L_2\left({\mathbb{R}}^2\right)$. The obtained estimates are key for further investigation of the qualitative properties of solutions and allow us to consider the equation under study in the context of a more general theory of third-order evolution equations with singular coefficients.

The presented results expand and complement previously known works devoted to Korteweg–de Vries-type equations with bounded coefficients by establishing resolvent estimates and maximal regularity for the case of unbounded lower-order terms. In contrast to classical settings, the analysis is carried out in a whole-line-in-time framework and under minimal assumptions on the coefficients.

Although the problem considered in this paper is linear, the obtained results provide a rigorous functional-analytic framework that is relevant for the study of nonlinear wave models. In particular, many nonlinear third-order equations, including those of Korteweg–de Vries, Gardner, and Burgers type, involve linearized operators whose structure is closely related to the operator studied in this work.

In this context, the derived coercive estimates and maximal $L_2$-regularity properties play an essential role in the analysis of nonlinear problems. They can be used, for example, in establishing well-posedness via perturbation and fixed-point methods, as well as in obtaining stability results and a priori bounds for nonlinear terms in appropriate function spaces.

Thus, the present work contributes to the nonlinear-waves theme of the special issue by providing mathematically justified analytical tools for the study of linearized operators with unbounded coefficients, which naturally arise in models of wave propagation in inhomogeneous or unbounded media.

At the same time, it should be emphasized that the extension of these results to fully nonlinear and quasilinear third-order equations requires additional analysis, particularly in connection with dispersive effects and nonlinear interactions. This direction will be the subject of future research.

The results obtained in this paper may also serve as a basis for further investigations of spectral properties, approximation methods, and the development of analytical techniques for nonlinear and quasilinear third-order differential equations.