## Abstract

In this paper, a priori estimate for a linear third pseudoparabolic operator with bound is established, and applying the above result, the existence and uniqueness theorem of solutions for a class of nonlinear pseudoparabolic equations is obtained with the help of the homeomorphism method and the initial value method. Furthermore, an existence and uniqueness theorem of the semilinear equation is obtained as a corollary.

## Introduction

Consider the initial-boundary value problem

where *Ω* is a connected bounded subset of *n*-dimensional space, the boundary of *Ω* is piecewise smooth and has nonnegative mean curvature everywhere, \(D=\varOmega\times[0,T]\) belongs to the Hilbert space \(W_{2}^{2,1}(D)\), and \(a_{i,j}\), \(b_{i}\) are bounded measurable functions.

Using a continuous method, Sigillito [1] explored the solution for the heat equation. Elcart and Sigillito [2, 3] had derived an explicit coercivity inequality and discussed the convergence of the algorithm for a semilinear third order pseudoparabolic equation of the following type:

since then, there have been some further studies of other forms of parabolic equations, most of the results have focused on the discussion of algorithms [4].

In 2004, Bouziani [5, 6] had derived an explicit coercivity inequality and given a sufficient condition for the existence and uniqueness of a solution to the first order parabolic equation.

Motivated by the spirit of this work and results by Brown and Lin [7], the explicit coercivity inequalities of a linear third pseudoparabolic operator with bound are obtained in Sect. 3. By using these estimates, we shall utilize the homeomorphism method and the initial value method to give a new set of sufficient conditions for the existence and uniqueness of the third order pseudoparabolic equation in this paper, which can be found in Sect. 4.

## Preliminaries and lemmas

In this section, we will state some lemmas which are useful to our results.

Firstly, we will give sufficient conditions for *f* to be a global homeomorphism of *D* onto *Y*.

### Definition 2.1

([8])

Let *X*, *Y* be Banach spaces, \(D\subseteq X\) be open and connected, the continuous mapping \(f:D\subset X\rightarrow Y\) satisfies condition (C) if and only if for any continuous function \(r:[0,a)\rightarrow D\subseteq X\) such that

where \(q:[0,1]\rightarrow Y\) is any line in *Y*, there is a sequence \(\{ t_{n}\}\) such that \(t_{n}\rightarrow a\), \(n\rightarrow\infty\) and

exists and is in *D*.

In the following for convenience, with no loss of generality, for the function *q* one may assume that \(q(t)=(1-t)f(x_{0})+ty\), \(t\in[0,1]\), for arbitrary \(x_{0}\in D\) and \(y\in Y\).

### Theorem 2.1

(Plasctock [8])

*Let*\(f:D\subset X\rightarrow Y\)*be a local homeomorphism*.

*Then**f**is a global homeomorphism of**D**onto**Y**if and only if**f**satisfies condition* (C).

Secondly, the comparison theorem plays an important role to prove the sufficient condition for the existence of a unique solution of the problem (1).

Let *E* be an open \((t,x)\)-set in \(R^{2}\) and \(g\in C[E,R]\). Consider the scalar differential equation with an initial condition

### Definition 2.2

([9])

Let \(y(t)\) be a solution of the scalar differential equation (4) on \({[t_{0},t_{0}+a)}\), then \(y(t)\) is said to be a maximal solution of (4) if, for every solution \(u(t)\) of (4) existing on \([t_{0},t_{0}+a)\), we have the inequality

holds.

### Theorem 2.2

(Plasctock [9])

*Let*\(g\in C[R_{0},R]\), *where*\(R_{0}\)*is the rectangle*\(t_{0}\leq t\leq t_{0}+a\), \(|u-u_{0}|\leq b\), *and*\(|g(t,u)|\leq M\)*on*\(R_{0}\).*Then there exist a maximal solution and a minimal solution of* (4) *on*\([t_{0},t_{0}+a]\), *where*\(\alpha=\min(a,b\setminus2(M+b))\).

### Theorem 2.3

(Comparison theorem [9])

*In the setting of the above*, *suppose that*\([t_{0},t_{0}+b)\)*is the largest interval in which the maximal solution*\(y(t)\)*of* (4) *exists*. *Let*

*and for a fixed Dini derivative*

*then*

*where**T**denotes an almost countable subset of*\(t\in[t_{0},t_{0}+b)\).

## The coercivity inequality

Let \(W_{0}(D)\) denote the Hilbert space with the norm

here \(|D^{2}u|^{2}\) represents the sum of the squares of all the second derivatives with respect to space variables. In this section we derive a coercivity inequality,

for the pseudoparabolic operator defined by

The norm \(|\|\cdot\||\) on \(W_{0}\) is defined by

where \(\|\cdot\|_{2}\) is the norm on \(W_{2}^{2}(\varOmega)\), \(\|\cdot\| \) is the norm on \(L_{2}(D)\), \(a:W_{0}(D)\rightarrow L_{2}(D)\) is continuous and a bounded function on \(t,x_{1},\ldots,x_{n},u\).

We assume that \(a_{ij}\) is a symmetric matrix of measurable functions satisfying the inequality

for some positive constant *τ*, all *n*-dimensional vectors *ξ* and all *x* in *D*. We also assume that the functions \(a_{ij}\) are sufficiently regular to ensure the validity of the identity

for *u* in \(W_{2}^{2}(\varOmega)\). From (5), we have

Using the inequality

and the inequality

for \(\varepsilon>0\), \(\alpha>0\), the inequality

is valid for *u* in \(W_{0}\).

The next two lemmas are obtained from (5) by evident choices of *ε* and *α*. In order to facilitate statements to be made below, we define \(S=\sup|b_{i}-(a_{i,j})_{x_{j}}|\), \(a_{0}=\inf_{D}a(x,t)\).

### Lemma 3.1

*The inequality*

*is valid for**u**in*\(W_{0}\).

### Lemma 3.2

*The inequality*

*is valid for**u**in*\(W_{0}\).

We define

and from [2] we have the inequality

From \(Pu=Qu-(a_{i,j})_{x_{j}}u_{x_{i}}\), \(u\in W_{0}\), we have

so

### Remark 3.1

Results analogous in the present situation are in Lemma 3.3.

### Lemma 3.3

*The inequality*

*is valid for**u**in*\(W_{0}\).

From \(Qu=Pu+(a_{i,j})_{x_{j}}u_{x_{i}}\), we also have

The inequalities (10) and (11) imply that

Denote

and by further application of the arithmetic–geometric mean inequality to

we obtain

Combining (8), (9), (12) and (13), we have Lemma 3.4.

### Lemma 3.4

*If*\(a_{0}-\frac{S^{2}}{4\tau^{2}}>0\), *the inequality*

*is valid for**u**in*\(W_{0}\), *where*

## The coercivity inequality

Denote

then *M* is a linear operator from \(W_{0}(D)\) to \(L_{2}(D)\). Now let us turn our attention to the following operator equation:

For all \(u,\phi\in W_{0}(D)\), we have

If \(\inf_{\varOmega}f_{u}>\frac{S^{2}}{4\tau^{2}}\), then zero is not an eigenvalue of \(M\phi-f_{u}(x,u(x))\phi\), so for every \(u\in W_{0}(D)\), the operator \(A'(u)=M-f_{u}I\) is invertible and *A* is a local homeomorphism from \(W_{0}(D)\) onto \(L_{2}(D)\), where *I* denotes the identical operator. Furthermore, an upper bound for \(|\|[A'(x)]^{-1}\| |\) is provided by Lemma 3.4 if the coefficient \(a(x)\) is identified with \(f_{u}(x,u)\), it implies that

for positive constant *α*, *β*.

Denote

then

We may express the first line of (1) in the form

for \(u,\phi\in W_{0}(D)\), we have

We can state and prove our main theorem.

### Theorem 4.1

*In the setting above*, *for Eqs*. (1), (2) *and* (3) *there exists a unique solution if the following conditions hold*:

- (1)
\(\inf_{\varOmega}f_{u}>\frac{S^{2}}{4\tau^{2}}\);

- (2)
*for each*\(\mu\in R\),*the maximum solution**y**of the initial value problem*$$\begin{aligned}& \textstyle\begin{cases} y'(t) = \mu\delta(y(t)),\quad t\in[0,a), \\ y(0)=0,\end{cases}\displaystyle \end{aligned}$$(15)*is defined on*\([0,a]\)*and there exists a sequence*\(t_{n}\rightarrow a\)*as*\(n\rightarrow\infty\)*such that*\(\lim_{n\rightarrow\infty }y(t_{n})=y^{\star}\)*is finite*; - (3)
*F**is continuously differentiable and*$$\begin{aligned} \biggl\vert \frac{\partial F_{i}}{\partial x_{j}}(w) \biggr\vert , \biggl\vert \frac {\partial F_{i}}{\partial y_{j}}(w) \biggr\vert \leq& \frac{c}{n\delta( \Vert w \Vert )},\quad c< 1,w \in R^{n}. \end{aligned}$$(16)

### Proof

Firstly, we prove \([M-f_{u}(u)+F_{u}(u)]^{-1}\leq\delta(\|u\|)\). For \(u,v\in D\), it is obvious that *F* is continuously Frechet differentiable with

It follows from the above assumption and (16) that

Now

Let \(Q:W_{0}(D)\rightarrow W_{0}(D)\) be defined by

So \(I+Q\) is invertible with

Hence, \(M-f_{u}(u)+F_{u}(u):D\rightarrow W_{0}(D)\) is invertible with

and so

Denote \(\delta(\|u\|)=\frac{\lambda(\|u\|)}{1-c}\), then

It implies that *P* is invertible at every \(u\in W_{0}(D)\), hence, *P* is a local homeomorphism of \(W_{0}(D)\).

Secondly, in view of Theorem 2.1, we need only show that *P* has the property (C) for any continuous function \(q:[0,1]\rightarrow L_{2}(D)\). For a given \(y\in L_{2}(D)\) and an arbitrary \(x_{0}\in W_{0}(D)\), let

suppose that there exists a continuous function \(r:[0,a)\rightarrow D\subseteq W_{0}(D)\) such that

Now we need to prove that there exists a real sequence \(\{t_{n}\}\) such that \(t_{n}\rightarrow a\), \(n\rightarrow\infty\) and

exists and is in \(W_{0}(D)\).

It is clear that *r* is differentiable in this case. We have from (18)

Denote by \(D\|r(t)\|\) the Dini derivative of \(\|r(t)\|\) and set \(\mu =P(r(t))-P(x_{0})\), and we have

By the assumption (2), we know the maximum solution \(y(t)\) of (15) is defined on \([0,a)\) and there exists a sequence \(t_{n}\rightarrow a \) as \(n\rightarrow\infty\) such that

is finite. It follows that \(y(t)\) is continuous on \([0,a]\) and there is a constant *K* such that \(|y(t)|\leq K\), \(t\in[0,a] \). By the comparison theorem, Theorem 2.3, we have

For \(t_{1},t_{2}\in[0,a)\), we have

So \(\{r(t_{n})\}\) is a Cauchy sequence and consequently for the real sequence \(t_{n}\rightarrow a\) as \(n\rightarrow\infty\),

exists. This proves that \(r^{\star}\in W_{0}(D)\) and *P* satisfies the condition (C). The theorem is proved. □

## Related results

Using a similar technique to the one of the theorem, we can prove the following conclusion as regards the initial-boundary problem for pseudoparabolic equations.

### Corollary 5.1

*Let the condition* (1) *and the condition* (2) *hold and*

- (3)
*F**be continuously differentiable and*$$\biggl\vert \frac{\partial F_{i}}{\partial x_{j}}(w) \biggr\vert , \biggl\vert \frac {\partial F_{i}}{\partial y_{j}}(w) \biggr\vert \leq\frac{c}{nW},\quad c< 1,w\in R^{n}, $$*where**W**is constant*.*Then Eq*. (1)*has a unique solution*.

### Corollary 5.2

*Let the condition* (1) *and the condition* (2) *hold*. *Assume that**f**is continuous and has continuous partial derivatives with respect to**u**through the third order*. *Then the semilinear equation*

*has a unique solution*.

Especially when \(a_{i,j}=1\), \(b_{i}=0\), we get the equation in [3]:

and the conclusion of [3] but with a different method.

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## Acknowledgements

The authors are grateful to the referees for their comments and references, which improved the paper.

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Feng, Y., Guo, L. & Wang, Z. The initial-boundary value problem for a class of third order pseudoparabolic equations.
*Bound Value Probl* **2020, **115 (2020). https://doi.org/10.1186/s13661-020-01407-3

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### Keywords

- Third order pseudoparabolic equations
- Homeomorphism method
- Initial-boundary value problem
- Existence and uniqueness theorem