On the oscillation of solutions of third-order differential equations with non-positive neutral coefficients

1.   Introduction

The author of this study, given $\Omega \in \mathrm{R}_N (N \ge 2)$, a bounded regular domain with Lipschitz boundary and ${\Omega _T} = [0, T] \times \Omega$, considers a kind of variation-inequality problem

with the non-Newtonian polytropic operator

Here, ${u_0} \in {H^1_0}(\Omega)$, $f$, $h$, and $\alpha$ have been used with different conditions in Sections 3 and 4, as specified in Theorem 3.1 and Theorem 4.1.

Variational inequalities, such as problem (1.1), have found widespread application in the field of finance. For example, ^[1] explores the investment-consumption model, while ^[2] analyzes dividend optimization and risk control problems through weak solutions of variation-inequality. In ^[3], a continuous-time, finite horizon, irreversible investment problem is examined, resulting in the emergence of a free boundary that represents the optimal investment boundary.

The behaviours of the free boundary and existence of a weak solution were studied by using the partial differential equation (PDE) approach. Moreover, the regularities of the value function and optimal investment and maintenance policies were considered in ^[4].

In recent years, there have been much literature on the theoretical research of variation-inequality problems.The authors in ^[5] studied the following variation-inequality initial-boundary value problems:

with fourth-order $p$-Laplacian Kirchhoff operators,

The existence, stability and uniqueness of solutions are mainly obtained using the Leray Schauder principle. Moreover, Li and Bi in ^[6] considered the two-dimensional case in ^[5]. The conditions to ensure the existence of weak solutions are given in ^[7]. The existence results of weak solutions of variational inequalities can also be found in ^[8,9,10,11]. For the uniqueness of weak solutions of variational inequalities, refer to ^[9,10,11,12]. In addition, the results about the stability of weak solutions on initial values are also worth studying ^[13]. At present, there are few studies on the regularity of solutions of variation-inequality problems.

In this paper, we study the regularity and blow-up of weak solutions of variational inequalities (1.1). First, we assume that $f \ge 0$ and $h \ge 0$ for any $(x, t) \in {\Omega _T}$, ${u_0} \in {H^1_0}(\Omega)$, ${u^m} \in L(0, T;{H^2}(\Omega))$ and $f \in L(0, T;{L^2}(\Omega))$. The weak solution equation is transformed into a difference equation by using the difference operator. Under the property of the difference operator, the $L(0, T;{H^3(\Omega ')})$ estimation inequality is obtained, which is the regularity of the weak solution. Second, we consider the blowup of weak solutions with the restriction that $f < 0$ for any $(x, t) \in {\Omega _T}$, $h$ is a negative constant and $\alpha > 1$. After defining the energy function $E (t)$, it is proved that the weak solution will blow up in finite time by using Hölder inequality and differential transformation techniques.

2.   Statement of the problem and its background

We first give an application of variational inequality in investment and consumption theory. In order to fit optimally the random demand of a good, a social planner needs to control its capacity production at time interval $[0, T]$. Let $\{ {D_t}, t \in [0, T]\}$ be the random demand of a good

where $\mu _1$ and $\sigma _1$ are the expected rate of return and volatility respectively. Further, process $\{ {C_t}, t \in [0, T]\}$ is the production capacity of the firm,

Here $\mu _2$ and $\sigma _2$ are the expected rate of return and volatility of the production process.

A planner is able to create a production plan $C_t$ at any point in time between 0 and $T$ to equilibrate uncertain demand $D_t$. As such, the planner can use a value function $V$ to determine an optimal policy that minimizes the anticipated total cost within a finite timeframe. According to literature ^[1,2,3], the value function $V$ satisfies

where $L_1 V$ is a two-dimensional parabolic operator with constant parameters,

Here, $r$ represents the risk-free interest rate of the bank. The cost function,

is designed to represent the potential expense associated with storing goods, where $p_1$ and $p_2$ indicate the per unit costs of having excessive supply and demand, respectively.

If transportation loss and storage costs are taken into account, sigma is dependent on ${\partial _c}V$, ${\partial _d}V$, and $V$ itself. This is illustrated by the well-known Leland model, which expresses $\sigma_1$ and $\sigma_2$ as

where $m > 0$, $i = 1, 2$, ${\sigma _{0, 1}}$ and ${\sigma _{0, 2}}$ represent the original volatility of $C_t$ and $D_t$, respectively, and Le is the Leland number.

When studying variation-inequality problems, this paper considers cases that are more complex than the example given in Eq 2.2. To do this, we introduce a set of maximal monotone maps that have been defined in previous works ^[1,2,3,5,6],

where $M_0$ is a positive constant.

Definition 2.1. A pair $(u, \xi)$ is said to be a generalized solution of variation-inequality (1.1), if $(u, \xi)$ satisfies $u \in {L^\infty }(0, T, {H^1}(\Omega)), \; {\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$ and $\xi \in G\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T}$,

(a) $u(x, t) \ge {u_0}(x), \; u(x, 0) = {u_0}(x)\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T}$,

(b) for every test-function $\varphi \in {C^1}({\bar \Omega _T})$, there admits the equality

By a standard energy method, the following existence theorem can be found in ^[5,6,14,15].

Theorem 2.2. Assume that ${u_0} \in {H^1_0}(\Omega)$, $f, h \in {L^\infty }(0, T;{L^2}(\Omega))$, $f(x, t) \ge 0$ and $h(x, t) \ge 0$ for any $(x, t) \in {\Omega _T}$. If $\alpha > 0, m > 0$, then (1) admits a solution $u$ within the class of Definition 2.1.

Note that from (1), it follows that $Lu \le 0$ and $L0 = 0$ for any $(x, t) \in \Omega _T$. Additionally, we have ${u_0} \ge 0$ in $\Omega$, and $u = 0$ on $\partial {\Omega _T}$. Therefore, by the extremum principle ^[16], we have

One purpose of this paper is the regularity of weak solutions, so we give some functions and their valuable results. Define the difference operator,

where ${e_i}$ is the unit vector in the direction ${x_i}$. According to literature ^[14], the difference operator has the following results.

Lemma 2.3. (1) Let $\Delta _{\Delta x}^{i*} = - \Delta _{ - \Delta x}^i$ be the conjugate operator of $\Delta _{\Delta x}^i$, then we have

in other words, $\int_{{{\rm{R}}_n}} {f(x)\Delta _{\Delta x}^ig(x){\rm{d}}x} = \int_{{{\rm{R}}_n}} {g(x)\Delta _{\Delta x}^{i*}f(x){\rm{d}}x}$.

(2) Operator $\Delta _{\Delta x}^i$ has the following commutative results

(3) If $u \in {W^{1, p}}(\Omega)$, for any $\Omega ' \subset \subset \Omega$,

(4) Assuming $u \in {L^p}(\Omega)$ with $p \ge 2$, if $h$ is sufficiently small such that $\int_\Omega{|\Delta_h^iu{|^p}{\rm{d}}x}\le C$, where $C$ is independent of $h$, then we have

3.   Regularity of solution

This section considers the regularity of weak solutions. Select the sub-region $\Omega ' \subset \subset \Omega$, define $d = {\rm{dist}}(\Omega ', \Omega)$ and let $\eta \in C_0^\infty (\Omega)$ be the cutoff factor of $\Omega '$ in $\Omega$, such that

Let $\Delta x < d$, define $\varphi = \Delta _{\Delta x}^{i*}({\eta ^2}\Delta _{\Delta x}^iu)$, and note that $u \in H_0^1(\Omega)$, then substituting $\varphi = \Delta _{\Delta x}^{i*}({\eta ^2}\Delta _{\Delta x}^iu)$ into the weak solution equation gives

Now we pay attention to $\int_{\Omega '} {{\partial _t}u\Delta _{\Delta x}^{i*}({\eta ^2}\Delta _{\Delta x}^iu){\rm{d}}x}$. Using differential transformation techniques,

Substitute (3.2) into (3.1), so that

Here we use the commutativity of conjugate operator $\Delta _{\Delta x}^{i*}$ in $\int {\int_{{{\Omega '}_T}} {\Delta {u^m}\Delta \Delta _{\Delta x}^{i*}({\eta ^2}\Delta _{\Delta x}^iu){\rm{d}}x{\rm{d}}t} }$. Further using the differential technique to expand $\Delta \Delta _h^i{u^m}\Delta ({\eta ^2}\Delta _h^iu)$, one can get

Combining formula (3.3) and (3.4), it is easy to verify that

By Hölder and Young inequalities,

Applying Hölder and Young inequalities again and combining with (3.1),

Substituting (3.6)-(3.9) to (3.5), it is clear to verify

Rearranging the above formula, such that

Using the relationship between difference and partial derivative,

Therefore,

Recall that sub-area $\Omega '$ belongs to $\Omega$. It follows from (4) of Lemma 2.3 that

If $\alpha \le 1$, using Hölder inequality gives

Theorem 3.1. Assume $f \ge 0$ and $h \ge 0$ for any $(x, t) \in {\Omega _T}$. If ${u_0} \in {H^1}(\Omega)$, ${u^m} \in L(0, T;{H^2}(\Omega))$ and $f \in L(0, T;{L^2}(\Omega))$, then for any sub-area $\Omega ' \subset \subset \Omega$, there holds $u \in L(0, T;{H^3}(\Omega '))$, and estimate (3.10). Moreover, if $\alpha \le 1$, (3.11) follows.

Using the finite cover principle and the flattening operator ^[14], we have the following global regularity result.

Theorem 3.2. Let $f \ge 0$ and $h \ge 0$ for any $(x, t) \in {\Omega _T}$. If ${u_0} \in {H^1}(\Omega)$, ${u^m} \in L(0, T;{H^2}(\Omega))$ and $f \in L(0, T;{L^2}(\Omega))$, then

If $\alpha \le 1$, we have

4.   Blowup of solution

This section discusses the blow-up properties of weak solutions to the variation-inequality problem (1.1), under the constraints that $\alpha \le 1$, $f < 0$, and $h < 0$. As $u > 0$ in $\Omega_T$, we define the function

for this purpose. Choosing the test function $\varphi = \frac{{{u^m}}}{{{u^m} + \varepsilon }}$ in weak equation, we have

It follows from $u \in {L^\infty }(0, T, {H^2}(\Omega)), \; {\partial _t}u \in {L^2}({\Omega _T})$ and $f \in L(0, T;{L^2}(\Omega))$ that

Recall that ${u^m} \ge 0$ and $\xi \ge 0$ for any $(x, t) \in {\Omega _T}$. In this section we consider the case that $f \le 0$ for any $(x, t) \in {\Omega _T}$ and $h$ is a negative constant, so

Substituting (4.2)-(4.5) to (4.1), one can have

Using Hölder inequality (here, we used the conditions $\alpha > 1$ and $h < 0$),

such that combining (4.6) and (4.7) gives

Applying variable separation techniques to above equation, and then integrating from 0 to $T$ gives

Rearranging (4.9), one can get

Note that $\alpha < 1$ and $h < 0$. As $t$ approaches $\frac{1}{{(\alpha - 1)}}{h^{ - 1}}|\Omega {|^{\alpha - 1}}E{(0)^{1 - \alpha }}$, $E(t)$ tends to infinity. This indicates that the weak solution of the equation will experience a finite-time blow up at $T^*$, and $T^*$ satisfies

Further, we analyze the rate of Blowup. Integrating the value of (4.8) from $t$ to $T^*$ gives

which (note that $E{({T^*})^{1 - \alpha }} = 0$) implies that

Rearranging (4.12), it is easy to see that

Theorem 4.1. Assume that $f < 0$ for any $(x, t) \in {\Omega _T}$ and $h$ is a negative constant. If $\alpha > 1$, then the weak solution $(u, \xi)$ of variation-inequality problem (1) at time ${T^*}$ in which ${T^*}$ is bounded by (4.13). Moreover, the rate of blowup is given by

where $C = {(\alpha - 1)^{\frac{1}{{1 - \alpha }}}}|h{|^{\frac{1}{{1 - \alpha }}}}|\Omega |$.

5.   Conclusions

This article investigates the global regularity and blow-up of weak solutions for the following variational inequality (1.1) with the non-Newtonian polytropic operator

Firstly, this article analyzes the $H^3(\Omega)$ regularity of weak solutions for variational inequality (1.1). We assume that $f \ge 0$ and $h \ge 0$ for any $(x, t) \in {\Omega _T}$, ${u_0} \in {H^1_0}(\Omega)$, ${u^m} \in L(0, T;{H^2}(\Omega))$ and $f \in L(0, T;{L^2}(\Omega))$. Since using $\partial _{xx} u$ as test function does not comply with the definition of weak solution, this article introduces spatial difference operator and constructs test functions with it to approximate the second-order spatial gradient of $u$. Additionally, with the aid of spatial cutoff factor, Hölder's inequality and Young's inequality, two $H^3(\Omega)$ regularity estimates for weak solutions of variational inequality (1.1) are obtained. The specific results can be seen in Theorem 3.1 and Theorem 3.2.

Secondly, we analyze the blow-up properties of weak solutions for variational inequality (1.1) within a finite time under the assumption that $f < 0$ for any $(x, t) \in {\Omega _T}$, $h$ is a negative constant and $\alpha \le 1$. Considering that $u$ is non-negative, we define an energy function

and obtain the differential inequality of the energy function, as shown in (4.8). By using differential transform techniques, we obtain the lower bound of the blow-up point and the blow-up rate. The results are presented in Theorem 4.1.

Currently, there are still some limitations in this article: (1) Equations (4.6) and (4.10) can only hold when $h$ is a non-negative parameter; (2) Equations (4.10)-(4.13) can only hold when $\alpha \le 1$. In future research, we will attempt to overcome these limitations.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions.

Conflict of interest

The authors declare that he has no conflict of interest.