Existence, Blow-Up, and Blow-Up Rate of Weak Solution to Fourth-Order Non-Newtonian Polytropic Variation-Inequality Arising from Consumption-Investment Models

Abstract

This article obtains the conditions for the existence and nonexistence of weak solutions for a variation-inequality problem. This variational inequality is constructed by a fourth-order non-Newtonian polytropic operator which is receiving much attention recently. Under the proper condition of the parameter, the existence of a solution is proved by constructing the initial boundary value problem of an ellipse by time discretization and some elliptic equation theory. Under the opposite parameter condition, we analyze the nonexistence of the solution. The results show that the weak solution will blow up in finite time. Finally, we give the blow-up rate and the upper bound of the blow-up time.

1. Introduction

We consider the existence and blow-up phenomenon for a kind of variation-inequality problemwith the fourth-order non-Newtonian polytropic operator

Here, , is a bounded domain with appropriately smooth boundary , , and satisfies .

Variational inequalities similar to model (1) have good applications in option pricing. Han and Yi in [1] studied a kind of irreversible investment problem of the firm on finite horizon in which they consider the following variation-inequality problem:where is a linear parabolic operator with constant parameters and represents the dividend rate of risk assets. The existence of the solution and the regularity of solution were studied. Moreover, the behaviors of free boundaries were considered in [2]. Chen et al. in [3] consider a class of singular control problems whose solution is governed by a time-dependent HJB equation with gradient constraints. The existence, uniqueness, and the behaviors of free boundary face were analyzed.

In recent years, the theoretical research on variation-inequality has also attracted the attention of scholars. In [4], the existence of a weak solution for a class of variational inequality in whole is studied, in which the author used the penalization method [5] and the Mountain Pass Theorem shown in [6]. Wang and Zhang in [7] considered the partial differential mixed variational inequality on Hilbert lattices using order-theoretic approaches. The existence of mild solutions to mixed variational inequality is established without Lipschitz continuity. There are some other new studies [8–11], which will not be repeated here.

Unlike classical partial differential equations, parabolic variational inequalities involve inequality constraints. As a result, we introduce a maximal monotone map to characterize these inequality constraints. Furthermore, the existence of weak solutions for parabolic variational inequalities is proven through an auxiliary problem. In this process, we construct a penalty function to approximate the maximal monotone map, which constitutes the first innovation of this paper. To demonstrate the explosive nature of weak solutions in a finite time, we initially analyze the non-negativity of the integral term constructed from the maximal monotone map and weak solution test functions. These test functions also happen to be suitable for constructing differential inequalities based on energy functions, thus completing the proof of the explosive nature of weak solutions, marking another innovative aspect of this paper.

This paper mainly considers the existence and blow-up phenomenon for a kind of variation-inequality problem (1) with the fourth-order non-Newtonian polytropic operator (2). When , we analyze the existence of weak solutions of variational inequality (1). The time interval is discretized, and several estimates of the auxiliary problem are given by using the theory of elliptic equations. Then the existence of weak solutions is given by a limit method. The result is shown in Section 3. When , we show that weak solutions of variational inequality (1) will blow up at finite time and do not exist. Using differential inequality and Sobolev inequality, the existence of blow-up is proved, and the upper bound estimate of blow-up time and blow-up rate is given. The results as well as relative proofs are shown in Section 4.

2. The Main Results and the Financial Background

Variational inequalities are often used to quantify consumption-investment models [12, 13] which contain two factors

Here, is the demand of a good in which and represent the expected rate of return and volatility, respectively. is the production capacity of the firm, and and are positive constants called the expected rate of return and volatility of the production process.

The aim of the consumption-investment model is to find an optimal policy to minimize the expected total cost over the finite horizon, such that the value functionsatisfies a variation-inequalitywith . Here, is the risk-free interest rate in the market.

The investment consumption model involves the specification of goods demand and production capacity of the firm . Investors must make decisions between current production and future production while considering future income, interest rates, and other economic factors. This balancing act can be seen as an optimization problem, where investors seek to maximize the company’s utility or wealth. This optimization process can be formalized as a variational inequality, in which investors strive to find the optimal production strategy to match future demand, maximizing utility or wealth, while satisfying budget constraints. Therefore, the investment consumption model is often described as an optimization problem involving variational inequalities.

This paper investigates a variational inequality that is more extensive than model (6). For this purpose, one gives the maximal monotone mapping .where is a positive constant.

Definition 1. we say that is a generalized solution of variation-inequality (1), if it satisfies(a),(b)for every test-function and for almost all , we getNote that when or , operator degenerates, so the following result of well-posedness for variation-inequality (1) can be obtained by the regular method, and we consider the following regular problem:with , the penalty map satisfies ,It is noteworthy that is used to describe inequality constraints, which transforms variation-inequality into parabolic regularized problem with local nonlinear term.
Denote the classical solution of problem (9) by , whose existence follows, for example, from Theorem 1.1 of [12], such that one can have the weak equationwith . Moreover, the solution of (9) from [10, 11] satisfiesFirst, we obtain estimates, independent of, for and its derivatives. In doing so, we discretize the problem (9) in time domain . Let be the uniform partition on the domain , where , . No matter how small is, we always choose the suitable to make sure that . We denote and introduce an approximate solutionThen, the classical theorem of elliptic equations (see Wu et al. [14]) ensures that problemhas a unique solution for any , , and satisfiesNext, we give some useful estimates for . Choosing in (15) givesApplying inequality , one can infer thatRecall that . We sum (17) from to ,Pass the limit and combine it with the definition of integralSince , it follows from (10) and (12) thatThen auxiliary problem, dropping the nonnegative term , has the following estimates:Finally, we consider the time gradient estimation of and can prove it in a rather simple way. Indeed choosing in (11) or taking in (14), it follows from Holder and Young inequalities that

3. Existence of a Generalized Solution

To discuss the existence of solutions, we need to use some weak limit results. Here, we give the details of the proof. (21)–(24) imply that, for any given , the sequence contains a subsequence (for the sake of simplicity, still denoted by itself) and three functions , and , such that

By (12), we derive that , a.e. in , so combining with (10) gives

Letting in (11) yields

Define with , and subtract (11) and (31), so

By the boundary condition in (9), it is easy to verify that

It follows from (25) that as ,

Combining above, it is clear to see that

On the contrary, it follows from [12] that

Note that , then

Subtracting (37) from (35), one can find that

Obliviously, if we subtract both hand side of (35) by , it is clear to find the invited inequality, such that

By the arbitrariness of , one can establish the following lemma.

Lemma 2. a.e. in .

Furthermore, taking , employing Holder inequality, and joining with (32), (36), and (37), it can be easily verified thatfor any given . For detailed proof, refer to [14]. Further, we consider the fact as , then combining with (40), we have the following result.

Lemma 3. If , converges to with norm , that is to say a.e. in .

Further, analyze the limit of . From (10) and (12), boundedness of gives

Here, , we will show it later in two cases: and . If , we have , so together with (7) it gives

If , one gets . Passing the limit , . Combining above, it is inferred thatfor all . Following a standard limit method, one can show the existence of generalized solution and summarize the following theorem.

Theorem 4. If , parameter given by (1) satisfies , then there exists a solution in the sense of Definition 1 and it satisfies

Note that the condition is used in Section 2, so is also required in Theorem 4.

4. Blow-Up of a Generalized Solution

This section is devoted to give the proof of blow-up of solution with the condition and . In doing so, we introduce the following function:

From Definition 1, we know that . Choosing in generalized equation gives

From (25), , as well as , it is clear to verify that

On the contrary, from (7), Definition 1, and the fact that , we can infer and which combining with (46)–(49) (note that ), we see that

Next, we reduce to make it a function of . Employing Sobolev inequality, one gets

Here, represents the Sobolev coefficient. According to [15], satisfies

Using Holder inequality with parameter , we infer for any thatthat is to say

This, combining with (50) and (51) implies that

Now, we analyze the above ordinary differential inequality and separate the variable to obtain

Integrating the above inequality over gives

In this section, we use the condition , such that

This means that generalized solution blows up at some finite time , and satisfies

Since , . Then integrating the value of (56) over , it is inferred that

Theorem 5. Assume that . If , , the generalized solution of variation-inequality (1) will blow up at finite time , and the upper bound of blow-up time is given by (42). Moreover, the blow-up rate of the generalized solution can be estimated by (43).

5. Conclusions

This paper concerns with the existence and nonexistence of weak solutions for a class of variation-inequality problems with the fourth-order non-Newtonian polytropic operator.

The existence of weak solutions is analyzed under the condition and . First of all, in order to solve the inequality constraints in problem (1), we introduce penalty functions to construct auxiliary problems. The auxiliary problem approaches the variational inequality (1) as . Thirdly, the limit of penalty term is analyzed, and the existence of weak solution is given by combining it with the limit method of other literature [12].

When and , we consider the nonexistence of weak solutions. We introduce the function and choose in generalized equation to obtain an ordinary differential inequality. Finally, the existence of the blow-up phenomenon and the upper bound estimate of blow-up rate and blow-up point are obtained by using inequality transformation technique and Sobolev inequality.

The condition needs to be supplemented in (56), otherwise the inequality under (57) cannot be obtained, such that the blow-up phenomenon of the generalized solution cannot be verified. We will study those focuses in the future.

Data Availability

No datasets were generated or analyzed during this current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Supply and Demand Docking Education Employment Practice Base Project of Education Ministry (Grant no. 20220106223) and the Supply and Demand Docking Education Oriented Talent Training Project of Education Ministry (Grant no. 20220104747).