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1.   Introduction

In this paper, we consider the following initial boundary value problem for higher-order nonlinear viscous parabolic type equations.

where $L, K \geq 1$ is an integer number, $R\geq$ max $\{2, 2a, 2H\}$ where $a > 0$ is a real number, and $\Omega \subseteq R^N(N\geq 1)$ is a bounded domain with a smooth boundary $\partial\Omega$.

Equation (1.1) includes many important physical models. In the absence of the memory term and dispersive term, and with $L = K = 1$ and $a = 0$, Eq (1.1) becomes the linear pseudo-parabolic equation

Showalter and Ting ^[1] and Gopala Rao and Ting ^[2] investigated the initial boundary value problem of the linear Eq (1.4) and proved the existence and uniqueness of solutions. Pseudo-parabolic equations appear in many applications for natural sciences, such as radiation with time delay ^[3], two-phase porous media flow models with dynamic capillarity or hysteresis ^[4], phase field-type models for unsaturated porous media flows ^[5], heat conduction models ^[6], models to describe lightning ^[7], and so on. A number of authors (Showalter ^[8,9], DiBenedetto and Showalter ^[10], Cao and Pop ^[11], Fan and Pop ^[12], Cuesta and Pop ^[13], Schweizer ^[14], Kaikina ^[15,16], Matahashi and Tsutsumi ^[17,18]) have considered this kind of equation by various methods and made a lot of progress. Not only were the existence, uniqueness, and nonexistence results for pseudo-parabolic equations were obtained, but the asymptotic behavior, regularity, and other properties of solutions were also investigated.

In 1972, Gopala Rao et al. ^[2,19] studied the equation $u_{t}-k\Delta u_{t}- \Delta u = 0$. They use the principle of maximum value to establish the uniqueness and the existence of solutions. Using the potential well method and the comparison principle, Xu and Su^[20] studied the overall existence, nonexistence, and asymptotic behavior of the solution of the equation $u_{t}-\Delta u_{t}-\Delta u = u^q$, and they also proved that the solution blows up in finite time when $J(u_{0}) > d$.

When $L = K = 1$, Eq (1.1) becomes

Equation (1.5) originates from various mathematical models in engineering and physical sciences, such as in the study of heat conduction in materials with memory. Yin ^[21] discussed the problem of initial boundary values of Eq (1.5) and obtained the global existence of classical solutions under one-sided growth conditions. Replacing the memory term $b(\cdot)$ in (1.5) by $-g(\cdot)$, Messaoudi^[22] proves the blow-up of the solution with negative and vanishing initial energies. When $f(u) = |u|^{q-2}u$, Messaoudi^[23] proved the result of the blow-up of solutions for this equation with positive initial energy under the appropriate conditions of $b$ and $q$. Sun and Liu ^[24] studied the equation

They applied the Galerkin method, the concavity method, and the improved potential well method to prove existence of a global solution and the blow-up results of the solution when the initial energy $J(u(0))\leq d(\infty)$, and Di et al. ^[25] obtained the blow-up results of the solution of Eq (1.6) when the initial energy is negative or positive and gave some upper bounds on the blow-up time, and they proved lower bounds on the blow up time by applying differential inequalities.

When $m > 1$, Cao and Gu^[26] studied the higher order parabolic equations

By applying variational theory and the Galerkin method, they obtained existence and uniqueness results for the global solution. When the initial value belongs to the negative index critical space $H^{-s, R^s}, R^s = \frac{n\alpha}{w-s\alpha}$, Wang^[27,28] proved the existence and uniqueness of the local and the global solutions of the Cauchy problem of Eq (1.7) by using $L^r-L^q$ estimates. Caristi and Mitidieri ^[29] applied the method in ^[30] to prove the existence and nonexistence of the global solution of the initial boundary value problem for higher-order parabolic equations when the initial value decays slowly. Budd et al. ^[31] studied the self-similar solutions of Eq (1.7) for $n = 1, k > 1$. Ishige et al. ^[32] proved the existence of solutions to the Cauchy problem for a class of higher-order semilinear parabolic equations by introducing a new majority kernel, and also gave the existence of a local time solution for the initial data and necessary conditions for the solution of the Cauchy problem, and determine the strongest singularity of the initial data for the solutions of the Cauchy problem.

When $K = L, g = 0$, problem (1.1) becomes the following $n$-dimensional higher-order proposed parabolic equation

Equation (1.8) describes some important physical problems ^[33] and has attracted the attention of many scholars. Xiao and Li ^[34] have proved the existence of a non-zero weak solution to the static problem of problem (1.8) by means of the mountain passing theorem, and, additionally, based on the method of potential well theory, they proved the existence of a global weak solution of the development in the equations.

Based on the idea of Li and Tsai ^[35], this paper discusses the property of the solution of problem (1.1)–(1.3) regarding the solution blow-up in finite time under different initial energies $E(0)$. An upper bound on the blow-up time $T^{*}$ is established for different initial energies, and, additionally, a lower bound on the blow-up time $T^{*}$ is established by applying a differential inequality.

2.   Preliminaries

To describe the main results of this paper, this section gives some notations, generalizations, and important lemmas. We adopt the usual notations and convention. Let $H^L(\Omega)$ denote the Sobolev space with the usual scalar products and norm, Where $H_{0}^L(\Omega)$ denotes the closure in $H_{0}^L(\Omega)$ of $C_{0}^{\infty}(\Omega)$. For simplicity of notation, hereafter we denote by $||.||_{p}$ the Lebesgue space $L^p(\Omega)$ norm, and by $||.||$ the $L^2(\Omega)$ norm; equivalently we write the norm $||D^L \cdot ||$ instead of the $H_{0}^L(\Omega)$ norm $||.||_{H_{0}^L(\Omega)}$, where $D$ denotes the gradient operator, that is, $D\cdot = \bigtriangledown \cdot = \left(\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, .... \frac{\partial}{\partial x_{n}}\right)$. Moreover, $D^L\cdot = \bigtriangleup^{j}\cdot$ if $L = 2j$, and $D^L\cdot = D \bigtriangleup^{j}\cdot$ if $L = 2j+1$.

To justify the main conclusions of this paper, the following assumptions are made on $K$ $L$, and the relaxation function $g(\cdot)$.

$(A_1)$ $1\leq K < L$ are integers with $2a \leq R < +\infty$ if $n < 2L$; $2a \leq R\leq \frac{2n}{n-2L}$ if $n > 2L$,

where $a > 1$

$(A_2)$ $g:R^+\rightarrow R^+$ is a $C^1$ function, satisfing

Define the energy functional of problem $(1.1)-(1.3)$ as

where $\left(g\circ D^L u \right)(t) = \int_0^t g(t-s)\parallel D^L u(t)-D^L u(s)\parallel^2 \, ds$.

Both sides of Eq $(1.1)$ are simultaneously multiplied by $u_{t}$ and integrated over $\Omega$, and from $(A_1)$ and $(2.1)$ we have that

Definition 2.1 We say that $u(x, t)$ is a weak solution of problem (1.1) if $u \in L^{\infty}([0, T);H_{0}^L(\Omega)), u_{t} \in L^2([0, T);H_{0}^L(\Omega))$, and $u$ satisfies

for all test functions $v \in H_{0}^L(\Omega)$ and $t\in [0, T]$.

Theorem 2.1 (Local existence) Suppose that (A1) and (A2) hold. If $(u_{0}, u_{1}) \in H_{0}^L(\Omega) \times L^2 (\Omega)$, then there exists $T > 0$ such that problem (1.1) admits a unique local solution $u(t)$ which satisfies

Moreover, at least one of the following statements holds true:

The existence and uniqueness of the local solution for problem (1.1) can be obtained by using Faedo-Galerkin methods and the contraction mapping principle in ^{[30,36,37,38]}.

Lemma 2.1^[39]. Let $q$ be a real number with $2\leq q\leq +\infty$ if $n\leq 2L$, and $2\leq q\leq \frac{2n}{n-2L}$ if $n > 2L$. Then there exists a constant $B$ dependent on $\Omega$ and $q$ such that

Remark 2.1. According to Eqs $(1.1)-(1.3)$ and Lemma 2.1, we get

Let $Q(\xi) = \frac{1}{2}\xi^2-\frac{a B^R}{R\beta^{\frac{R}{2}}}\xi^R, \; \xi = \left(\beta\|D^L u\|^2+(g\circ D^L u)(t)\right)^{\frac{1}{2}} > 0.$ A direct calculation yields that $Q'(\xi) = \xi-\frac{a B^R}{\beta^{\frac{R}{2}}}\xi^{R-1}, Q''(\xi) = 1-\frac{a(R-1)B^R}{\beta^{\frac{R}{2}}}\xi^{R-2}.$ From $Q'(\xi) = 0,$ we get that $\xi_1 = \left(\frac{\beta}{aB^2}\right)^{\frac{R}{2(R-2)}}$. When $\xi = \xi_1$, direct calculation gives $Q''(\xi) = 2-R < 0$. Therefore, $Q(\xi)$ is maximum at $\xi_1$, and its maximum value is

Lemma 2.2. Let conditions $(A_1), (A_2)$ hold, $u$ be a solution of ($(1.1-(1.3)$), $E(0) < H$, and $\beta^{\frac{1}{2}}\|D^L u_0\| > \xi_1$. Then there exists $\xi_2 > \xi_1,$ such that

Proof. From Remark 2.1, $Q(\xi)$ is increasing on $(0, \xi_1)$ and decreasing on $(\xi_1, +\infty)$. $Q(\xi)\rightarrow -\infty, (\xi\rightarrow \infty).$ According to $E(0) < H$, there exists $\xi'_2, \xi_2$ such that $\xi_1\in(\xi'_2, \xi_2),$ and $Q(\xi'_2) = Q(\xi_2) = E(0).$ To prove Eq $(2.7)$, we use the converse method. Assume that there exists $t_0 > 0$ such that

1) If $\xi'_2 < \left(\beta\|D^L u(t_0)\|^2+(g\circ D^L u)(t_0)\right)^{\frac{1}{2}} < \xi_2$, then

This contradicts $(2.5)$.

2) If $\left(\beta\|D^L u(t_0)\|^2+(g\circ D^L u)(t_0)\right)^{\frac{1}{2}}\leq\xi'_2.$

As $\beta^{\frac{1}{2}}||D^L u_0|| > \xi_1$, according to $(2.5)$, $Q\left(\beta^{\frac{1}{2}}\|D^L u_0\| \right) < E(0) = Q(\xi_2)$, which implies that $\beta^{\frac{1}{2}}\|D^L u_0\| > \xi_2.$ Applying the continuity of $\left(\beta\|D^L u(t_0)\|^2+(g\circ D^L u)(t_0)\right)^{\frac{1}{2}}$, we know that there exists a $t_1\in(0, t_0)$ such that $\xi'_2 < \left(\beta\|D^L u(t_1)\|^2+(g\circ D^L u)(t_1)\right)^{\frac{1}{2}} < \xi_2.$ hence, we have $Q(\left(\beta\|D^L u(t_1)\|^2+(g\circ D^L u)(t_1)\right)^{\frac{1}{2}}) > E(0)\geq E(t_0)$, which contradicts $(2.5)$.

The following lemma is very important and is similar to the proof of Lemma 4.2 in ^[35]. Here, we make some appropriate modifications

Lemma 2.3^[40]. Let $\Gamma(t)$ be a nonincreasing function of $[t_0, \infty], t_0\geq 0$. Satisfying the differential inequality

where $\rho > 0, \psi < 0$, there exists a positive number $T^{*}$ such that

The upper bound for $T^{*}$ is

where $\Gamma(t_0) < min \lbrace 1, \sqrt{\frac{\rho}{-\psi}}\rbrace$, and $T_{max}$ denotes the maximal existence time of the solution

3.   Upper bound on blow-up time

In this section, we will give some blow-up results for solutions with initial energy $(i)$ $E(0) < 0$; $(ii)$ $0\leq E(0) < \frac{w}{R-2}H$; and $(iii)$ $\frac{w}{R-2}H\leq E(0) < \frac{||u_{0}||^2+||D^K u_{0}|||^2}{\mu}$. Moreover, some upper bounds for blow-up time $T^*$ depending on the sign and size of initial energy $E(0)$ are obtained for problem (1.1)–(1.3).

Define the functionals

where $\frac{1}{\beta}\leq\varepsilon\leq \frac{R-2a}{2a}$, and $T_0$ is positive.

Lemma 3.1. Let $X, Y$, and $\phi$ be positive, with $p, q\geq 1, \frac{1}{p}+\frac{1}{q} = 1$. Then,

Lemma 3.2. Let $(A_1), (A_2)$ hold, $u_0\in H_0^L(\Omega)$, and $u$ be a solution of $(1.1)-(1.3)$. Then, we have

where $\Pi(t) = -4(1+\varepsilon)E(0)+w[\beta\|D^L u\|^2+(g\circ D^L u)], \; w = 2\varepsilon-\frac{1}{2\beta} > 0.$

Proof. From $(3.1)$, a direct calculation yields that

We infer from $(2.2), (2.3)$, and $(3.6)$ that

Applying Lemma 3.1 yields

Combining $(3.7)$ and $(3.8)$, we get

where $w = 2\varepsilon-\frac{1}{2\beta}$.

Therom 3.1. Let assumptions $(A_1)$ and $(A_2)$ hold, and $T_0 < \frac{1}{\|u_0\|^2+\|D^Ku_0\|^2}$. In addition, it is assumed that one of the following conditions holds true:

Then, the solution of problem $(1.1)-(1.3)$ blows up in finite time, which means the maximum time $T^{*}$ of $u$ is finite and

Case (1). if $E(0) < 0$, an upper bound on the blow-up time $T^{*}$ can also be estimated according to the sign and size of energy $E(0)$. Then,

Case (2). if $0 < E(0) < \frac{w}{R-2}H$, and $\xi_1 < \beta^{\frac{1}{2}}\|D^Lu_0\|$, then

Case (3). if $\frac{w}{R-2}H\leq E(0) < \frac{\|u_0\|^2+\|D^Ku_0\|^2}{\mu}$, then

where $\chi(0) = \|u_0\|^2+\|D^K u_0\|^2-\mu E(0) = \Phi'(0)-\mu E(0)$, $\mu = \frac{4(1+\delta)}{\Lambda}, \Lambda = w\beta\frac{1}{B}$.

Case (1). if $E(0) < 0$, from $(3.9)$ we infer that

Thus, it follows that $\Phi'(t)$ is monotonically increasing. Therefore, $\Phi'(t) > \Phi'(0) = \|u_0\|^2+\|D^K u_0\|^2$

and the second derivative of Eq $(3.2)$ gives

where

From Lemma 3.2, we have

Therefore,

Applying Holder's inequality, Lemma 3.1 yields

Substituting $(3.14)-(3.18)$ into $(3.13)$ yields

From the definitions of $(3.12)$, $(3.19)$, and $\Pi(t)$, it follows that

From $\Phi'(t) > \Phi'(0) = \|u_0\|^2+\|D^Ku_0\|^2 > 0$ and $(3.12)$, we get $\Gamma'(t) < 0$, $\Gamma'(0) = 0$. $(3.20)$ multiplied by $\Gamma'(t)$ and integrated over $(0, t)$ gives

where

where $\Gamma(0) = \left[T_0(\|u_0\|^2+\|D^Ku_0\|^2)\right]^{-\varepsilon}$.

Combining $(3.21)-(3.23)$ and Lemma 2.3 shows that there exists $T^{*}$ such that $\lim_{t\rightarrow T^{*}}\Gamma(t) = 0.$ I.e.

Furthermore, according to Lemma 2.3, the upper bound on the blow-up is given by

Case (2). if $0 < E(0) < \frac{w}{R-2}H$, and $\beta^{\frac{1}{2}}\|D^Lu_0\| > \xi_{1}$, by Lemma 2.2 and the definition of $\xi_{1}$

Substituting $(3.25)$ into (3.9) yields

Hence, $\Phi'(t) > \Phi'(0) = \|u_0\|^2+\|D^Ku_0\|^2 \geq 0.$

Similar to case (1), we get

From (3.25) and $(3.27)$, we get

Similar to case (1), we have $\Gamma'(t) < 0$, $\Gamma(0) = 0$. $(3.28)$ Multiply by $\Gamma'(t)$ and integratig over $(0, t)$ gives

where

By Lemma 2.3 and $(3.29)-(3.31)$, there exists $T^*$ such that

and

Case (3) : $\frac{w}{R-2}H\leq E(0) < \frac{\|u_0\|^2+\|D^K u_0\|^2}{\mu}$.

Define

where $\mu = \frac{4(1+\varepsilon)}{\Lambda}, \; \Lambda = w\beta\frac{1}{B}.$

According to (3.31) and

we have $\frac{d}{dt}\chi(t)\geq\Lambda \chi(t)$, i.e., $\chi(t)\geq \chi(0)e^{\Lambda t}$. Thereby, we have

By $(3.34)-(3.36)$, we obtain

Thus, we get

Similar to the process in case (1), it is possible to derive

By $(3.34)-(3.36)$, we conclude that

Consequently,

Multiplying both sides of $(3.39)$ by $\Gamma(t)$, and integrating over $[0, t]$, we have

By Lemma 2.3 and $(3.40)-(3.41)$, there exists a time $T^*$ such that

and

4.   Lower bound on blow-up time

This section investigates a lower bound on the blow-up time $T^*$ when the solution of Eqs $(1.1)-(1.3)$ occurs in finite time.

Theorem 4.1. Let $A_1$ and $A_2$ hold, $u_0\in H_0^L(\Omega)$, and $u$ be a solution of Eqs $(1.1)-(1.3)$. If $u$ blows up in the sense of $H_{0}^L(\Omega)$, then the lower bound $T^{*}$ of the blow-up can be estimated as

Proof. Let

Differentiating $(1.5)$ with respect to $t$, we know from (1.1) that

By Lemma 3.1, we have

Substituting $(4.5)$ into $(1.6)$ yields

Integrating $(4.6)$ over $[0, t]$ yields

If $u$ blows up with $H_0^L$, then $T^*$ has a lower bound

which thereby completes the proof of Theorem 4.1.

5.   Conclusions

By using concavity analysis, we get the blow-up results of the solution when the initial energy is negative or positive and an upper bound on the blow-up time $T^*$. In addition, a lower bound on the blow-up time $T^*$ is obtained by applying differential inequalities in the case where the solution has a blow-up.

Use of AI tools declaration

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

Conflict of interest

The authors declare there is no conflict of interest.