A within-host model on the interaction dynamics between innate immune cells and Mycobacterium tuberculosis

1.   Introduction

In this paper, we study the initial boundary value problem of the nonlinear viscoelastic hyperbolic problem with variable exponents:

where $\Omega\subset R^{n}(n\geq1)$ is a bounded domain in $R^n$ with a smooth boundary $\partial \Omega$, $\nu$ is the unit outer normal to $\partial\Omega$, the exponents $m(x)$ and $p(x)$ are continuous functions on $\overline{\Omega}$ with the logarithmic module of continuity:

where

In addition to this condition, the exponents satisfy the following:

$g: R^{+} \, \rightarrow R^{+}$ is a $C^{1}$ function satisfying

The equation of Problem $(1.1)$ arises from the modeling of various physical phenomena such as the viscoelasticity and the system governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann's model, or electro-rheological fluids, viscoelastic fluids, processes of filtration through a porous medium, and fluids with temperature-dependent viscosity and image processing which give rise to equations with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. More details on these problems can be found in previous studies ^{[1,2,3,4,5,6]}.

When $m(x)$ and $p(x)$ are constants, Messaoudi ^[7] discussed the nonlinear viscoelastic wave equation

he proved that any weak solution with negative initial energy blows up in finite time if $p > m$, and a global existence result for $p\leq m$. The results were improved later by Messaoudi ^[8], where the blow-up result in finite time with positive initial energy was obtained. Moreover, Song ^[9] showed the finite-time blow-up of some solutions whose initial data had arbitrarily high initial energy. In the same year, Song ^[10] studied the initial-boundary value problem

and proved the nonexistence of global solutions with positive initial energy. Cavalcanti, Domingos, and Ferreira ^[11] were concerned with the non-linear viscoelastic equation

and proved the global existence of weak solutions. Moreover, they obtained the uniform decay rates of the energy by assuming a strong damping $\triangle u_{t}$ acting in the domain and providing the relaxation function which decays exponentially.

In 2017, Messaoudi ^[12] considered the following nonlinear wave equation with variable exponents:

where $a, b$ are positive constants. By using the Faedo$-$Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents $m(x)$ and $p(x)$. Then this paper also proved the finite-time blow-up of solutions and gave a two-dimensional numerical example to illustrate the blow up result. Park ^[13] showed the blow up of solutions for a viscoelastic wave equation with variable exponents

where the exponents of nonlinearity $p(x)$ and $m(x)$ are given functions and $a, b > 0$ are constants. For nonincreasing positive function $g$, they prove the blow-up result for the solutions with positive initial energy as well as nonpositive initial energy. Alahyane ^[14] discussed the nonlinear viscoelastic wave equation with variable exponents

where $\mu$ is a nonnegative constant and the exponent of nonlinearity $p(x)$ and $g$ are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function $g$, they prove a finite-time blow-up result and give some numerical applications to illustrate their theoretical results. Ouaoua and Boughamsa ^[15] considered the following boundary value problem:

the authors established the local existence by using the Faedo$-$Galerkin method with positive initial energy and suitable conditions on the variable exponents $m(x)$ and $r(x)$. In addition, they also proved that the local solution is global and obtained the stability estimate of the solution. Ding and Zhou ^[16] considered a Timoshenko-type equation

they prove that the solutions blow up in finite time with positive initial energy. Therefore, the existence of finite-time blow-up solutions with arbitrarily high initial energy is established, and the upper and lower bounds of the blow-up time are derived. More related references can be found in ^{[17,18,19,20,21,22]}.

Motivated by ^[7,13,14], we considered the existence of the solutions and their blow-up for the nonlinear damping and viscoelastic hyperbolic problem with variable exponents. Our aim in this work is to prove the existence of the weak solutions and to find sufficient conditions on $m(x)$ and $p(x)$ for which the blow-up takes place.

This article consists of three sections in addition to the introduction. In Section 2, we recall the definitions and properties of $L^{p(x)}(\Omega)$ and the Sobolev spaces $W^{1, p(x)}(\Omega)$. In Section 3, we prove the existence of weak solutions for Problem (1.1). In Section 4, we state and prove the blow-up result for solutions with positive initial energy as well as nonpositive initial energy.

2.   Preliminaries

In this section, we review some results regarding Lebesgue and Sobolev spaces with variable exponents first. All of these results and a comprehensive study of these spaces can be found in ^[23]. Here $(\cdot, \cdot)$ and $\langle \cdot, \cdot \rangle$ denote the inner product in space $L^{2}(\Omega)$ and the duality pairing between $H^{-2}(\Omega)$ and $H_{0}^{2}(\Omega)$.

The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined by

this space is endowed with the norm

The variable exponent Sobolev space $W^{1, p(x)}(\Omega)$ is defined by

the corresponding norm for this space is

define$\ W_0^{1, p(x)}(\Omega)$ as the closure of $\ C_0^\infty(\Omega)$ with respect to the$\ W^{1, p(x)}(\Omega)$ norm. The spaces $\ L^{p(x)}(\Omega), W^{1, p(x)}(\Omega)$ and $W_0^{1, p(x)}(\Omega)$ are separable and reflexive Banach spaces when $1 < p^-\leq p^+ < \infty$, where $p^-: = ess\inf\limits_{\Omega} p(x)$ and $p^+: = ess\sup\limits_{\Omega} p(x).$ As usual, we denote the conjugate exponent of $p(x)$ by $p'(x) = p(x)/(p(x)-1)$ and the Sobolev exponent by

Lemma 2.1. If $p_{1}(x), \ p_{2}(x)\in C_{+}(\overline{\Omega}) = \{h\in C(\overline{\Omega}):\min\limits_{x\in \overline{\Omega}} h(x) > 1\}$, $p_{1}(x)\leq p_{2}(x)$ for any $x\in\Omega$, then there exists the continuous embedding $L^{p_{2}(x)}(\Omega)\hookrightarrow L^{p_{1}(x)}(\Omega)$, whose norm does not exceed $|\Omega|+1$.

Lemma 2.2. Let $p(x), \ q(x)\in C_{+}(\overline{\Omega}).$ Assuming that $q(x) < p^{\ast}(x)$, there is a compact and continuous embedding $W^{k, p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega).$

Lemma 2.3. (Hölder's inequality) ^[24] For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$, then the following inequality holds:

Lemma 2.4. For $u\in L^{p(x)}(\Omega)$, the following relations hold:

Next, we give the definition of the weak solution to Problem $(1.1)$.

Definition 2.1. A function u(x, t) is called a weak solution for Problem $(1.1)$, if $u\in C(0, T;H_{0}^{2}(\Omega))$ $\cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega))$ with $u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega))$ and u satisfies the following conditions:

(1) For every $\omega \in H_{0}^{2}(\Omega)$ and for $a.e. \, t\in (0, T)$

(2) $u(x, 0) = u_{0}(x)\in H_{0}^{2}(\Omega), \, u_{t}(x, 0) = u_{1}(x)\in H_{0}^{2}(\Omega).$

3.   The local existence of weak solution

In this section, we prove the existence of a weak solution for Problem $(1.1)$ by making use of the Faedo–Galerkin method and the contraction mapping principle. For a fixed $T > 0$, we consider the space $\mathscr H = C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega))$ with the norm $||v||_{\mathscr H}^{2} = \max\limits_{0\leq t\leq T} (||\triangle v_{t}||_{2}^{2}+l||\triangle v||_{2}^{2})$.

Lemma 3.1. Assume that $(1.4)$, $(1.5)$, and $(1.6)$ hold, let $(u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega)$, for any $T > 0$, $v\in \mathscr H$, then there exists $u\in C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega))$ with $\ u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega))$ satisfying

Proof. Let $\{\omega_{j}\}_{j = 1}^{\infty}$ be the orthogonal basis of $H\mathcal{}_{0}^{2}(\Omega)$, which is the standard orthogonal basis in $L^{2}(\Omega)$ such that

we denote by $V_{k} = {\rm span}\{\omega_{1}, \omega_{2}, \cdot\cdot\cdot, \omega_{k}\}$ the subspace generated by the first $k$ vectors of the basis $\{\omega_{j}\}_{j = 1}^{\infty}$. By normalization, we have $||\omega_{j}||_{2} = 1$. For all $k\geq1$, we seek $k$ functions $c_{1}^{k}(t), c_{2}^{k}(t), \ldots, c_{k}^{k}(t)\in C^{2}[0, T]$ such that

satisfying the following approximate problem

where

thus, $(3.2)$ generates the initial value problem for the system of second-order differential equations with respect to $c_{i}^{k}(t)$:

where

and

by Peano's Theorem, we infer that the Problem $(3.3)$ admits a local solution $c_{i}^{k}(t)\in C^{2}[0, T]$.

$\textbf{The}$ $\textbf{first}$ $\textbf{estimate}.$ Multiplying $(3.2)$ by $c_{it}^{k}(t)$ and summing with respect to $i$, we arrive at the relation

By simple calculation, we have

where

inserting $(3.5)$ into $(3.4)$, using Hölder's inequality and Young's inequality, we obtain

using the embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{2(p(x)-1)}(\Omega)$ and Lemma $2.4$, we easily obtain

where $C$ is a positive constant. We denote by $C$ various positive constants that may be different at different occurrences.

Combining $(3.6)$ and $(3.7)$, we obtain

by Gronwall's inequality, there exists a positive constant $C_{T}$ such that

therefore, there exists a subsequence of $\{u^{k}\}_{k = 1}^{\infty}$, which we still denote by $\{u^{k}\}_{k = 1}^{\infty}$, such that

$\textbf{The}$ $\textbf{second}$ $\textbf{estimate}.$ Multiplying $(3.2)$ by $c_{itt}^{k}(t)$ and summing with respect to $i$, we obtain

Note that we have the estimates for $\varepsilon > 0$

and

similar to $(3.6)$ and $(3.7)$, from $H_{0}^{2}(\Omega)\hookrightarrow L^{2}(\Omega)$, we have

Taking into account $(3.10)-(3.14)$, we obtain

integrating $(3.15)$ over $(0, t)$, we obtain

taking $\varepsilon$ small enough in (3.16), for some positive constant $C_{T}$, we obtain

we observe that estimate $(3.17)$ implies that there exists a subsequence of $\{u^{k}\}_{k = 1}^{\infty}$, which we still denote by $\{u^{k}\}_{k = 1}^{\infty}$, such that

In addition, from $(3.9)$, we have

Next, we will deal with the nonlinear term. Combining $(3.9)$, $(3.18)$, and Aubin–Lions theorem ^[25], we deduce that there exists a subsequence of $\{u^{k}\}_{k = 1}^{\infty}$ such that

then

using the embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{2(m(x)-1)}(\Omega)$ and Lemma $2.4$, we have

hence, using $(3.21)$ and $(3.22)$, we obtain

Setting up $k\rightarrow \infty$ in $(3.2)$, combining with $(3.9)$, $(3.18)$, $(3.19)$, and $(3.23)$, we obtain

To handle the initial conditions. From $(3.9)$ and Aubin–Lions theorem, we can easily get $u^{k}\rightarrow u$ in $C(0, T;L^{2}(\Omega))$, thus $u^{k}(0)\rightarrow u(0)$ in $L^{2}(\Omega)$, and we also have that $u^{k}(0) = u_{0}^{k}\rightarrow u_{0}$ in $H_{0}^{2}(\Omega)$, hence $u(0) = u_{0}$ in $H_{0}^{2}(\Omega)$. Similarly, we get that $u_{t}(0) = u_{1}$.

Uniqueness. Suppose that $(3.1)$ has solutions $u$ and $z$, then $\omega = u-z$ satisfies

Multiplying the first equation of Problem $(3.1)$ by $\omega_{t}$ and integrating over $\Omega$, we have

from the inequality

for all $a, b\in R^{n}$ and a.e. $x\in\Omega$, we obtain

which implies that $\omega = 0$. This completes the proof. □

Theorem 3.1. Assume that $(1.4)$ and $(1.6)$ hold, let the initial date $(u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega)$, and

then there exists a unique local solution of Problem $(1.1)$.

Proof. For any $T > 0$, consider $M_{T} = \{u\in\mathscr H:u(0) = u_{0}, u_{t}(0) = u_{1}, ||u||_{\mathscr H}\leq M\}$. Lemma $3.1$ implies that for $\forall v\in M_{T}$, there exists $u = S(v)$ such that $u$ is the unique solution to Problem $3.1$. Next, we prove that for a suitable $T > 0$, $S$ is a contractive map satisfying $S(M_{T})\subset M_{T}$.

Multiplying the first equation of the Problem $(3.1)$ by $u_{t}$ and integrating it over $(0, t)$, we obtain

using Hölder's inequality and Young's inequality, we have

thus, $(3.25)$ becomes

hence, we have

where $\lambda_{0} = ||u_{1}||_{2}^{2}+||\triangle u_{1}||_{2}^{2}+||\Delta u_{0}||_{2}^{2}$, choosing $\gamma = \frac{1}{2T}$ such that

For any $v\in M_{T}$, by choosing $M$ large enough so that

and $T > 0$, sufficiently small so that

we obtain $||u||_{\mathscr H}\leq M$, which shows that $S(M_{T})\subset M_{T}$.

Let $v_{1}, v_{2}\in M_{T}, u_{1} = S(v_{1}), u_{2} = S(v_{2}), u = u_{1}-u_{2}$, then $u$ satisfies

Multiplying by $u_{t}$ and integrating over $\Omega\times(0, t)$, we obtain

where $f(v) = |v|^{p(x)-2}v$. From $(1.6)$ and $(3.24)$, we obtain

Now, we evaluate

where $v = v_{1}-v_{2}$ and $\xi = \alpha v_{1}+(1-\alpha)v_{2}$, $0\leq\alpha\leq1$. Thanks to Young's inequality and Hölder's inequality, we have

Inserting $(3.28)$ into $(3.27)$, choosing $\delta$ small enough, we obtain

taking $T$ small enough so that $\frac{(p^{+}-1)^{2}CT}{\delta}(M^{2(p^{-}-2)}+M^{2(p^{+}-2)}) < 1$, we conclude

thus, the contraction mapping principle ensures the existence of a weak solution to Problem $(1.1)$. This completes the proof. □

4.   The blow-up the solution

In this section, we show that the solution to Problem $(1.1)$ blows up in finite time when the initial energy lies in positive as well as nonpositive. For this task, we define

by the definition of $E(t)$, we also have

Now, we set

and

where the constant $E_{2}\in(E(0), E_{1})$ will be discussed later, and $B$ is the best constant of the Sobolev embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega)$. It follows from $(4.2)$ that

and $H(t)$ is a non$-$decreasing function.

To prove Theorem $4.1$, we need the following two lemmas:

Lemma 4.1. Suppose that $(1.6)$ holds and the exponents $m(x)$ and $p(x)$ satisfy condition $(1.4)$ and $(1.5)$. Assume further that

then there exists a constant $\lambda_{2} > \lambda_{1}$ such that

Proof. Using $(1.6)$, $(4.1)$, Lemma $2.4$, and the embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega)$, we find that

where $\lambda: = \lambda(t) = B_{1}^{2}l||\triangle u||_{2}^{2}.$ Analyzing directly the properties of $G(\lambda)$, we deduce that $G(\lambda)$ satisfies the following properties:

It is easily verified that $G(\lambda)$ is strictly increasing for $0 < \lambda < \lambda_{1}$, strictly decreasing for $\lambda_{1} < \lambda$, $G(\lambda)\rightarrow -\infty$ as $\lambda\rightarrow +\infty$, and $G(\lambda_{1}) = E_{1}$. Since $E(0) < E_{1}$, there exists a $\lambda_{2} > \lambda_{1}$ such that $G(\lambda_{2}) = E(0)$. By $(4.6)$, we see that $G(\lambda(0))\leq E(0) = G(\lambda_{2})$, which implies $\lambda(0)\geq\lambda_{2}$ since the condition $\lambda(0) > \lambda_{1}.$ To prove $(4.5)$, we suppose by contradiction that for some $t_{0} > 0$, $\lambda_{t_{0}} = B_{1}^{2}l||\triangle u(t_{0})||_{2}^{2} < \lambda_{2}$. The continuity of $B_{1}^{2}l||\triangle u||_{2}^{2}$ illustrates that we could choose $t_{0}$ such that $\lambda_{1} < \lambda_{t_{0}} < \lambda_{2}$, then we have $E(0) = G(\lambda_{2}) < G(\lambda_{t_{0}})\leq E(t_{0})$. This is a contradiction. The proof is completed. □

Lemma 4.2. Let the assumption in Lemma $4.1$ be satisfied. For $t\in[0, T)$, we have

Proof. $(4.4)$ indicates that $H(t)$ is nondecreasing with respect to $t$, thus

It follows from $(1.6)$, $(4.1)$, and Lemma $4.1$ that

The proof is completed.

Our blow-up result reads as follows:

Theorem 4.3. Suppose that

and

hold, if the following conditions

are satisfied, then there exists $T^{\ast} < +\infty$ such that

Proof. Assume by contradiction that $(4.8)$ does not hold true, then for $\forall T^{\ast} < +\infty$ and all $t\in[0, T^{\ast}]$, we get

where $C_{\ast}$ is a positive constant.

Now, we define $L(t)$ as follows:

where $\varepsilon > 0$, small enough to be chosen later, and

The remaining proof will be divided into two steps.

$\textbf{Step}$ $\textbf{1:}$ $\textbf{Estimate}$ $\textbf{for}$ $\textbf{L'(t)}.$ By taking the derivative of $(4.10)$ and using $(1.1)$, we obtain

applying Hölder's inequality and Young's inequality, we have

where $0 < \varepsilon_{1} < \frac{p^{-}-2}{p^{-}}$, then

rewriting $(4.7)$ to $(\frac{p^{-}}{2}-1)l-\frac{1}{2p^{-}}(1-l) > 0$, using $(4.1)$ and $(4.3)$ to substitute for $(g\diamond \bigtriangleup u)$, choosing $\varepsilon_{1} > 0$ sufficiently small, we obtain

$\textbf{Step}$ $\textbf{1.1:}$ $\textbf{Estimate}$ $\textbf{for}$ $\epsilon\frac{\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon p^{-}(1-\varepsilon_{1})E_{2}.$ It follows from the condition in Theorem $3.1$ that

here, we can take $\varepsilon_{1} > 0$ sufficiently small and choose $E_{2}\in(E(0), E_{1})$ sufficiently close to $E(0)$ such that

Therefore, we obtain by combining $(4.11)$ and $(4.12)$,

$\textbf{Step}$ $\textbf{1.2:}$ $\textbf{Estimate}$ $\textbf{for}$ $-\epsilon \int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx.$ Applying Young's inequality with $\varepsilon_{2} > 1$, the embedding $L^{p(x)}(\Omega)\hookrightarrow L^{m(x)}(\Omega)$, Lemma $2.4$ and Lemma $4.2$, we easily have

where $C_{1} = \min\left\{H(0), 1\right\}$, $C_{2} = 2(1+|\Omega|)^{m^{+}}C_{1}^{\alpha(m^{-}-m^{+})}$. Next, we have

and

which illustrate

where $C_{3} = 2\min\left\{p^{-}H(0), 1\right\}^{\frac{m^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}}$. Recalling $0 < \alpha\leq\frac{p^{-}-m^{+}}{p^{-}(m^{+}-1)}$ and Lemma $4.2$, apparently,

it follows from $(4.13)$, $(4.14)$, and $(4.15)$ that

let us fix the constant $\varepsilon_{2}$ so that

and then choose $\epsilon$ so small that $1-\alpha > \epsilon\varepsilon_{1}$. Therefore, we obtain

where

Inequalities $(4.16)$ and Lemma $4.2$ imply $L(t)\geq L(0).$ Therefore, for a sufficiently small $\epsilon$, we have

$\textbf{Step}$ $\textbf{2:}$ $\textbf{A}$ $\textbf{differential}$ $\textbf{inequality}$ $\textbf{for}$ $\textbf{L(t)}.$ Applying Hölder's inequality, Young's inequality and the embedding $L^{p(x)}(\Omega)\hookrightarrow L^{2}(\Omega)$, we easily obtain

where $\frac{1}{\mu}+\frac{1}{\nu} = 1.\ $Choosing $\mu = 2(1-\alpha) > 1$, then $\nu = \frac{2(1-\alpha)}{2(1-\alpha)-1}, \ $further, $(4.17)$ can be rewritten as

recalling $0 < \alpha < \frac{p^{-}-2}{2p^{-}}$, we obtain

with $C_{4} = \min\{p^{-}H(0), 1\}^{\frac{2-p^{+}[2(1-\alpha)-1]}{p^{+}[2(1-\alpha)-1]}}$. Inserting $(4.19)$ into $(4.18)$, we obtain

We now estimate

therefore, combining $(4.20)$ and $(4.21)$, we obtain

where

Combining $(4.16)$ and $(4.22)$, we arrive at

A simple integration of $(4.23)$ over $(0, t)$ yields

this shows that $L(t)$ blows up in finite time

furthermore, one gets from $(4.22)$ that

it easily follows that

and using Lemma $4.2$, we obtain

this leads to a contradiction with $(4.9)$. Thus, the solution to Problem $(1.1)$ blows up in finite time. □

Author contributions

Ying Chu: Methodology, Wring-original draft, Writing-review editing; Bo Wen and Libo Cheng: Methodology, Writing-original draft.

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 express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work was supported by Science and Technology Development Plan Project of Jilin Province, China (20240101307JC).

Conflict of interest

The authors declare there is no conflict of interest.