Accurate determination of the onset time in acute ischemic stroke (AIS) patients helps to formulate more beneficial treatment plans and plays a vital role in the recovery of patients. Considering that the whole brain may contain some critical information, we combined the Radiomics features of infarct lesions and whole brain to improve the prediction accuracy. First, the radiomics features of infarct lesions and whole brain were separately calculated using apparent diffusion coefficient (ADC), diffusion-weighted imaging (DWI) and fluid-attenuated inversion recovery (FLAIR) sequences of AIS patients with clear onset time. Then, the least absolute shrinkage and selection operator (Lasso) was used to select features. Four experimental groups were generated according to combination strategies: Features in infarct lesions (IL), features in whole brain (WB), direct combination of them (IW) and Lasso selection again after direct combination (IWS), which were used to evaluate the predictive performance. The results of ten-fold cross-validation showed that IWS achieved the best AUC of 0.904, which improved by 13.5% compared with IL (0.769), by 18.7% compared with WB (0.717) and 4.2% compared with IW (0.862). In conclusion, combining infarct lesions and whole brain features from multiple sequences can further improve the accuracy of AIS onset time.
Citation: Jiaxi Lu, Yingwei Guo, Mingming Wang, Yu Luo, Xueqiang Zeng, Xiaoqiang Miao, Asim Zaman, Huihui Yang, Anbo Cao, Yan Kang. Determining acute ischemic stroke onset time using machine learning and radiomics features of infarct lesions and whole brain[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 34-48. doi: 10.3934/mbe.2024002
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Accurate determination of the onset time in acute ischemic stroke (AIS) patients helps to formulate more beneficial treatment plans and plays a vital role in the recovery of patients. Considering that the whole brain may contain some critical information, we combined the Radiomics features of infarct lesions and whole brain to improve the prediction accuracy. First, the radiomics features of infarct lesions and whole brain were separately calculated using apparent diffusion coefficient (ADC), diffusion-weighted imaging (DWI) and fluid-attenuated inversion recovery (FLAIR) sequences of AIS patients with clear onset time. Then, the least absolute shrinkage and selection operator (Lasso) was used to select features. Four experimental groups were generated according to combination strategies: Features in infarct lesions (IL), features in whole brain (WB), direct combination of them (IW) and Lasso selection again after direct combination (IWS), which were used to evaluate the predictive performance. The results of ten-fold cross-validation showed that IWS achieved the best AUC of 0.904, which improved by 13.5% compared with IL (0.769), by 18.7% compared with WB (0.717) and 4.2% compared with IW (0.862). In conclusion, combining infarct lesions and whole brain features from multiple sequences can further improve the accuracy of AIS onset time.
As we all know, Keller and Segel [1] first proposed the classical chemotaxis model (hereafter called K-S model), which has been widely applied in biology and medicine. The model can be given by the following:
{v1t=Δv1−χ∇⋅(v1∇v2)+f(v1), x∈Ω, t>0,τv2t=Δv2−v2+v1, x∈Ω, t>0, | (1.1) |
where v1 is the cell density, v2 is the concentration of the chemical signal, and f(v1) is the logistic source function. For the case of τ=1 and f(v1)=0, it has been proven that the classical solutions to system (1.1) always remain globally bounded when n=1 [2]. A critical mass phenomenon of system (1.1) has been shown in a two-dimensional space. Namely, if the initial data v10 satisfies ‖v10‖L1(Ω)<4πχ, then the solution (v1,v2) is globally bounded [3]. Alternatively, if the initial data v10 satisfies ‖v10‖L1(Ω)>4πχ, then the solution (v1,v2) is unbounded in finite or infinite time, provided Ω is simply connected [4,5]. In particular, for a framework of radially symmetric solutions in a planar disk, the solutions blow up in finite time if ‖v10‖L1(Ω)>8πχ [6]. When f(v1)=0, Liu and Tao [7] changed τv2t=Δv2−v2+v1 to v2t=Δv2−v2+g(v1) with 0≤g(v1)≤Kvα1 for K,α>0, and obtained the global well-posedness of model (1.1) provided that 0<α<2n. Later on, the equation τv2t=Δv2−v2+v1 was changed to 0=Δv2−ϖ(t)+g(v1) with ϖ(t)=1|Ω|∫Ωg(v1(⋅,t)) for g(v1)=vα1. Winkler [8] deduced that for any v10, the model (1.1) is globally and classical solvable if α<2n; conversely, if α>2n, then the solutions are unbounded in a finite-time for any ∫Ωv10=m>0. For τ=0, when f(v1)≤v1(c−dv1) with c,d>0, Tello and Winkler [9] deduced the global well-posedness of model (1.1) provided that d>n−2nχ. Afterwards, when f(v1)=cv1−dvϵ1 with ϵ>1,c≥0,d>0, Winkler [10] defined a concept of very weak solutions and observed that these solutions are globally bounded under some conditions. For more results on (1.1), the readers can refer to [11,12,13,14].
Considering the volume filling effect [15], the self-diffusion functions and chemotactic sensitivity functions may have nonlinear forms of the cell density. The general model can be written as follows:
{v1t=∇⋅(ψ(v1)∇v1−ϕ(v1)∇v2)+f(v1), x∈Ω, t>0,τv2t=Δv2−v2+v1, x∈Ω, t>0. | (1.2) |
Here, ψ(v1) and ϕ(v1) are nonlinear functions. When τ=1 and f(v1)=0, for any ∫Ωv10=M>0, Winkler [16] derived that the solution (v1,v2) is unbounded in either finite or infinite time if ϕ(v1)ψ(v1)≥cvα1 with α>2n,n≥2 and some constant c>0 for all v1>1. Later on, Tao and Winkler [17] deduced the global well-posedness of model (1.5) provided that ϕ(v1)ψ(v1)≤cvα1 with α<2n,n≥1 and some constant c>0 for all v1>1. Furthermore, in a high-dimensional space where n≥5, Lin et al. [18] changed the equation τv2t=Δv2−v2+v1 to 0=Δv2−ϖ(t)+v1 with ϖ(t)=1|Ω|∫Ωv1(x,t)dx, and showed that the solution (v1,v2) is unbounded in a finite time.
Next, we introduce the chemotaxis model that involves an indirect signal mechanism. The model can be given by the following:
{v1t=∇⋅(ψ(v1)∇v1−ϕ(v1)∇v2)+f(v1), x∈Ω, t>0,τv2t=Δv2−v2+w, x∈Ω, t>0,τwt=Δw−w+v1, x∈Ω, t>0. | (1.3) |
For τ=1, when ψ(v1)=1,ϕ(v1)=v1 and f(v1)=λ(v1−vα1), the conclusion in [19] implied that the system is globally classical solvable if α>n4+12 with n≥2. Furthermore, the authors in [20,21,22] extended the boundedness result to a quasilinear system. Ren [23] derived the global well-posedness of system (1.3) and provided the qualitative analysis of such solutions. For τ=0, when ψ(s)≥c(s+1)θ and |ϕ(s)|≤ds(s+1)κ−1 with s≥0,c,d>0 and θ,κ∈R, Li and Li [24] obtained that the model (1.3) is globally classical solvable. Meanwhile, they also provided the qualitative analysis of such solutions. More results of the system with an indirect signal mechanism can be found in [25,26,27,28].
Considering that the cell or bacteria populations have a tendency to move towards a degraded nutrient, the authors obtain another well-known chemotaxis-consumption system:
{v1t=Δv1−χ∇⋅(v1∇v2), x∈Ω, t>0,v2t=Δv2−v1v2, x∈Ω, t>0, | (1.4) |
where v1 denotes the cell density, and v2 denotes the concentration of oxygen. If 0<χ≤16(n+1)‖v20‖L∞(Ω) with n≥2, then the results of [29] showed that the system (1.4) is globally classical solvable. Thereafter, Zhang and Li [30] deduced the global well-posedness of model (1.4) provided that n≤2 or 0<χ≤16(n+1)‖v20‖L∞(Ω),n≥3. In addition, for a sufficiently large v10 and v20, Tao and Winkler [31] showed that the defined weak solutions globally exist when n=3. Meanwhile, they also analyzed the qualitative properties of these weak solutions.
Based on the model (1.4), some researchers have considered the model that involves an indirect signal consumption:
{v1t=Δv1−χ∇⋅(v1∇v2), x∈Ω, t>0,v2t=Δv2−v1v2, x∈Ω, t>0,wt=−δw+v1, x∈Ω, t>0, | (1.5) |
where w represents the indirect signaling substance produced by cells for degrading oxygen. Fuest [32] obtained the global well-posedness of model (1.5) provided that n≤2 or ‖v20‖L∞(Ω)≤13n, and studied the convergence rate of the solution. Subsequently, the authors in [33] extended the boundedness conclusion of model (1.5) using conditions n≥3 and 0<‖v20‖L∞(Ω)≤π√n. For more results on model (1.5), the readers can refer to [34,35,36,37,38,39].
Inspired by the work mentioned above, we find that there are few papers on the quasilinear chemotaxis model that involve the nonlinear indirect consumption mechanism. In view of the complexity of the biological environment, this signal mechanism may be more realistic. In this manuscript, we are interested in the following system:
{v1t=∇⋅(ψ(v1)∇v1−χϕ(v1)∇v2)+λ1v1−λ2vβ1, x∈Ω, t>0,v2t=Δv2−wθv2, x∈Ω, t>0,0=Δw−w+vα1, x∈Ω, t>0,∂v1∂ν=∂v2∂ν=∂w∂ν=0, x∈∂Ω, t>0,v1(x,0)=v10(x),v2(x,0)=v20(x), x∈Ω, | (1.6) |
where Ω⊂Rn(n≥1) is a bounded and smooth domain, ν denotes the outward unit normal vector on ∂Ω, and χ,λ1,λ2,θ>0,0<α≤1θ,β≥2. Here, v1 is the cell density, v2 is the concentration of oxygen, and w is the indirect chemical signal produced by v1 to degrade v2. The diffusion functions ψ,ϕ∈C2[0,∞) are assumed to satisfy
ψ(s)≥a0(s+1)r1 and 0≤ϕ(s)≤b0s(s+1)r2, | (1.7) |
for all s≥0 with a0,b0>0 and r1,r2∈R. In addition, the initial data v10 and v20 fulfill the following:
v10,v20∈W1,∞(Ω) with v10,v20≥0,≢0 in Ω. | (1.8) |
Theorem 1.1. Assume that χ,λ1,λ2,θ>0,0<α≤1θ, and β≥2, and that Ω⊂Rn(n≥1) is a smooth bounded domain. Let ψ,ϕ∈C2[0,∞) satisfy (1.7). Suppose that the initial data v10 and v20 fulfill (1.8). It has been proven that if r1>2r2+1, then the problem (1.6) has a nonnegative classical solution
(v1,v2,w)∈(C0(ˉΩ×[0,∞))∩C2,1(ˉΩ×(0,∞)))2×C2,0(ˉΩ×(0,∞)), |
which is globally bounded in the sense that
‖v1(⋅,t)‖L∞(Ω)+‖v2(⋅,t)‖W1,∞(Ω)+‖w(⋅,t)‖W1,∞(Ω)≤C, |
for all t>0, with C>0.
Remark 1.2. Our main ideas are as follows. First, we obtain the L∞ bound for v2 by the maximum principle of the parabolic equation. Next, we establish an estimate for the functional y(t):=1p∫Ω(v1+1)p+12p∫Ω|∇v2|2p for any p>1 and t>0. Finally, we can derive the global solvability of model (1.6).
Remark 1.3. Theorem 1.1 shows that self-diffusion and logical source are advantageous for the boundedness of the solutions. In this manuscript, due to the indirect signal substance w that consumes oxygen, the aggregation of cells or bacterial is almost impossible when self-diffusion is stronger than cross-diffusion, namely r1>2r2+1. We can control the logical source to ensure the global boundedness of the solution for model (1.6). Thus, we can study the effects of the logistic source, the diffusion functions, and the nonlinear consumption mechanism on the boundedness of the solutions.
In this section, we first state a lemma on the local existence of classical solutions. The proof can be proven by the fixed point theory. The readers can refer to [40,41] for more details.
Lemma 2.1. Let the assumptions in Theorem 1.1 hold. Then, there exists Tmax such that the problem (1.6) has a nonnegative classical solution (v_{1}, v_{2}, w) that satisfies the following:
\begin{equation*} (v_{1},v_{2},w)\in\left(C^{0}(\bar{\Omega}\times[0,T_{\max}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{\max}))\right)^{2} \times C^{2,0}(\bar{\Omega}\times(0,T_{\max})). \end{equation*} |
Furthermore, if T_{\max} < \infty, then
\begin{equation*} \label{3.13z} {\limsup\limits_{t\nearrow T_{\max}}} \left(\|v_{1}(\cdot,t)\|_{L^{\infty}(\Omega)} +\|v_{2}(\cdot,t)\|_{W^{1,\infty}{(\Omega)}}\right) = \infty. \end{equation*} |
Lemma 2.2. (cf. [42]) Let \Omega \subset \mathbb{R}^{n} (n\geq 1) be a smooth bounded domain. For any s\geq 1 and \epsilon > 0, one can obtain
\begin{equation} \int_{\partial \Omega}|\nabla z|^{2s-2}\frac{\partial|\nabla z|^{2}}{\partial \nu} \leq \epsilon\int_{\Omega}|\nabla z|^{2s-2}|D^{2}z|^{2}+C_{\epsilon}\int_{\Omega}|\nabla z|^{2s}, \end{equation} |
for all z\in C^{2}(\bar{\Omega}) fulfilling \frac{\partial z}{\partial \nu}\big{|}_{\partial \Omega} = 0, with C_{\epsilon} = C(\epsilon, s, \Omega) > 0.
Lemma 2.3. (cf. [43]) Let \Omega \subset \mathbb{R}^{n} (n\geq 1) be a bounded and smooth domain. For s\geq 1, we have
\begin{equation*} \int_{ \Omega}|\nabla z|^{2s+2} \leq 2(4s^{2}+n)\|z\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}|\nabla z|^{2s-2}|D^{2}z|^{2}, \end{equation*} |
for all z\in C^{2}(\bar{\Omega}) fulfilling \frac{\partial z}{\partial \nu}\big{|}_{\partial \Omega} = 0.
Lemma 2.4. Let \Omega \subset \mathbb{R}^{n} (n\geq 1) be a bounded and smooth domain. For any z\in C^{2}(\Omega), one has the following:
\begin{equation*} (\Delta z)^{2}\leq n|D^{2}z|^{2}, \end{equation*} |
where D^{2}z represents the Hessian matrix of z and |D^{2}z|^{2} = \sum_{i, j = 1}^{n}z_{x_{i}x_{j}}^{2}.
Proof. The proof can be found in [41, Lemma 3.1].
Lemma 2.5. (cf. [44,45]) Let a_{1}, a_{2} > 0. The non-negative functions f\in C([0, T))\cap C^{1}((0, T)) and y\in L_{loc}^{1}([0, T)) fulfill
\begin{equation*} f'(t)+a_{1}f(t)\leq y(t), \ \ t\in (0,T), \end{equation*} |
and
\begin{equation*} \int_{t}^{t+\tau}y(s)ds\leq a_{2}, \ \ t\in (0,T-\tau), \end{equation*} |
where \tau = \min\{1, \frac{T}{2}\} and T\in(0, \infty]. Then, one deduces the following:
\begin{equation*} f(t)\leq f(0)+2a_{2}+\frac{a_{2}}{a_{1}}, \ \ t\in (0,T). \end{equation*} |
In this section, we provide some useful Lemmas to prove Theorem 1.1.
Lemma 3.1. Let \beta > 1, then, there exist M, \; M_{1}, \; M_{2} > 0 such that
\begin{align} \|v_{2}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq M\ for\ all \ t\in (0,T_{\max}), \end{align} | (3.1) |
and
\begin{align} \int_{\Omega}v_{1}\leq M_{1} \ for\ all\ t\in (0,T_{\max}). \end{align} | (3.2) |
Proof. By the parabolic comparison principle for v_{2t} = \Delta v_{2}-w_{1}^{\theta}v_{2}, we can derive (3.1). Invoking the integration for the first equation of (1.6), one has the following:
\begin{align} \frac{d}{dt}\int_{\Omega}v_{1} = \lambda_{1}\int_{\Omega}v_{1} -\lambda_{2}\int_{\Omega}v_{1}^{\beta} \ \ \mbox{for all}\ t\in (0,T_{\max}). \end{align} | (3.3) |
Invoking the Hölder inequality, we obtain the following:
\begin{equation} \frac{d}{dt}\int_{\Omega}v_{1}\leq \lambda_{1}\int_{\Omega}v_{1} -\frac{\lambda_{2}}{|\Omega|^{\beta-1}} \left(\int_{\Omega}v_{1}\right)^{\beta}. \end{equation} | (3.4) |
We can apply the comparison principle to deduce the following:
\begin{align} \int_{\Omega}v_{1}\leq \max\bigg\{\int_{\Omega}v_{10}, \left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{\frac{1}{\beta-1}}|\Omega|\bigg\} = M_{1}. \end{align} | (3.5) |
Thereupon, we complete the proof.
Lemma 3.2. For any \gamma > 1, we have the following:
\begin{align} \int_{\Omega}w^{\gamma}\leq C_{0}\int_{\Omega}v_{1}^{\alpha\gamma} \ \ for\ all\ \ t\in (0,T_{\max}), \end{align} | (3.6) |
where C_{0} = \frac{2^{\gamma}}{1+\gamma} > 0.
Proof. For \gamma > 1, multiplying equation 0 = \Delta w-w+v_{1}^{\alpha} by w^{\gamma-1}, one obtain the following:
\begin{align} 0& = -\left(\gamma-1\right)\int_{\Omega}w^{\gamma-2}|\nabla w|^{2}-\int_{\Omega}w^{\gamma} +\int_{\Omega}v_{1}^{\alpha}w^{\gamma-1} \\&\leq\int_{\Omega}v_{1}^{\alpha}w^{\gamma-1} -\int_{\Omega}w^{\gamma} \ \ \mbox{for all}\ t\in (0,T_{\max}). \end{align} | (3.7) |
By Young's inequality, it is easy to deduce the following:
\begin{align} \int_{\Omega}v_{1}^{\alpha}w^{\gamma-1}\leq \frac{\gamma-1}{2\gamma}\int_{\Omega}w^{\gamma} +2^{\gamma-1}\cdot\frac{1}{\gamma}\int_{\Omega}v_{1}^{\alpha\gamma}. \end{align} | (3.8) |
Thus, we arrive at (3.6) by combining (3.7) with (3.8).
Lemma 3.3. Let the assumptions in Lemma 2.1 hold. For any p > \max\{1, \frac{1}{\theta}-1\}, there exists C > 0 such that
\begin{align} \frac{1}{2p}\frac{d}{dt}\int_{\Omega}|\nabla v_{2}|^{2p}+\frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p}+\frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}\leq C \int_{\Omega}v_{1}^{\theta\alpha(p+1)}+C, \end{align} | (3.9) |
for all t\in (0, T_{\max}) .
Proof. Using the equation v_{2t} = \Delta v_{2}-w_{1}^{\theta}v_{2} , we obtain the following:
\begin{align} \nabla v_{2}\cdot\nabla v_{2t}& = \nabla v_{2}\cdot \nabla\Delta v_{2}-\nabla v_{2}\cdot\nabla\left(w^{\theta}v_{2}\right) \\ & = \frac{1}{2}\Delta|\nabla v_{2}|^{2}-|D^{2}v_{2}|^{2}-\nabla v_{2}\cdot\nabla\left(w^{\theta}v_{2}\right), \end{align} | (3.10) |
where we used the equality \nabla v_{2}\cdot \nabla\Delta v_{2} = \frac{1}{2}\Delta|\nabla v_{2}|^{2}-|D^{2}v_{2}|^{2}. Testing (3.10) by |\nabla v_{2}|^{2p-2} and integrating by parts, we derive the following:
\begin{align} \frac{1}{2p}&\frac{d}{dt}\int_{\Omega}|\nabla v_{2}|^{2p}+\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+ \frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p} \\ & = \frac{1}{2}\int_{\Omega}|\nabla v_{2}|^{2p-2}\Delta|\nabla v_{2}|^{2}+ \int_{\Omega}|\nabla v_{2}|^{2p}-\int_{\Omega}|\nabla v_{2}|^{2p-2}\nabla v_{2}\cdot\nabla\left(w^{\theta}v_{2}\right) \\ & = I_{1}+ \frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p}+I_{2}. \end{align} | (3.11) |
Using Lemma 2.3 and (3.1), one has the following:
\begin{align} \int_{\Omega}|\nabla v_{2}|^{2p+2}\leq C_{1}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2} \ \ \mbox{for all}\ t\in (0,T_{\max}), \end{align} | (3.12) |
where C_{1} = 2(4p^{2}+n)M^{2}. In virtue of Lemma 2.2, Young's inequality, and (3.12), an integration by parts produces the following:
\begin{align} I_{1}+ \frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p}& = \frac{1}{2}\int_{\Omega}|\nabla v_{2}|^{2p-2}\Delta|\nabla v_{2}|^{2}+\frac{1}{2p} \int_{\Omega}|\nabla v_{2}|^{2p} \\ & = \frac{1}{2}\int_{\partial\Omega}|\nabla v_{2}|^{2p-2}\frac{\partial|\nabla v_{2}|^{2}}{\partial \nu} -\frac{1}{2}\int_{\Omega}\nabla|\nabla v_{2}|^{2p-2}\cdot\nabla|\nabla v_{2}|^{2}+\frac{1}{2p} \int_{\Omega}|\nabla v_{2}|^{2p} \\ &\leq \frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+C_{2}\int_{\Omega}|\nabla v_{2}|^{2p} -\frac{p-1}{2}\int_{\Omega}|\nabla v_{2}|^{2p-4}\bigg|\nabla|\nabla v_{2}|^{2}\bigg|^{2} \\ &\leq \frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2} +\frac{1}{4C_{1}}\int_{\Omega}|\nabla v_{2}|^{2p+2}+C_{3} \\ &\leq \frac{1}{2}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+C_{3} \ \ \mbox{for all}\ t\in (0,T_{\max}), \end{align} | (3.13) |
with C_{2}, C_{3} > 0. Due to |\Delta v_{2}|\leq \sqrt{n}|D^{2}v_{2}|, we can conclude from (3.1) and the integration by parts that
\begin{align} I_{2}& = -\int_{\Omega}|\nabla v_{2}|^{2p-2}\nabla v_{2}\cdot\nabla\left(w^{\theta}v_{2}\right) = \int_{\Omega}w^{\theta}v_{2}\nabla\cdot\left(\nabla v_{2}|\nabla v_{2}|^{2p-2}\right) \\ &\leq\int_{\Omega}w^{\theta}v_{2}\left(\Delta v_{2}|\nabla v_{2}|^{2p-2}+(2p-2)|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|\right) \\ &\leq \int_{\Omega}\left(\sqrt{n}+2(p-2)\right) Mw^{\theta}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}| \\ & = C_{4}\int_{\Omega}w^{\theta}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|\ \ \mbox{for all}\ t\in (0,T_{\max}), \end{align} | (3.14) |
with C_{4} = (\sqrt{n}+2(p-2))M > 0 . Due to p > \max\{1, \frac{1}{\theta}-1\}, we have \theta(p+1) > 1. With applications of Young's inequality, (3.12), and Lemma 3.2, we obtain the following from (3.14):
\begin{align} &C_{4}\int_{\Omega}w^{\theta}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}| \leq \frac{1}{8}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+C_{5} \int_{\Omega}w^{2\theta}|\nabla v_{2}|^{2p-2} \\ &\leq \frac{1}{8}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2} +\frac{1}{8C_{1}}\int_{\Omega}|\nabla v_{2}|^{2p+2} +C_{6}\int_{\Omega}w^{\theta(p+1)} \\ &\leq \frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+C_{7} \int_{\Omega}w^{\theta(p+1)} \\ &\leq \frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+C_{8} \int_{\Omega}v_{1}^{\theta\alpha(p+1)}, \end{align} | (3.15) |
with C_{5}, C_{6}, C_{7}, C_{8} > 0. Substituting (3.13) and (3.15) into (3.11), we derive the following:
\begin{align} \frac{1}{2p}\frac{d}{dt}\int_{\Omega}|\nabla v_{2}|^{2p}+\frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p}+\frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}\leq C_{8}\int_{\Omega}v_{1}^{\theta\alpha(p+1)}+C_{3}, \end{align} | (3.16) |
for all t\in (0, T_{\max}). Thereupon, we complete the proof.
Lemma 3.4. Let the assumptions in Lemma 2.1 hold. If r_{1} > 2r_{2}+1, then for any p > 1, we obtain the following:
\begin{align} &\frac{1}{p}\frac{d}{dt} \int_{\Omega}(v_{1}+1)^{p}+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p} \\ &\leq\frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+ (C+\lambda_{1}+\frac{1}{p})\int_{\Omega}(v_{1}+1)^{p} -\lambda_{2}\int_{\Omega}v_{1}^{p+\beta-1}+C, \end{align} | (3.17) |
for all t\in (0, T_{\max}), with C > 0.
Proof. Testing the first equation of problem (1.6) by (v_{1}+1)^{p-1}, one can obtain the following:
\begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega} (v_{1}+1)^{p}+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p} = &-(p-1)\int_{\Omega}(v_{1}+1)^{p-2}\psi(v_{1})|\nabla v_{1}|^{2}+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p} \\ &+\chi(p-1)\int_{\Omega}(v_{1}+1)^{p-2}\phi(v_{1})\nabla v_{1}\cdot\nabla v_{2} \\ &+\lambda_{1}\int_{\Omega}v_{1}(v_{1}+1)^{p-1}-\lambda_{2} \int_{\Omega}v_{1}^{\beta}(v_{1}+1)^{p-1}, \end{align} | (3.18) |
for all t\in(0, T_{\max}). In view of (1.7), the first term on the right-hand side of (3.18) can be estimated as follows:
\begin{align} -(p-1)\int_{\Omega}(v_{1}+1)^{p-2}\psi(v_{1})|\nabla v_{1}|^{2}\leq -(p-1)a_{0}\int_{\Omega}(v_{1}+1)^{p+r_{1}-2}|\nabla v_{1}|^{2}. \end{align} | (3.19) |
For the second term on the right-hand side of (3.18), we can see that
\begin{align} \chi(p-1)\int_{\Omega}(v_{1}+1)^{p-2} \phi(v_{1})\nabla v_{1}\cdot\nabla v_{2} \leq \chi(p-1)b_{0}\int_{\Omega}v_{1}(v_{1}+1)^{p+r_{2}-2}\nabla v_{1}\cdot\nabla v_{2}. \end{align} | (3.20) |
We can obtain the following from Young's inequality:
\begin{align} &\chi(p-1)b_{0}\int_{\Omega}v_{1}(v_{1}+1)^{p+r_{2}-2}\nabla v_{1}\cdot\nabla v_{2} \\ &\leq\chi(p-1)b_{0}\int_{\Omega} (v_{1}+1)^{p+r_{2}-1}\nabla v_{1}\cdot\nabla v_{2} \\ &\leq(p-1)a_{0}\int_{\Omega} (v_{1}+1)^{p+r_{1}-2}|\nabla v_{1}|^{2} +C_{1}\int_{\Omega}(v_{1}+1)^{p+2r_{2}-r_{1}}|\nabla v_{2}|^{2}, \end{align} | (3.21) |
with C_{1} > 0. Utilizing Young's inequality and (3.12), one has the following:
\begin{align} C_{1}\int_{\Omega}(v_{1}+1)^{p+2r_{2}-r_{1}}|\nabla v_{2}|^{2} &\leq \frac{1}{8(4p^{2}+n)M^{2}}\int_{\Omega}|\nabla v_{2}|^{2(p+1)}+C_{2}\int_{\Omega}(v_{1}+1)^{\frac{(p+1)(p+2r_{2}-r_{1})}{p}} \\ &\leq \frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+ C_{2}\int_{\Omega}(v_{1}+1)^{\frac{(p+1)(p+2r_{2}-r_{1})}{p}}, \end{align} | (3.22) |
where C_{2} > 0. Due to r_{1} > 2r_{2}+1, for any p > 1 > \frac{r_{1}-2r_{2}}{2r_{2}-r_{1}+1}, we can obtain \frac{(p+1)(p+2r_{2}-r_{1})}{p} < p. Applying Young's inequality, we obtain the following:
\begin{align} C_{2}\int_{\Omega}(v_{1}+1)^{\frac{(p+1)(p+2r_{2}-r_{1})}{p}} \leq C_{3}\int_{\Omega}(v_{1}+1)^{p}+C_{3}, \end{align} | (3.23) |
where C_{3} > 0. Hence, substituting (3.19)–(3.23) into (3.18), one obtains the following:
\begin{align} &\frac{1}{p}\frac{d}{dt} \int_{\Omega}(v_{1}+1)^{p}+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p} \\ &\leq\frac{1}{4}\int_{\Omega}|\nabla v_{2}|^{2p-2}|D^{2}v_{2}|^{2}+ (C_{3}+\lambda_{1}+\frac{1}{p})\int_{\Omega}(v_{1}+1)^{p} -\lambda_{2}\int_{\Omega}v_{1}^{p+\beta-1}+C_{4}, \end{align} | (3.24) |
for all t\in (0, T_{\max}), where C_{4} > 0.
Lemma 3.5. Let the assumptions in Lemma 2.1 hold. If r_{1} > 2r_{2}+1, then for any p > \max\{1, \frac{1}{\theta}-1\}, we obtain the following:
\begin{align} \int_{\Omega} (v_{1}+1)^{p}+\int_{\Omega}|\nabla v_{2}|^{2p}\leq C, \end{align} | (3.25) |
where C > 0.
Proof. We can combine Lemma 3.3 with Lemma 3.4 to infer the following:
\begin{align} &\frac{d}{dt} \big(\frac{1}{p}\int_{\Omega}(v_{1}+1)^{p}+\frac{1}{2p} \int_{\Omega}|\nabla v_{2}|^{2p}\big)+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p}+\frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p} \\ &\leq C_{1}\int_{\Omega}v_{1}^{\theta\alpha(p+1)}+ (C_{1}+\lambda_{1}+\frac{1}{p})\int_{\Omega}(v_{1}+1)^{p} -\lambda_{2}\int_{\Omega}v_{1}^{p+\beta-1}+C_{1}, \end{align} | (3.26) |
where C_{1} > 0. Due to 0 < \alpha\leq\frac{1}{\theta} and \beta\geq2, we can obtain \theta\alpha(p+1)\leq p+1\leq p+\beta-1. Using Young's inequality, we can obtain the following:
\begin{align} C_{1}\int_{\Omega}v_{1}^{\theta\alpha(p+1)} \leq \frac{\lambda_{2}}{2}\int_{\Omega}v_{1}^{p+\beta-1}+C_{2}, \end{align} | (3.27) |
where C_{2} > 0. By the inequality (w+s)^{\kappa}\leq 2^{\kappa}(w^{\kappa}+s^{\kappa}) with w, s > 0 and \kappa > 1, we deduce the following:
\begin{align} (C_{1}+\lambda_{1}+\frac{1}{p})\int_{\Omega}(v_{1}+1)^{p} \leq \frac{\lambda_{2}}{2}\int_{\Omega}v_{1}^{p+\beta-1}+C_{3}, \end{align} | (3.28) |
where C_{3} > 0, where we have applied Young's inequality. Thus, we obtain the following:
\begin{align} \frac{d}{dt} \big(\frac{1}{p}\int_{\Omega}(v_{1}+1)^{p}+\frac{1}{2p} \int_{\Omega}|\nabla v_{2}|^{2p}\big)+\frac{1}{p}\int_{\Omega} (v_{1}+1)^{p}+\frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p} \leq C_{4}, \end{align} | (3.29) |
where C_{4} > 0. Therefore, we can obtain (3.25) by Lemma 2.5. Thereupon, we complete the proof.
The proof of Theorem 1.1. Recalling Lemma 3.5, for any p > \max\{1, \frac{1}{\theta}-1\}, and applying the L^{p}- estimates of elliptic equation, there exists C_{1} > 0 such that
\begin{align} \sup\limits_{t\in(0,T_{\max})}\|w(\cdot,t)\|_{W^{2,\frac{p}{\alpha}}{(\Omega)}} \leq C_{1} \ \ \mbox{for all}\ t\in (0,T_{\max}). \end{align} | (3.30) |
The Sobolev imbedding theorem enables us to obtain the following:
\begin{align} \sup\limits_{t\in(0,T_{\max})}\|w(\cdot,t)\|_{W^{1,\infty}{(\Omega)}}\leq C_{2} \ \ \mbox{for all}\ t\in (0,T_{\max}), \end{align} | (3.31) |
with C_{2} > 0. Besides, using the well-known heat semigroup theory to the second equation in system (1.6), we can find C_{3} > 0 such that
\begin{align} \|v_{2}(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C_{3} \ \ \mbox{for all}\ t\in (0,T_{\max}). \end{align} | (3.32) |
Therefore, using the Moser-iteration[17], we can find C_{4} > 0 such that
\begin{align} \|v_{1}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C_{4} \ \ \mbox{for all}\ t\in (0,T_{\max}). \end{align} | (3.33) |
Based on (3.31)–(3.33), we can find C_{5} > 0 that fulfills the following:
\begin{equation} \|v_{1}(\cdot,t)\|_{L^{\infty}{(\Omega)}} +\|v_{2}(\cdot,t)\|_{W^{1,\infty}{(\Omega)}} +\|w(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C_{5}, \end{equation} | (3.34) |
for all t\in (0, T_{\max}). According to Lemma 2.1, we obtain T_{\max} = \infty. Thereupon, we complete the proof of Theorem 1.1.
In this manuscript, based on the model established in [35], we further considered that self-diffusion and cross-diffusion are nonlinear functions, as well as the mechanism of nonlinear generation and consumption of the indirect signal substance w. We mainly studied the effects of diffusion functions, the logical source, and the nonlinear consumption mechanism on the boundedness of solutions, which enriches the existing results of chemotaxis consumption systems. Compared with previous results [29,32], the novelty of this manuscript is that our boundedness conditions are more generalized and do not depend on spatial dimension or the sizes of \|v_{20}\|_{L^{\infty}(\Omega)} established in [32], which may be more in line with the real biological environment. In addition, we will further explore interesting problems related to system (1.6) in our future work, such as the qualitative analysis of system (1.6), the global classical solvability for full parabolic of system (1.6), and so on.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was partially supported by the National Natural Science Foundation of China (No. 12271466, 11871415).
The authors declare there is no conflict of interest.
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