Abstract
We study herein the initial boundary value problem for a two-species chemotaxis competition system with loop
under the homogeneous Neumann boundary condition, where \(\Omega \subset {\mathbb {R}}^{n} (n\ge 3)\) is a smooth and bounded domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\) \((i, j=1, 2)\). In the radial symmetric setting, for any \(T>0\) and \(L>0\), it is proved that there exists positive initial data such that the corresponding solution \((u_1, u_2, v_1, v_2)\) satisfies
Moreover, when \(\chi _{11}=\chi _{12}, \chi _{21}=\chi _{22}\), \(\mu =\max \{\mu _1, \mu _2\}\in (0, 1)\), one can find initial data \( (u_{10}, u_{20}, v_{10}, v_{20})\in \left( C^0({\overline{\Omega }})\right) ^2\times \left( W^{1, \infty }(\Omega )\right) ^2\), which is irrelevant to \(\mu \), such that for all \(\mu \in (0, 1)\), the corresponding solution \((u_{1, \mu }, u_{2, \mu }, v_{1, \mu }, v_{2, \mu })\) fulfills
In particular, it is proved that blowup for one of the chemotactic species implies also blowup for the other one at the same time.
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1 Introduction
Chemotaxis, the property such that species move toward or away from the higher concentration of the chemical substance, has attracted many mathematicians’ interests in recent years. So far, there is a considerable amount of research on the classical Keller–Segel system [10]
with \(\tau \in \{0, 1\}\). For the case \(f(u)=0, k=1\), there is no possibility that the solution of (1.1) blows up when \(n = 1\); when \(n = 2\), the system of (1.1) exhibits a remarkable feature: critical mass [6, 7, 23, 24]. As for the case \(n\ge 3\), one can refer to [42, 44] for the blowup results. For the case \(f(u)=ru-\mu u^2, k=1\), when \(n=2\), it was proved that the solution of (1.1) with \(\tau =1\) is globally bounded for any \(r\in {\mathbb {R}}\) and \(\mu >0\) [27]; in the smooth and bounded convex domain \(\Omega \subset {\mathbb {R}}^n (n\ge 3)\), Lankeit obtained the existence of global weak solutions to (1.1) with \(\tau =1\) for arbitrary \(\mu >0\) [12], especially, in the three-dimensional setting, they further proved that these solutions become classical solutions after some time; what is more, under the condition that \(\mu \) is sufficiently large, the large time behavior of solutions to (1.1) with \(\tau =1\) was considered in the convex domain [45]. For the case \(f(u)\le a-bu^2, k=1\), in the convex domain \(\Omega \subset {\mathbb {R}}^n (n\ge 3)\), the global solvability for \(\tau =1\) was obtained when b is sufficiently large [43]; when it is supposed that either \(n\le 2\) or \(b>\frac{n-2}{n}\chi _{11}\), Tello and Winkler [32] showed that the system (1.1) with \(\tau =0\) admits globally bounded classical solutions; afterward, their result was extended to the borderline case \(b=\frac{n-2}{n}\chi _{11}\) by Xiang [50]. Besides these, one can refer to [5, 8, 9] for more global boundedness results of (1.1) when \(f(u)\ne 0\). Recently, some new progress on the blowup phenomenon has been made, and Winkler [47] investigated the blowup result of (1.1) for low-dimensional spatial settings; in addition, replacing the second equation by \(0=\Delta v-{\tilde{\mu }}(t)+g(u)\) (\({\tilde{\mu }}(t)=\frac{1}{|\Omega |}\int _{\Omega }g(u(\cdot , t))dx\)), they also obtained a finite-time blowup result without considering the logistic source [48].
After the pioneering works above, the following two-species and one-stimuli chemotaxis model [49]
where \(\tau \in (0, 1)\), which is a variant of the classical Keller–Segel chemotaxis system, has been studied intensively. For the case \(\tau =1\), when \(h(u_1, u_2, v)\) is some function under the explicit conditions, the works related to global dynamics can be found in [18, 22, 25, 26, 39]. When \(h(u_1, u_2, v)=-\gamma v+\alpha _{1}u_{1}+\alpha _{2}u_{2}\), Bai and Winkler [1] proved the global boundedness of solutions for (1.2) when \(n\le 2\), moreover, they gave the asymptotic stabilization of globally bounded solutions when \(n\ge 1\); then the global boundedness result was further extended to the case \(n=3\) in [13]; recently, the conditions for asymptotic stabilization assumed in [1] were improved by Mizukami [19, 21]. Without the kinetic terms, the finite-time blowup of (1.2) in higher dimension was considered in [14]. When considering the strong logistic diffusion, Tu et al. [37] investigated the finite time blowup of solutions. Very recently, considering the competitive kinetics in (1.2), Li [15] established the large densities and simultaneous blowup result for the chemotactic species when \(n\ge 3\). As for the parabolic–parabolic–elliptic case, the global solvability and stabilization of (1.2) was shown in [2, 20, 30, 33].
To better understand the chemotactic interaction when there are two chemicals, the two-species and two-stimuli chemotaxis system is considered,
for the case \(\tau _1=\tau _2=0\), the boundedness versus blowup was systematically investigated in [31, 52] when \(\mu _1=\mu _2=0\); when \(\mu _1, \mu _2>0\), the global dynamics of solutions were established in [34, 41, 54, 55]. For the case \(\tau _1=\tau _2=1\), in the absence of the Lotka–Volterra-type competition, Xie and Wang [51] investigated the global boundedness of solutions under the condition that both \(\Vert u_{10}\Vert _{L^1(\Omega )}\) and \(\Vert u_{20}\Vert _{L^1(\Omega )}\) are appropriately small for \(n=2\); when \(\mu _1, \mu _2>0\), the uniform boundedness and asymptotic behavior of solutions for \(n\le 2\) were detected by Black [3]; thereafter, this result was improved to the 3D case [28, 40].
In the present study, we consider the following two-species and two-stimuli system with loop,
which stems from [11] by Knútsdóttir et al., as a mathematical biological model to describe the metastatic process for the breast cancer. Here, \(\Omega \subset {\mathbb {R}}^n (n\ge 3)\) is a bounded domain with smooth boundary \(\partial \Omega \), \(\frac{\partial }{\partial \nu }\) represents differentiation with respect to the outward normal on \(\partial \Omega \), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\), \(\alpha _{ij}>0\), \(\lambda _{i}>0\) \((i, j=1, 2)\). \(u_1, u_2\) denote the densities of macrophages and tumor cells, and the concentration of the chemical signals \(v_i (i=1, 2)\) is secreted by \(u_1\) and \(u_2\). Let the initial data \(u_{10}, u_{20}, v_{10}\) and \(v_{20}\) satisfy
For the parabolic–elliptic case, the global dynamics of (1.4) was established for sufficiently large \(\mu _i (i=1, 2)\) [35], and regardless of the Lotka–Volterra-type competition, the simultaneous blowup, global boundedness, gradient estimates of solutions in \({\mathbb {R}}^2\) were investigated in [4, 16, 17, 53]. Nevertheless, the studies on the fully parabolic system of (1.4) are only devoted to the global solvability, and it was shown that the solution exists globally without any restriction on \(\mu _i (i=1, 2)\) in the 2D setting [36]; and the validity of global boundedness was guaranteed for the 3D setting under the condition that \(\mu _i (i=1, 2)\) are sufficiently large [38]; apart from the global boundedness, the large time behavior of solution was also given in both studies.
In this paper, we investigate a phenomenon which is weaker than blowup but still significant, that is, the solution exhibits unbounded peculiarity for the proper choice of initial data. Apart from that, we also show the simultaneous blowup for both chemotactic species once the blowup happens. Our work is enlightened by the one-specie one-stimuli and two-specie one-stimuli case in [14, 15, 46], we shall construct an appropriate functional and use the contradictory argument to address this issue. Compared with the previous works, our difficulty consists in two aspects: (1) the disposal of the functional \(\mathrm {F}(u_1, u_2, v_1, v_2)\) denoted in the following (1.6), the functional in our paper contains more terms than the previous one, this leads to the emergence of some extra cross-terms (such as \(u_i|\nabla v_j|^2, \nabla u_i\cdot \nabla v_j, \frac{|\nabla u_i|^2}{u_i}\), \(i, j=1, 2\)) in the later calculations, while the previous functional given in [15, 46] can be directly handled by completing the square, it is not so trivial to handle these additional terms; (2) the complicated interaction among the loop, multi-stimuli and competitive kinetics, which makes the computation and analysis more delicate.
To this end, for \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\), we define the following functional
with \(\chi _1=\min \{\chi _{11}, \chi _{12}\}\), \(\chi _2=\min \{\chi _{21}, \chi _{22}\}\), which plays an important role in our later proof.
Now we state our main results as follows.
Theorem 1.1
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\) \((i, j=1, 2)\). Then for all \({\mathcal {M}}>0, {\mathcal {N}}>0\) and each \(L>0, T>0\), one can find some positive constant \({\mathcal {P}}= {\mathcal {P}}( {\mathcal {M}}, {\mathcal {N}}, L, T, R, n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, \mu _1, \mu _2, a_1, a_2)\), whenever \((u_{10}, u_{20}, v_{10}, v_{20})\) is from the set
there exists some \((x_{L}, t_{L})\in \Omega \times (0, T)\) which makes the corresponding classical solution \((u_1, u_2, v_1, v_2)\) of (1.4) fulfill
Corollary 1.1
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{11}=\chi _{12}>0, \chi _{21}=\chi _{22}>0\), \(a_i>0\), \(\mu _{i}\in (0, 1)\) \((i=1, 2)\), \(\mu =\max \{\mu _1, \mu _2\}\). Then for all \({\mathcal {M}}>0, {\mathcal {N}}>0\) and each \(L>0, T>0\), there exists positive constant \(K= K( {\mathcal {M}}, {\mathcal {N}}, L, T, R, n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, a_1, a_2)\), provided that the initial data is from the set \({\mathcal {A}}_2\) denoted by
for all \(\mu \in (0, 1) \), one can find \( (x_{\mu }, t_{\mu })\in \Omega \times (0, T)\) such that the corresponding classical solution \((u_{1, \mu }, u_{2, \mu }, v_{1, \mu }, v_{2, \mu })\) of (1.4) satisfies
Remark 1.1
The above sets \({\mathcal {A}}_i({\mathcal {M}}, {\mathcal {N}}, {\mathcal {L}}) \ne \emptyset (i=1, 2)\), which is a straightforward result from Lemma 6.1 in [44].
Remark 1.2
From Corollary 1.1, we notice that the lower bound of \( u_{1, \mu }(x_{\mu }, t_{\mu }) \) or \( u_{2, \mu }(x_{\mu }, t_{\mu })\) can be sufficiently large as long as we control the value of \(\mu \). In addition, for the case \( \chi _{11}=\chi _{12}, \chi _{21}=\chi _{22}\), problem (1.4) can be reduced to the two-species and one-stimuli case, so our Corollary 1.1 extends the main result in [15].
The following consequence underlines that once the blowup happens, it is simultaneous for the two species.
Theorem 1.2
Let \(\Omega \subset {\mathbb {R}}^n(n\ge 3)\) be a smooth, bounded, and convex domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\) \((i, j=1, 2)\), and let \(u_{10}, u_{20}, v_{10}\) and \(v_{20}\) satisfy (1.5). If the corresponding solution \((u_1, u_2, v_1, v_2)\) of (1.4) satisfies
Then
This paper is organized as follows. In Sect. 2, we derive the local existence and uniqueness result, as well as some important inequalities. In Sect. 3, we focus on deriving a differential inequality related to the function \(\mathrm {F}(u_1, u_2, v_1, v_2)\) denoted by (1.6), and in light of the contradictory arguments, we accomplish the proof of Theorem 1.1 and Corollary 1.1. Section 4 is devoted to investigating the simultaneous blowup for both species.
2 Local existence and basic estimates
Our goal in this section is to present some useful inequalities and the local well-posedness results for (1.4). First of all, we recall the following lemma, which is a direct result of Lemma 2.1 in [36].
Lemma 2.1
Let \(\Omega \subset {\mathbb {R}}^n(n\ge 1)\) be a smooth and bounded domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)), and assume that \((u_{10},u_{20}, v_{10}, v_{20})\) satisfies (1.5). Then there exists a maximal \(T_{\max }\in (0,\infty ]\) such that the system (1.4) has a uniquely determined nonnegative solution \((u_1,u_2,v_1,v_2)\)
which solves (1.4) classically and satisfies
Moreover, the classical solution of (1.4) fulfills
and
as well as
and
with \(\tau :=\min \{1, \frac{T_{\mathrm{max}}}{2}\}\).
Next, our attention is turned to deriving the estimate for \(v_1, v_2\), which plays a significant role in our later proof.
Lemma 2.2
Suppose that the assumptions in Lemma 2.1 are satisfied. Then for any classical solution \((u_1, u_2, v_1, v_2)\) of (1.4), one can find \(M_1, M_2, C_1, {\hat{C}}_1>0\) such that
and
as well as
Proof
Using the fact that \(\int _{\Omega }u_i\mathrm{d}x (i=1, 2)\) are bounded, integrating the third and fourth equations of (1.4) over \(\Omega \) we find
by a direct computation we can obtain (2.5). Testing the third equation of (1.4) by \(-\Delta v_1\), and in light of the Young inequality, it is clear that
then utilizing (2.3) and (2.4), it follows from Lemma 3.4 in [29] that there exists a positive constant \(C_1=C_1(m_1, m_2, \int _{\Omega }|\nabla v_{10}(x, t)|^2\mathrm{d}x)\) such that
In quite a similar manner, we can derive (2.7). \(\square \)
With the aid of Lemmas 2.1 and 2.2, we now establish the pointwise upper bounds for radial solutions \(v_1, v_2\) in the following lemma. The proof of the following lemma is similar to [14, 44], and hence we omit it here for brevity.
Lemma 2.3
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(p\in (1, \frac{n}{n-1})\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)). Then for any radially symmetric initial data \((u_{10},u_{20}, v_{10}, v_{20})\) satisfies (1.5), there exists \(C(p)>0\) such that the corresponding classical solution of (1.4) satisfies
and
for all \((r, t)\in (0, R)\times (0, T_{\mathrm{max}})\).
Remark 2.1
From Lemmas 2.1–2.3, we find that any radially symmetric classical solutions \((u_1, u_2, v_1, v_2)\) from Lemma 2.1 has the property
and
with \(m=\max \{m_1, m_2\}\), \(M=\max \{M_1, M_2\}\),
where \(p\in (1, \frac{n}{n-1})\), this implies \(k>n-2\).
Moreover, we introduce the space S(M, B, k), which is defined by
This thus implies that any radially symmetric classical solutions from Lemma 2.1 have the property \((u_1, u_2, v_1, v_2)\in S(M, B, k)\) for all \(t\in (0, T_{\mathrm{max}})\).
3 Emergence of large population densities
Motivated by the contradictory strategy in [14, 15, 44], for \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}(\overline{\Omega }\times (0,T_{\max }))\right) ^4\), we introduce the energy function \(\mathrm {F}(u_1, u_2, v_1, v_2)\),
with \(\chi _1=\min \{\chi _{11}, \chi _{12}\}\), \(\chi _2=\min \{\chi _{21}, \chi _{22}\}\). Our aim in this section is to establish the following differential inequality
under the hypothesis that \(u_1, u_2\) are bounded, this would lead to an absurd conclusion, then we can obtain Theorem 1.1 and Corollary 1.1. To this end, for \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\), we introduce the following functions,
where \({\hat{\chi }}_1= \frac{\chi _{11}+\chi _{12}}{2}\), \({\hat{\chi }}_2=\frac{\chi _{21}+\chi _{22}}{2}\).
The following differential inequality of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) is essential for our proof.
Lemma 3.1
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)). Suppose that \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\) is a classical solution of (1.4), and \((u_1, u_2, v_1, v_2)\in S(M, B, k)\). Then for all \(t\in (0, T_{\mathrm{max}})\), one obtains
where \(\mathrm {F}\) and \(\mathrm {D}\) are defined in (3.1) and (3.2), \(\chi _1=\min \{\chi _{11}, \chi _{12}\}\), \(\chi _2=\min \{\chi _{21}, \chi _{22}\}\).
Proof
Testing the third and fourth equations of (1.4) by \(v_{1t}, v_{2t}\), respectively, to obtain
and
Together with (3.7) and (3.8), we find
For the last two terms of (3.9), utilizing the first and second equations of (1.4), it follows that
for \(t\in (0, T_{\mathrm{max}})\). For \(\chi _1=\min \{\chi _{11}, \chi _{12}\}\), \(\chi _2=\min \{\chi _{21}, \chi _{22}\}\), a direct calculation reveals that
for \(t\in (0, T_{\mathrm{max}})\). Similarly, we can deduce that
for \(t\in (0, T_{\mathrm{max}})\). Thus, a combination of (3.9)–(3.12) yields (3.6). \(\square \)
In the next four lemmas, we focus on dealing with the term \(\int _{\Omega }(u_1+u_2)(v_1+v_2)\mathrm{d}x\) in \(\mathrm {F}(u_1, u_2, v_1, v_2)\), our aim is to seek the constants \(\vartheta \in (\frac{1}{2}, 1)\) and \(C>0\) such that
where D is introduced in (3.2), which acts a pivotal part in the latter proof.
Lemma 3.2
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\). Suppose that \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\) is a classical solution of (1.4), and \((u_1, u_2, v_1, v_2)\in S(M, B, k)\). Let \(f_i\) be given by (3.3) (\(i,j=1,2\)), then for any \(\epsilon \in (0, 1)\), there exist \({\mathcal {C}}_1(M), \hat{{\mathcal {C}}}_1(M)>0\) such that
and
Proof
From (3.3), for \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\), we have
testing the above equation by \(v_1\), we find
it follows from the Gagliardo–Nirenberg inequality that
with \(\frac{1}{2}=a(\frac{1}{2}-\frac{1}{n})+\frac{1-a}{1}\), i.e., \(a=\frac{n}{n+2}\). In light of (2.10), one can find \(c_1(M)>0\) such that
thereupon, invoking the Young inequality, we find
with \(c_2(M)>0\), which combined with (3.17) result in
In view of (3.17) and the Young inequality, for any \(\epsilon >0\), we see that
with \(c_3(M), c_4(M)>0\). Therefore, a combination of (3.15), (3.18) and (3.19) yields (3.13). In a similar way, we can derive (3.14). \(\square \)
The following lemma is an extension of Lemma 4.3 in [44].
Lemma 3.3
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\). Suppose that \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\) is a classical solution of (1.4), and \((u_1, u_2, v_1, v_2)\in S(M, B, k)\). Let \(f_i\) be as in (3.3) (\(i,j=1,2\)), then for any \(\epsilon \in (0, 1)\) and \(r_0\in (0, R)\), there exists \(C=C(\epsilon , m_1, m_2, M, B, k)\) such that
and
for all \(t\in (0, T_{\mathrm{max}})\).
Proof
Fixing any \(\xi \in (0, 1)\), testing \(f_1\) by \(v_1^{\xi }\), it is easy to obtain that
for all \(t\in (0, T_{\mathrm{max}})\). In light of (2.11), one obtains
Based on the above two inequalities, we arrive at
for all \(t\in (0, T_\mathrm{max})\). To achieve our goal, we need to handle the right hand side of (3.24), for \(\frac{B^{1-\xi }}{\xi }\cdot r_0^{-(1-\xi )k} \int _{\Omega }(u_1+u_2)v_1^{\xi }\mathrm{d}x\), it follows from the Young inequality that for any \(\epsilon >0\), one can find \(C(\epsilon , B)\) satisfies
then the upper bound of \(\int _{\Omega }u_1\mathrm{d}x, \int _{\Omega }u_2\mathrm{d}x\) guarantees that
As for \(\frac{B^{1-\xi }}{\xi }\cdot r_0^{-(1-\xi )k}\int _{\Omega }f_1v_1^{\xi }\mathrm{d}x\), utilizing the estimates (3.17) and (3.25), it is clear that
for all \(t\in (0, T_{\mathrm{max}})\), and applying the Young inequality, we get
Thus, plugging (3.26)–(3.28) into (3.24), it is inferred that
for all \(t\in (0, T_{\mathrm{max}})\). And we notice that
this implies (3.20) holds. In view of a similar proof, one can attain (3.21). \(\square \)
Inspired by the idea in Lemma 4.4 of [44], we generalize the above lemma to the whole spherical region.
Lemma 3.4
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\), \(i, j=1, 2\), \({\hat{\chi }}_1= \frac{\chi _{11}+\chi _{12}}{2}\), \({\hat{\chi }}_2=\frac{\chi _{21}+\chi _{22}}{2}\). Suppose that \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\) is a classical solution of (1.4), and \((u_1, u_2, v_1, v_2)\in S(M, B, k)\). Let \(f_i\) and \(g_i (i=1,2)\) be as in (3.3)–(3.5), then one can find \(C=C(m_1, m_2, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\) such that
holds for any \(r_0\in (0, R)\) and \(t\in (0, T_{\mathrm{max}})\).
Proof
From the definition of \(f_1, f_2\), we have \(f_1:=-\Delta v_1+v_1-u_1-u_2, f_2:=-\Delta v_2+v_2-u_1-u_2\), which implies that
and
Applying (3.4) and (3.5), there appears the relation
Multiplying (3.31), (3.32) with \(r^{n-1}v_{1r}\) and \(r^{n-1}v_{2r}\) respectively, and putting them together, one can select \(\delta \in (0, \frac{2n-2}{R}]\) such that
where we have used the Young inequality and (3.33). Let
along with (3.34), it is clear that
for all \(r\in (0, R)\). Since \(y(0)=0\), then integrating (3.35) over (0, r), one achieves
For \(I_1\), in view of integration by parts, we find
for all \(r\in (0, R)\). In a similar way, we have
For \(I_2\), using the fact that
where the abbreviation \(w_n:=n |B_1(0)|\), then we can derive
for all \(r\in (0, R)\), where we have used the fact that \(\Vert u_1\Vert _{L^1(\Omega )}\le m_1\). Analogously, one can deduce
And a simple computation leads to
Once more, applying a integration by parts, utilizing the fact that \(\delta \in (0, \frac{2n-2}{R}]\), we can see that
for all \(r\in (0, R)\), and
From (3.36)–(3.44), it is now clear that there exists \(c_5=c_5(m_1, m_2, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\) such that
Multiplying (3.45) with \(\frac{1}{r^{n-1}}\) and integrating it over \((0, r_0)\), it follows that
for any \(r_0\in (0, R)\), here, we can use Fubini’s theorem to handle the first term on the right side of (3.46),
which yields (3.30). \(\square \)
Thereupon, utilizing the above three lemmas, we deduce the following relation between \(\int _{\Omega }(u_1+u_2)(v_1+v_2)dx\) and \(\mathrm {D}(u_1, u_2, v_1, v_2)\).
Lemma 3.5
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)). Suppose that \((u_1, u_2, v_1, v_2)\in \left( C^{2,1}( \overline{\Omega }\times (0,T_{\max }))\right) ^4\) is a classical solution of (1.4), and \((u_1, u_2, v_1, v_2)\in S(M, B, k)\). Then for \(\mathrm {D}(u_1, u_2, v_1, v_2)\) given by (3.2), one can find \(C=C( m_1, m_2, M, B, k, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)\) and \(\vartheta = \frac{1}{1+\frac{n}{(2n+4)k}} \in (\frac{1}{2}, 1)\) such that
Proof
In view of Lemmas 3.3 and 3.4, for all \(t\in (0, T_{\mathrm{max}})\), there exist \({\mathfrak {C}}_1={\mathfrak {C}}_1(\epsilon , m_1, m_2, M, B, k)\) and \({\mathfrak {C}}_2={\mathfrak {C}}_2(m_1, m_2, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)\) satisfying
and
putting together the above two estimates we arrive at
where
and it follows from (3.18) that
with \({\mathfrak {C}}_3={\mathfrak {C}}_3(m_1, m_2, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\). For \(\phi \), it is easy to obtain that
where \({\mathfrak {C}}_4={\mathfrak {C}}_4(\epsilon , m_1, m_2, M, B, k), {\mathfrak {C}}_5={\mathfrak {C}}_5(m_1, m_2, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\). To further estimate \(\phi \), we select \(r_0=\min \{ \frac{R}{2}, (\Vert f_1\Vert _{L^2(\Omega )}+\Vert f_2\Vert _{L^2(\Omega )})^{-\frac{2}{\varrho +1}} \}\), and we divide the following discussion into two cases.
For the case \(\frac{R}{2}\le (\Vert f_1\Vert _{L^2(\Omega )}+\Vert f_2\Vert _{L^2(\Omega )})^{-\frac{2}{\varrho +1}}\), we have
which implies
For the case \(\frac{R}{2}> (\Vert f_1\Vert _{L^2(\Omega )}+\Vert f_2\Vert _{L^2(\Omega )})^{-\frac{2}{\varrho +1}}\), it means that \(r_0=(\Vert f_1\Vert _{L^2(\Omega )}+\Vert f_2\Vert _{L^2(\Omega )})^{-\frac{2}{\varrho +1}}\), then
for all \(t\in (0, T_{\mathrm{max}})\), therefore, in light of (2.12) and \(p\in (1, \frac{n}{n-1})\), it is inferred that
and
which guarantees that
this makes it possible to use Young’s inequality in (3.56), that is,
with \({\mathfrak {C}}_6={\mathfrak {C}}_6(\epsilon , m_1, m_2, M, B, k, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2), {\mathfrak {C}}_7={\mathfrak {C}}_7(\epsilon , m_1, m_2, M, B, k, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\). Let \(\vartheta =\frac{\varrho }{\varrho +1}=\frac{1}{1+\frac{n}{(2n+4)k}}\in (\frac{1}{2}, 1)\), by adding up (3.51), (3.53), (3.56) and (3.58), for any \(\epsilon >0\), one finds \({\mathfrak {C}}_8={\mathfrak {C}}_8(\epsilon , m_1, m_2, M, B, k, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\) such that
It follows from Lemma 3.2 and \(2\vartheta >\frac{2n+4}{n+4}\) that
for all \(t\in (0, T_{\mathrm{max}})\), with \({\mathfrak {C}}_9(M)>0\). Selecting \(\epsilon \in (0, \frac{1}{4})\), inserting (3.60) into (3.59) we attain
for all \(t\in (0, T_{\mathrm{max}})\), with \({\mathfrak {C}}_{10} ={\mathfrak {C}}_{10}( m_1, m_2, M, B, k, R, n, {\hat{\chi }}_1, {\hat{\chi }}_2)>0\), which directly yields (3.48). \(\square \)
With (3.48) at hand, now our eyes are turned to constructing the differential inequality for \(\mathrm {F}(u_1, u_2, v_1, v_2)\).
Lemma 3.6
Let \(\Omega =B_R(0)\subset {\mathbb {R}}^n(n\ge 3, R>0)\), \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)), \(L>0\), T>0. And let \((u_1, u_2, v_1, v_2)\) be a classical solution of (1.4), \((u_1, u_2, v_1, v_2)\in S(M, B, k)\) with the property
Then there exist \(T^{*}\in (0, T]\) and positive constants \({\overline{C}}={\overline{C}}( m_1, m_2, M, L, T, B, k, R, n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22},\) \( \mu _1, \mu _2, a_1, a_2)\) as well as \(C:=C( m_1, m_2, M, L, B, k, R, n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, \mu _1, \mu _2, a_1, a_2)\) whenever
the following inequality holds true for all \(t\in (0, T^{*})\),
where \(\mathrm {F}(u_1, u_2, v_1, v_2)\), \(\mathrm {D}(u_1, u_2, v_1, v_2)\), \(\vartheta \) are defined in (3.1), (3.2) and Lemma 3.5 respectively. Furthermore, \(-\mathrm {F}(u_1, u_2, v_1, v_2)\) is nondecreasing for all \(t\in (0, T^{*})\).
Proof
Step 1 With the boundedness restriction of \(u_1, u_2\), we claim that
for all \(t\in (0, T)\), where \({\mathcal {C}}_1=\mu _1+\mu _2\), \({\mathcal {C}}_0={\mathcal {C}}_0( \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, \mu _1, \mu _2, m_1, m_2, M, L, a_1, a_2 )\). Indeed, recalling Lemma 3.1, we have
In light of (2.6), (2.7), (2.10) and (3.61), it follows that
When \(\chi _{11}=\chi _{12}\), we have \(\left( \frac{\chi _{11}}{\chi _1}-1\right) \int _{\Omega }\nabla u_1\nabla v_1\mathrm{d}x + \left( \frac{\chi _{12}}{\chi _1}-1\right) \int _{\Omega }\nabla u_1\nabla v_2\mathrm{d}x -\left( \frac{1}{\chi _1}-\frac{2}{\chi _{11}+\chi _{12}}\right) \int _{\Omega }\frac{|\nabla u_1|^2}{u_1}\mathrm{d}x=0\); when \(\chi _{11}\ne \chi _{12}\), the Young inequality ensures that there exists \(0<\eta \le \min \left\{ \frac{1}{\chi _1}-\frac{2}{\chi _{11}+\chi _{12}}, \frac{1}{\chi _2}-\frac{2}{\chi _{21}+\chi _{22}}\right\} \) such that
A similar calculation reveals that
And utilizing the \(L^1\) bound for \(u_i, v_i (i=1, 2)\), there appears the relation
for all \(t\in (0, T)\). Furthermore, the definition of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) in (3.1) implies
where we have used the inequality \(\rho \ln \rho \ge -\frac{1}{e}\) for all \(\rho >0\). As for the rest terms in (3.65), the inequality \(\rho ^2\ln \rho \ge -\frac{1}{2e}\) directly yields
It then follows from (3.65)–(3.71) that (3.64) holds for all \(t\in (0, T)\).
Step 2 Now our attention is turned to showing that there exists \(T^*\in (0, T]\) such that
It follows from the definition of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) at (3.1) that
hence, by Lemma 3.5 and the Young inequality, one obtains
where \(\mathcal {\tilde{ C}}_1=\mathcal {\tilde{ C}}_1( m_1, m_2, M, L, B, k, R, n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, \mu _1, \mu _2, a_1, a_2)\), \({\mathcal {C}}_2={\mathcal {C}}_2( m_1, m_2, M, L, B, k, R, \) \(n, \chi _{11}, \chi _{12}, \chi _{21}, \chi _{22}, \mu _1, \mu _2, a_1, a_2))\) are positive constants.
For any \({{\overline{C}}}>0\), it follows from Lemma 6.1 in [44] that one can always find radially symmetric initial data such that \(\mathrm {F}(u_{10}, u_{20}, v_{10}, v_{20})< -{{\overline{C}}}\). We fix
when \(\mathrm {F}(u_{10}, u_{20}, v_{10}, v_{20})< -{{\overline{C}}}\), the continuous property of \(\mathrm {F}\) guarantees that the set
then we can define
From the definition of \(T^*\) and (3.74), we have \(-\mathrm {F}(u_1, u_2, v_1, v_2)\ge {{\overline{C}}}> \frac{{\mathcal {C}}_0+{\mathcal {C}}_2}{{\mathcal {C}}_1}\) and \( -\mathrm {F}(u_1, u_2, v_1, v_2)\ge {{\overline{C}}}> \frac{2|\Omega |}{e}\left( \frac{1}{\chi _1}+\frac{1}{\chi _2}\right) +\frac{2\mathcal {\tilde{C}}_1}{{\mathcal {C}}_0} \) for all \(t\in (0, T^*)\), which warrants that
and
along with (3.73), one obtains
thereupon, by virtue of (3.64) and (3.73), fixing \({\overline{\epsilon }} \in (0, \frac{1}{4})\), it is clear that
inserting (3.78) into (3.79), we arrive at
for all \( t\in (0, T^*).\) In light of (3.76), (3.77) and (3.80), we can estimate
Thus, we obtain (3.63), and the nondecreasing property of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) is also proved. \(\square \)
In what follows, we shall deduce a contradictory result, which verifies Theorem 1.1.
Proof of Theorem 1.1
To obtain that for any \(0<L<\infty \), \(T>0\), there exists some \(({\hat{x}}, {\hat{t}})\in \Omega \times (0, T)\) such that the corresponding solution \(( u_1, u_2, v_1, v_2)\) of (1.4) satisfies \( u_1({\hat{x}}, {\hat{t}})>L\) or \( u_2({\hat{x}}, {\hat{t}})> L\). We assume, on the contrary,
Let \(\Psi (t):=-\mathrm {F}( u_1(\cdot , t), u_2(\cdot , t), v_1(\cdot , t), v_2(\cdot , t))\), from Lemma 3.6, we notice that \(\Psi (t)\) is nondecreasing, that is \(\Psi (t)\ge \Psi (0)>{{\overline{C}}}\) for all \(t\in (0, T^*)\), where
the continuity and the nondecreasing property of \(\mathrm {F}( u_1, u_2, v_1, v_2)\) actually warrant that \(T^*=T\). Accordingly, from Lemma 3.6, we know that \(\Psi \) satisfies
with \(\delta :=\frac{1}{\vartheta }>1\) and \(C=\frac{3}{4}\left( \frac{{\mathcal {C}}_1}{2\mathcal {\tilde{ C}}_1} \right) ^{\frac{1}{\vartheta }}>0\). For the above ordinary differential equation, it is easy to check that
therefore,
utilizing (3.83), we find
then we achieve a contradiction. Hence, we obtain Theorem 1.1. \(\square \)
Now we extend the method in [46] to prove Corollary 1.1.
Proof of Corollary 1.1
Given any \(L>0\) and \(T>0\), we assume that the claimed property in Corollary 1.1 is false, then there exists \(\mu =\max \{\mu _1, \mu _2\}\in (0, 1)\) such that the corresponding solution \( (u_{1}, u_{2}, v_{1}, v_{2}):= (u_{1, \mu }, u_{2, \mu }, v_{1, \mu }, v_{2, \mu })\) satisfies
where \(\mu _1\in (0, 1), \mu _2\in (0, 1)\).
Recalling Lemma 3.1, in light of the condition \(\chi _{11}=\chi _{12}, \chi _{21}=\chi _{22}\), we have
where \(\chi _1=\chi _{11}=\chi _{12}, \chi _2=\chi _{21}=\chi _{22}\). Utilizing (2.10), (3.88) and the condition \(\mu =\max \{\mu _1, \mu _2\}\in (0, 1)\), a straightforward computation shows that
for all \(t\in (0, T)\). Making use of the definition of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) in (3.1) and the inequality \(\rho \ln \rho \ge -\frac{1}{e}\) for all \(\rho >0\), we can estimate
And it follows from the inequalities \(\rho ^2\ln \rho \ge -\frac{1}{2e}\) and \(\rho \ln \rho \ge -\frac{1}{e}\) that
Therefore, plugging the above (3.90), (3.91) and (3.92) into (3.89), one obtains
with
The definition of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) in (3.1) guarantees that
it is inferred from (3.94) that
and from Lemma 3.5, there exist \(C=C( m_1, m_2, M, B, k, R, n, \chi _1, \chi _2)>0\) and \(\vartheta \in (\frac{1}{2}, 1)\) such that
then together with (3.95) and (3.96), applying the Young inequality, we can derive
where \({\check{C}}_2:={\check{C}}_2( m_1, m_2, M, B, k, R, n, \chi _1, \chi _2, a_1, a_2, L ), {\check{C}}_3:={\check{C}}_3( m_1, m_2, M, B, k, R, n, \chi _1, \chi _2, a_1, a_2, L )\) are positive constants.
Choosing
from Lemma 6.1 in [44], one can always find radially symmetric initial data such that \(\mathrm {F}(u_{10}, u_{20}, v_{10}, v_{20})\) \(< -C^*\). When \(\mathrm {F}(u_{10}, u_{20}, v_{10}, v_{20})< -C^*\), the continuous property of \(\mathrm {F}(u_1, u_2, v_1, v_2)\) guarantees that the set
is not empty, then we can denote
Hence,
for all \(t\in (0, {\mathcal {T}}^*)\), which results in
and
Thanks to (3.97), we can derive
and
Fixing \(\varepsilon \in (0, \frac{1}{2})\), using the definition of \({\check{C}}_1\), and by virtue of (3.93) as well as (3.103), one obtains
In light of (3.102) and (3.104), we can see that
for all \(t\in (0, {\mathcal {T}}^*)\). Therefore, a combination of (3.100), (3.101) and (3.105) yields
The monotonicity and the continuous property of \(\mathrm {F}\) guarantee that \({\mathcal {T}}^*=T\). Let \(Y(t)=-\mathrm {F}\), then
where \({\check{C}}=\frac{{\check{C}}_2}{{\check{C}}_1}\), \(\vartheta \in (0, 1)\), a direct calculation warrants that
where we have used \(Y(0)=\mathrm {F}(u_{10}, u_{20}, v_{10},v_{20})>\left[ \left( \frac{2{\check{C}}_2}{{\check{C}}_1}\right) ^{\frac{1}{\vartheta }}\frac{4\vartheta }{(1-\vartheta )T}\right] ^{\frac{\vartheta }{1-\vartheta }}=\left[ \left( 2{\check{C}}\right) ^{\frac{1}{\vartheta }}\frac{4\vartheta }{(1-\vartheta )T} \right] ^{\frac{\vartheta }{1-\vartheta }}\) and \(0<\vartheta <1\), thus,
providing a contradiction. This completes the proof of Corollary 1.1. \(\square \)
4 Simultaneous blowup
In this section, our attention is turned to investigating whether the blowup of two species occurs at the same time when the blowup happens. We define the blowup time by \(T_{\mathrm{max}}\), to obtain Theorem 1.2, it suffices to prove that the boundedness of \(u_1\) holds when \(u_2\) is bounded, which contradicts with the hypothesis that \(\Vert u_1\Vert _{L^{\infty }(\Omega )}+\Vert u_2\Vert _{L^{\infty }(\Omega )} \rightarrow \infty \) as \(t\rightarrow T_{\mathrm{max}}\). To begin with, we give the following lemma which is essential for our latter proof.
Lemma 4.1
Let \(\Omega \subset {\mathbb {R}}^n(n\ge 3)\) be a smooth and bounded domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)). And let \((u_1, u_2, v_1, v_2)\) be a classical solution of (1.4), \(q>\max \{ 1, \frac{n}{2}-1 \}\), \(p>\max \{ 1, \frac{2q(n-2)}{2q+n-2} \}\) with the property
Then for any \(\zeta >0\), \(t\in (0, T_{\mathrm{max}})\), one can find \(\tilde{C}:=\tilde{C}(\zeta , p, q)>0\) such that
and
Proof
This lemma is a straightforward result from Lemmas 4.4 and 4.5 in [15], for completeness, we provide the detailed proof. Due to the fact that the estimates for (4.2) and (4.3) are similar, so we just consider the case (4.2). It follows from \(n\ge 3, p>1, q>1\) that
and from \(q>\frac{n}{2}-1\) we see that
Thanks to (4.4), one can apply the H\(\ddot{o}\)lder inequality to estimate
then utilizing (4.5), the Gagliardo–Nirenberg inequality and the boundedness of \(\int _{\Omega }udx\), there exist \(\breve{C}_1, \breve{C}_2>0\) such that
with
applying the Sobolev embedding, the Gagliardo–Nirenberg inequality, (2.6) and (2.7), one obtains
where \(\breve{C}_3, \breve{C}_3^{'}, \breve{C}_4, \breve{C}_5>0\), \(\lambda =\frac{\frac{q}{2}-\frac{1}{2}}{\frac{q}{2}-\frac{1}{2}+\frac{1}{n}}=\frac{n(q-1)}{n(q-1)+2}\in (0, 1)\). Therefore, combing (4.6), (4.7) with (4.9), and using the Young inequality, for any \(\zeta >0\), we can find \(\breve{C}_6, \breve{C}_7>0\) such that
In light of the first inequality in (4.1), it is clear that
hence, using the Young inequality once again, we arrive at
where \(\breve{C}_8>0\). As for (4.3), it can be derived by a similar method. \(\square \)
In the following lemma, we utilize Lemma 4.1 to derive the \(L^p\) estimate for \(u_1\) under the condition that \(\Vert u_2\Vert _{L^{\infty }(\Omega )}\) is bounded.
Lemma 4.2
Let \(\Omega \subset {\mathbb {R}}^n(n\ge 3)\) be a smooth, bounded and convex domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)), \(q>\max \{ 1, \frac{n}{2}-1 \}\) and \(p>\max \{ 1, \frac{2q(n-2)}{2q+n-2} \}\) satisfy (4.1). \((u_1, u_2, v_1, v_2)\) is a classical solution of (1.4). If there exists \(L_1>0\) such that
Then
where \(L_2>0\) depends on \(L_1\).
Proof
Testing the first equation of (1.4) by \(u_1^{p-1}\), it follows that
For \(\chi _{11}p(p-1)\int _{\Omega }u_1^{p-1}\nabla u_1\nabla v_1\mathrm{d}x\), in light of the Young inequality, a simple calculation reveals that
for all \(t\in (0, T_{max})\). Similarly, we can deduce that
for all \(t\in (0, T_{\mathrm{max}})\). And by virtue of the Young inequality, one can find \(\tilde{C}_1>0\) such that
then it follows from (4.14)–(4.17) that
and applying (4.2), we can find \(\tilde{C}_2>0\) such that
To estimate the terms \(\int _{\Omega }|\nabla |\nabla v_i|^q|^2\mathrm{d}x (i=1,2)\) in (4.19), one needs to cope with the following energy-type inequality concerning \(\int _{\Omega }|\nabla v_i|^{2q}\mathrm{d}x(i=1, 2)\). We notice that these two terms can be estimated by the similar method, so we just consider the case \(\int _{\Omega }|\nabla |\nabla v_1|^q|^2\mathrm{d}x\).
By a direct calculation, we find
To deal with \(J_1\), we notice that the domain \(\Omega \subset {\mathbb {R}}^n(n\ge 3)\) is convex, which implies \(\frac{\partial }{\partial \nu }|\nabla v_i|^2\le 0 (i=1, 2)\), then applying the equality \(2\nabla v_i\cdot \nabla \Delta v_i=\Delta |\nabla v_i|^2-2|D^2v_i|^2 (i=1, 2)\), one can infer that
For \(J_2\), employing Young’s inequality and the inequality \(|\Delta v_i|^2\le n|D^2v_i|^2 (i=1, 2)\), we arrive at
Similarly, we have
Putting the above four inequalities together, and using the identity \(|\nabla v_1|^{2q-4}|\nabla |\nabla v_1|^2|^2=\frac{4}{q^2}|\nabla |\nabla v_1|^q|^2\), we find
In the same vein, for \(|\nabla v_2|^{2q}\), one obtains
Let \(C^*=\max \left\{ \chi _{11}^2p(p-1), \chi _{12}^2p(p-1), n+4(q-1) \right\} \), combining (4.19), (4.24) with (4.25), and utilizing (4.3), we can see that
applying \(\Vert u_2\Vert _{L^{\infty }(\Omega )}<L_1\) and the Young inequality, we can find \(\tilde{C}_4>0, \tilde{C}_5>0\) such that
and
Hence, choosing \(0<\zeta \le \min \left\{ \frac{p-1}{2pC^*}, \frac{q-1}{q^2C^*} \right\} \), the above three inequalities warrant that
thereupon, with the aid of an application of ODE comparison argument, we can obtain (4.13) immediately. \(\square \)
With the above preparations, now we extend the method in [1, 15] to prove the boundedness of \(\Vert u_1\Vert _{L^{\infty }(\Omega )}\).
Lemma 4.3
Let \(\Omega \subset {\mathbb {R}}^n(n\ge 3)\) be a smooth and bounded domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_{i}>0\) (\(i,j=1,2\)), \(p>\frac{n}{2}\). Assume that \((u_1, u_2, v_1, v_2)\) is a classical solution of (1.4) with the property
Then \(T_{\mathrm{max}}=\infty \) and
Proof
We define
and let \(m:=\frac{1}{|\Omega |}\int _{\Omega }u_{10}\mathrm{d}x\), \(t_0=(t-1)_+\). For \(u_1\), we find
In light of the \(L^p-L^q\) estimates for the heat semigroup (see Lemma 1.3 in [42]), for all \(t\in (0, T_{\mathrm{max}})\), it is clear that
with \({\mathcal {K}}_1, {\mathcal {K}}_2, {\mathcal {K}}_1^{'}, {\mathcal {K}}_2^{'}>0\). To bound \(I_2\), we need introduce some new parameters. Since \(p>\frac{n}{2}\), we have \(n<\frac{np}{n-p}\), which guarantees that there exists \(\sigma >1\) and \(r>n\) such that
where \(r>n\) ensures \(\frac{1}{2}+\frac{n}{2r}<1\). And let \(\iota =1-\frac{\sigma -1}{r\sigma }\), it is obvious that \(0<\iota <1\). Then applying the \(L^p-L^q\) estimates of the heat semigroup and the H\(\ddot{o}\)lder inequality, we conclude
for all \(t\in (0, T_{\mathrm{max}})\), where \({\tilde{\lambda }}\) denotes the first nonzero eigenvalue of \(-\Delta \) in \(\Omega \), and \({\mathcal {K}}_3>0\). For \(\Vert \nabla v_1\Vert _{L^{r\sigma }(\Omega )}\), again, making use of the \(L^p-L^q\) estimates, we can see that
where we have used \(\frac{1}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{r\sigma })<1\), which is guaranteed by (4.33). Therefore, (4.34) and (4.35) indicate that
with \({\mathcal {K}}_6>0\), similarly, for \({\hat{I}}_3\), we have
with \({\mathcal {K}}_6^{'}>0\). As for \({\hat{I}}_4\), we note that \(u_1(1-u_1-a_1u_2)\le \frac{1}{4}\), which warrants the boundedness of \({\hat{I}}_4\). Thus, for \(0<\iota <1\), it follows from (4.31)–(4.37) that
with \({\mathcal {K}}_7>0\), this implies that \(T_{max}=\infty \) and (4.29) holds. \(\square \)
Proof of Theorem 1.2
Theorem 1.2 is a direct result of Lemma 4.2 and Lemma 4.3.
\(\square \)
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Acknowledgements
We would like to thank the anonymous reviewers for their valuable suggestions and fruitful comments which lead to significant improvement of this work. This work is funded by Chongqing Post-doctoral Innovative Talent Support program, China Postdoctoral Science Foundation under Grant 2020M673102, the Fundamental Research Funds for the Central Universities under Grants XDJK2020C054, 2020CDJQY-Z001 and 2019CDJCYJ001, the NSFC under Grants 11771062, 11971393 and 11971082, the Natural Science Foundation of Chongqing, China under Grant cstc2020jcyj-bshX0071.
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Tu, X., Tang, CL. & Qiu, S. The phenomenon of large population densities in a chemotaxis competition system with loop. J. Evol. Equ. 21, 1717–1754 (2021). https://doi.org/10.1007/s00028-020-00650-6
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DOI: https://doi.org/10.1007/s00028-020-00650-6