Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions

Since the pioneering work of Keller and Segel in the early 1970s [10, 11], the mathematical investigation of chemotaxis – the directed movement of organisms in response to chemical gradients – has significantly enhanced the understanding of various biological mechanisms. Of particular interest is the chemotactic behavior of aerobic bacteria that consume oxygen as a nutrient, a subject that has attracted considerable attention due to its complex dynamics observed in both experimental and simulation settings [6, 20]. Accordingly, the following equations have been derived and subjected to rigorous analysis as a fundamental model of this nutrient-based chemotaxis interaction [15]:

where $u$ denotes the density of the organisms and $v$ stands for the concentration of nutrients, such as oxygen.

When $D$, $S$, and $f$ are positive and appropriately chosen, it is well-known that the Neumann problem for (1) possesses an energy dissipation structure, arising from the interplay between the consumption term $-uf(v)$ and the chemotaxis term $-\nabla\cdot(uS(u,v)\nabla v)$, of the form in its simplest prototype case where $D(u)=1$, $S(u,v)=1$, and $f(v)=v$:

The energy dissipation structure (2) has played a crucial role in demonstrating global-in-time boundedness, stability, and asymptotic behaviors of solutions as well as global solvability (c.f. [15]).

Specifically, beyond the small initial data case where the global existence and asymptotic behaviors converging to a constant steady state have been proved using weighted $L^{p}$-estimates [16, 31], global large-data classical and weak solutions have been achieved via exploiting (2) in the two and three dimensions, respectively [25] (see [9, 18] for eventual smoothness and asymptotic behaviors of such weak solutions in three dimensions).

Moreover, variants of the dissipation structure (2), capable of allowing singular behaviors of $S(v)=\frac{1}{v^{\alpha}}$, have been employed to see the global solvability and asymptotic behaviors of solutions. Indeed, the global existence of the classical solution has been proved for general $f$ and $S$ involving $f(v)=v$ and $S(v)=\frac{1}{v^{\alpha}}$ with $0<\alpha<-\frac{1}{2}+\frac{1}{2}\left(\frac{8\sqrt{2}+11}{7}\right)^{\frac{1}{2}}\cong 0.3927$ for $n=2$ [2]. Later, this result has been improved to $0<\alpha<-\frac{1}{2}+\frac{1}{2}\sqrt{\frac{n+8}{n}}$ for $n=2,3$ [12]. For $0<\alpha<1$ and $n=2$, weak solutions have been constructed when the initial data is sufficiently small [21]. In the case of $\alpha=1$, another dissipation structure that provides weak regularity than (2) has been employed to prove global solvability of generalized solutions for $n=2$ [26] (c.f. [14]), and of renormalized solutions for $n\geq 2$ [27].

Furthermore, similar quantitative results have been considered when $D$ is of porous medium type, $u^{m-1}$, or its nondegenerate form, $(u+1)^{m-1}$ with $m>1$, by using this enhancing diffusivity or variants of (2). Exploiting the diffusivity of $D$, in presence of general $S\geq 0$ and $f>0$, the existence of global bounded solutions has been shown for $m>1$ and $n=2$ in a weak sense [19], and for $m>2-\frac{2}{n}$ and $n\geq 2$ in the classical sense [23]. In the case of $S=1$, the global solvability has been shown for $m>2-\frac{6}{n+4}$ and $n\geq 3$ using a variant of (2) [22]. Later, this result has been improved by [7], where the global well-posedness and asymptotic behaviors are established for $m>\frac{3}{2}-\frac{1}{n}$ and $n\geq 3$ [7]. In the case of $S(v)=\frac{1}{v}$, the global solvability has been proven for $m>1+\frac{4}{n}$ and $n\geq 2$ [13], and for $m>\frac{3}{2}-\frac{1}{n}$ and $n\geq 2$ in a generalized concept [29]. The latter has been extended to the tensor-valued $S(u,v,x)$ satisfying $|S(u,v,x)|\leq\frac{1}{v^{\alpha}}$ with $\alpha\in[0,1)$ for $m>\frac{3}{2}-\frac{1}{n}$ [28].

For parabolic-elliptic simplification of (1) with $D=1$, $S=1$, and $f(v)=v$, the global well-posedness and asymptotic behavior have been shown under Robin boundary condition by Fuest-Lankeit-Mizukami [8] when $n\geq 1$ (see also [1]). We refer to [30] for its Dirichlet counterpart.

Remarkably, the parabolic-elliptic counterpart of (1) may exhibit blow-up behavior when $S$ is of repulsion type, $S<0$. Indeed, under No-flux/Dirichlet boundary conditions, finite-time blow-up solutions have been found for $D(u)\leq(u+1)^{m-1}$, $0<m<1$, and $S(v)=-\frac{1}{v}$ by Wang-Winkler [24] when $n\geq 2$. This result has also been verified for $S=-1$ and $n=2$ [3]. Additionally, in the latter case, the global existence of bounded solutions has been established when $D$ has a positive lower bound. However, to the best of our knowledge, there are no results concerning the existence or blow-up of solutions for $S=-1$ and $n\geq 3$.  

Our main objective in this paper is to either establish the global existence of bounded solutions or construct unbounded solutions for the repulsion-type chemotaxis-consumption system in dimensions three and higher. To be more precise, we consider the following initial-boundary value problem

In our analysis, the spatial domain $\Omega$ is specifically chosen as $B_{R}$, a ball of radius $R>0$ centered at the origin in $\mathbb{R}^{n}$. The diffusion coefficient $D$ is chosen to extend the prototype choice as follows:

Throughout this paper, the initial function $u_{0}$ is assumed to satisfy

Before presenting the main results, we begin by outlining the local existence and blow-up criteria for (3). Although these topics have been widely explored in the literature for more generalized frameworks [5], this paper specifically addresses our particular case.

The first objective of this study is to investigate conditions on the diffusion rate that ensure the existence of globally bounded regular solutions. The following theorem demonstrates that, within this radially symmetric setting, the existence of a global-in-time bounded solution is assured for certain values of $m$, specifically $m>1$ or $m=1$ with a constraint on $M$.

A subsequent natural question is under what conditions the boundedness or unboundedness of the solution is determined for the case $0<m<1$. It might be expected to experts that small data implies the existence of global bounded solutions. However, we can not find it in the literature and therefore, for clarity, we first demonstrate that bounded solutions can be constructed globally in time when the Dirichlet data $M$ is sufficiently small.

Finally, we establish that when $M$ is sufficiently large, blow-up of the solution may occur, in particular, in a more restricted range $m\in(0,\frac{2}{n})$.

Plan of the paper Theorem 1 extends the result in [3] to higher dimensions. Whereas the authors in [3] exploited the uniform $L^{2}$ smallness of $\nabla v$ near the origin to ensure uniform $L^{1}$ boundedness of $u\ln u$, thereby achieving global boundedness of $u$ for $n=2$, our proof begins with a pointwise estimate of $\nabla v$ near the origin (Lemma 2). This estimate is derived from the uniform boundedness of $v$ and the assumption of radial symmetry. Employing this approach, we can control the mass accumulation function $U$ near the origin through a comparison method for the equation of $U$ (Lemma 4). This control leads to an improved regularity of $\nabla v$ (Lemma 5) near the origin, which, when combined with estimates away from the origin (Lemma 1), ensures global boundedness of $u$ through $L^{p}$-estimates in higher dimensions (Lemma 6).

For Theorem 2, we first exploit the smallness assumption on $M$, depending on $u_{0}$, to obtain the behavior of $U$ via a comparison argument similar to that used in Lemma 4 (Lemma 7). This behavior of $U$, in conjunction with a Hardy-type inequality (Lemma 8) and a one-dimensional variant of the Gagliardo-Nirenberg inequality, allows us to derive $L^{p}$-estimates for $u$ (Lemma 9).

In Theorem 3, the method follows the approach employed in [3] (see also [24]). This method involves tracking the time evolution of the quantity $\phi(t):=\int_{0}^{R}s^{-\alpha}U(s,t)ds$ with appropriately chosen $\alpha$. Obtaining a suitable lower bound for $\nabla v$ is crucial in this process. The authors in [3] provided this lower bound by appropriately estimating $\nabla\ln v$ and $v$, respectively. However, while the lower bound for $\nabla\ln v$ can be efficiently estimated even in higher dimensions using a variant of the ODE comparison method (Lemma 10, and see [24, Lemma 3.1] for the proof), demonstrating the polynomial decay of $v$, which was straightforward in dimensions two [3], seems difficult to demonstrate for dimensions three and higher. As a result, it is unlikely to obtain $\phi^{\prime}(t)\gtrsim\phi(t)$ as in [3]. To overcome the difficulty, instead, we make full use of the second equation of (3) and the estimate of $v$ at the spatial origin (Lemma 11) to obtain an appropriate ordinary differential inequality (ODI) for $\phi(t)$ (Lemma 12), which is a technical novelty compare to [3]. It turns out that such ODI satisfies $\phi^{\prime}(t)\gtrsim 1$, namely, $\phi(t)$ has a linear growth in time, which implies blow-up in finite or infinite time. On the other hand, in three dimensions, since it is shown that $\phi(t)$ is uniformly bounded, blow-up must occur in a finite time.

For clarity throughout this paper, we abbreviate $\|\cdot\|_{L^{p}(\Omega)}$ as $\|\cdot\|_{p}$ when no confusion arises. Constants are denoted by $C_{i}$ with $i=1,2,...$, where the subscript serves to distinguish these constants within each chapter. Additionally, $C(x,y,...)$ indicates a dependency of the constant $C$ on variables $x,y$, and others. Finally, we denote by $\sigma_{n}$ the measure of the $n-1$ dimensional unit sphere $\partial B_{1}$.

Hereafter, we fix $n\geq 3$, $\Omega=B_{R}\subset\mathbb{R}^{n}$ with $R>0$, and assume that $D$ satisfies (5). we denote by $(u,v)$ the solution in $\Omega\times(0,T_{max})$ as in Proposition 1, and denote

Utilizing the radial symmetry, the equations in (3) can be rewritten as the following scalar equations with a slight abuse of notation $(u,v)(x,t)=(u,v)(|x|,t)$, where $|x|=s\in(0,R)$ and $t\in(0,T_{max})$:

Hereafter, we fix $n\geq 3$, $\Omega=B_{R}\subset\mathbb{R}^{n}$ with $R>0$, and assume that $D$ satisfies (5).