Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

Abstract

The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

1 Introduction

The purpose of the current work is to study an activator-inhibitor system, introduced by Gierer and Meinhardt (1972) to describe the phenomenon of morphogenesis in hydra, on an isotropically evolving domain. In particular, a singular Gierer–Meinhardt system on a stationary domain $\Omega \subset \mathbb {R}^N, N\ge 1$ with smooth boundary is given by (Gierer and Meinhardt 1972),

$$\begin{aligned} u_t= & {} D_1 \Delta u-u+\displaystyle \frac{u^p}{v^q}, \qquad x\in \Omega ,\quad t\in (0,T), \end{aligned}$$

(1.1)

$$\begin{aligned} \tau v_t= & {} D_2 \Delta v-v+\displaystyle \frac{u^r}{v^s}, \qquad \; x\in \Omega ,\quad t\in (0,T), \end{aligned}$$

(1.2)

$$\begin{aligned} \frac{\partial u}{\partial \nu }= & {} \frac{\partial v}{\partial \nu }=0 \qquad \; x\in \partial \Omega _t,\quad t\in (0,T), \end{aligned}$$

(1.3)

$$\begin{aligned} u(x,0)= & {} u_0(x)>0,\quad v(x,0)=v_0(x)>0, \quad x\in \Omega _0\subset \mathbb {R}^N, \end{aligned}$$

(1.4)

where u(x, t) stands for the concentration of the activator, at a spatial point $x\in \Omega$ at time $t\in [0,T], T>0,$ which enhances its own production and that of the inhibitor whose concentration is denoted by v(x, t) according to (1.1). On the other hand, the presence of the inhibitor suppresses its own production as well as that of the activator as it is described by (1.2). Also $\nu$ denotes the unit normal vector on $\partial \Omega .$

Here, $D_1$ and $D_2$ are the diffusion coefficients of the activator and inhibitor, respectively; $\tau$ represents the response of the inhibitor to the activator’s growth. Moreover, the exponents satisfying the conditions: $p>1,\; q$, $r,>0,\;\text{ and }\; s>-1,$ measure the interactions between morphogens. The dynamics of system (1.1)–(1.4) is controlled by two values: the net self-activation index $\psi =( p-1)/r$ and the net cross-inhibition index $\gamma =q/(s+1).$ Index $\xi$ correlates the strength of self-activation of the activator with the cross-activation of the inhibitor. Thus, if $\xi$ is large, then the net growth of the activator is large no matter the growth of the inhibitor. The parameter $\gamma$ measures how strongly the inhibitor suppresses the production of the activator and that of itself. If $\gamma$ is large, then the production of the activator is strongly suppressed by the inhibitor. Finally, the parameter $\tau$ quantifies the inhibitor’s response against the activator’s growth, cf. Gierer and Meinhardt (1972). Guided by biological interpretation as well as by mathematical reasons, we assume that the parameters p, q, r, s satisfy the condition

$$\begin{aligned} p-r\gamma <1, \end{aligned}$$

(1.5)

which in the literature is known as the Turing condition. Indeed, as it is pointed in the seminal paper (Gierer and Meinhardt 1972), condition (1.5) guarantees the occurrence of patterns, induced by diffusion, for the solutions of system (1.1)–(1.4), see also Ni et al. (2006) and Ni (2011).

Apart from its biological importance, system (1.1)–(1.4) has a very rich mathematical structure including emerging singularities and thus its dynamics has been extensively study the last few years. More precisely, a thorough study of the structure of its stationary solutions is given in Ni et al. (2006), whilst some global-in-time existence results were proven in Jiang (2006), Li et al. (1995), Masuda and Takahashi (1987) and Rothe (1984) among others. The author in Jiang (2006) proved that under the condition $\psi =\frac{p-1}{r}<1$, a global-in-time solution exists, which is an almost optimal result, also taking into consideration the results in Ni et al. (2006). Moreover, in Karali et al. (2013) one can find an investigation of the asymptotic behaviour of the solution of (1.1)–(1.4). The occurrence of finite-time blow-up, which actually implies unlimited growth for the activator, was first established in Li et al. (1995) and later in Karch et al. (2016), Li et al. (2017) and Zou (2015), whereas the case of non-diffusing activator finite-time blow-up was investigated in Karch et al. (2016). The existence and stability of spiky stationary solutions was thoroughly studied in the survey paper (Wei 2008).

Now, in the case that the domain of the interaction of activator and inhibitor, denoted by $\Omega _t,$ is evolving in time, then the dynamics of this interaction can be described by the following reaction–diffusion system

$$\begin{aligned}&u_t+\nabla \cdot (\overrightarrow{\alpha } u)= D_1 \Delta u-u+\displaystyle \frac{u^p}{v^q}, \qquad x\in \Omega _t,\quad t\in (0,T), \end{aligned}$$

(1.6)

$$\begin{aligned}&\tau v_t+\nabla \cdot (\overrightarrow{\alpha } v) = D_2 \Delta v-v+\displaystyle \frac{u^r}{v^s}, \qquad \; x\in \Omega _t,\quad t\in (0,T), \end{aligned}$$

(1.7)

$$\begin{aligned}&\frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0 \qquad \; x\in \partial \Omega _t,\quad t\in (0,T), \end{aligned}$$

(1.8)

$$\begin{aligned}&u(x,0)=u_0(x)>0,\quad v(x,0)=v_0(x)>0, \quad x\in \Omega _0\subset \mathbb {R}^N, \end{aligned}$$

(1.9)

where $\overrightarrow{\alpha }\in \mathbb {R}^N$ stands for the convection velocity, induced by the material deformation due to the evolution of the domain and $\Omega _0\subset \mathbb {R}^N$ is the initial domain profile which has smooth boundary $\partial \Omega _0$. The initial datum $u_0, v_0$ are considered bounded, i.e.

$$\begin{aligned} u_0,\;v_0 \in L^{\infty }(\Omega _0). \end{aligned}$$

(1.10)

In the current work, we will only consider the case of an isotropic flow on an evolving domain, whilst the anisotropic case will be investigated in a forthcoming paper. Thus, for any $x\in \Omega _t$ we have:

$$\begin{aligned} x=\rho (t) \xi ,\quad \text{ for }\quad \xi \in \Omega _0\subset \mathbb {R}^N, \end{aligned}$$

(1.11)

where $\Omega _0$ is an open and bounded $C^1-$ domain of $\mathbb {R}^N.$ Uniform isotropic growth is a plausible biological assumption whereby the domain is assumed to expand uniformly at the same rate in all directions at all times. Examples illustrating isotropically evolving biological surfaces include the famous Nature paper by Kondo and Asai (1995) that depicted mode doubling in pigmentation patterns of the angelfish Pomacanthus as it grows from juvenile to adulthood.

To proceed, we take $\rho (t)$ to be a $C^1-$function with $\rho (0)=1.$ In the case of a growing domain, we have ${\dot{\rho }}(t)=\frac{\mathrm{d} \rho }{\mathrm{d}t}>0,$ whilst when the domain shrinks or for domain contraction ${\dot{\rho }}(t)=\frac{\mathrm{d} \rho }{\mathrm{d}t}<0.$ Furthermore, the following equality holds

$$\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}t}=\overrightarrow{\alpha }(x,t). \end{aligned}$$

(1.12)

Setting ${\hat{u}}(\xi ,t)=u(\rho (t)\xi ,t),\; {\hat{v}}(\xi ,t)=v(\rho (t)\xi ,t),$ and then using the chain rule as well as (1.11) and (1.12), see also Madzvamuse and Maini (2007), we obtain:

$$\begin{aligned}&{\hat{u}}_t-\overrightarrow{\alpha }\cdot \nabla _x u=u_t,\quad \nabla _x u=\frac{1}{\rho (t)}\nabla _{\xi }{\hat{u}}\\&\quad \Delta _x u= \frac{1}{\rho ^2(t)}\Delta _{\xi }{\hat{u}},\quad \nabla _x \cdot \left( \overrightarrow{\alpha } u\right) =\overrightarrow{\alpha }\cdot \nabla _x u+N\,u \frac{{\dot{\rho }}(t)}{\rho (t)}, \end{aligned}$$

whilst similar relations hold for v as well. Therefore, (1.6)–(1.9) is reduced to the following reaction–diffusion system on a reference stationary domain $\Omega _0$

$$\begin{aligned} {\hat{u}}_t= & {} \frac{D_1}{\rho ^2(t)} \Delta _{\xi } {\hat{u}}-\left( 1+N\frac{{\dot{\rho }}(t)}{\rho (t)}\right) {\hat{u}}+\displaystyle \frac{{\hat{u}}^p}{{\hat{v}}^q}, \qquad \xi \in \Omega _0,\quad t\in (0,T), \end{aligned}$$

(1.13)

$$\begin{aligned} \tau {\hat{v}}_t= & {} \frac{D_2}{\rho ^2(t)} \Delta _{\xi } {\hat{v}}-\left( 1+N\frac{{\dot{\rho }}(t)}{\rho (t)}\right) {\hat{v}}+\displaystyle \frac{{\hat{u}}^r}{{\hat{v}}^s}, \qquad \; \xi \in \Omega _0,\quad t\in (0,T), \end{aligned}$$

(1.14)

$$\begin{aligned} \frac{\partial {\hat{u}}}{\partial \nu }= & {} \frac{\partial {\hat{v}}}{\partial \nu }=0 \qquad \; \xi \in \partial \Omega _0,\quad t\in (0,T), \end{aligned}$$

(1.15)

$$\begin{aligned} {\hat{u}}(\xi ,0)= & {} {\hat{u}}_0(\xi )>0,\quad {\hat{v}}(\xi ,0)={\hat{v}}_0(\xi )>0, \quad \xi \in \Omega _0, \end{aligned}$$

(1.16)

where $\Delta _{\xi }$ represents the Laplacian on the reference static domain $\Omega _0.$ Henceforth, without any loss of generality we will omit the index $\xi$ from the Laplacian.

Defining a new time scale (Labadie (2008)),

$$\begin{aligned} \sigma (t)=\int _0^t\frac{1}{\rho ^2(\theta )}\,\mathrm{d}\theta , \end{aligned}$$

(1.17)

and setting ${\tilde{u}}(\xi ,\sigma )={\hat{u}}(\xi , t), {\tilde{v}}(\xi ,\sigma )={\hat{v}}(\xi , t),$ then system (1.13)–(1.16) can be written as

$$\begin{aligned} {\tilde{u}}_{\sigma }= & {} D_1 \Delta _{\xi } {\tilde{u}}\nonumber \\&-\,\left( \phi ^2(\sigma )+N\frac{{\dot{\phi }}(\sigma )}{\phi (\sigma )}\right) {\tilde{u}}+\phi ^2(\sigma )\displaystyle \frac{{\tilde{u}}^p}{{\tilde{v}}^q}, \quad \xi \in \Omega _0,\; \sigma \in (0,\Sigma ), \end{aligned}$$

(1.18)

$$\begin{aligned} \tau {\tilde{v}}_{\sigma }= & {} D_2 \Delta _{\xi } {\tilde{v}}\nonumber \\&-\,\left( \phi ^2(\sigma )+N\frac{{\dot{\phi }}(\sigma )}{\phi (\sigma )}\right) {\tilde{v}}+\phi ^2(\sigma )\displaystyle \frac{{\tilde{u}}^r}{{\tilde{v}}^s}, \quad \; \xi \in \Omega _0,\; \sigma \in (0,\Sigma ), \end{aligned}$$

(1.19)

$$\begin{aligned} \frac{\partial {\tilde{u}}}{\partial \nu }= & {} \frac{\partial {\tilde{v}}}{\partial \nu }=0, \qquad \; \xi \in \partial \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.20)

$$\begin{aligned} {\tilde{u}}(\xi ,0)= & {} {\hat{u}}_0(\xi )>0,\quad {\tilde{v}}(\xi ,0)={\hat{v}}_0(\xi )>0, \quad \xi \in \Omega _0, \end{aligned}$$

(1.21)

where $\rho (t)=\phi (\sigma ),$ and thus ${\dot{\rho }}(t)=\frac{{\dot{\phi }}(\sigma )}{\phi ^2(\sigma )},$ and $\Sigma =\sigma (T).$

Typically, in cellular biology, molecular species resident in the cytosol are known to diffuse a lot faster than those molecular species resident in the cell membrane [see Cusseddu et al. (2019)] and references therein). Hence, if we assume $D_1\ll D_2,$ where the inhibitor diffuses much faster than the activator, then system (1.18)–(1.21) can be fairly approximated by an ODE-PDE system with a non-local reaction term. We will denote the new approximation by shadow system as coined in Keener (1978). Below, we provide a rather rough derivation of the shadow system, while for a more rigorous approach one can appeal to the arguments in Bobrowski and Kunze (2019). Indeed, dividing (1.19) by $D_2$ and taking $D_2\rightarrow +\infty ,$ see also Ni (2011), then it follows that ${\tilde{v}}$ solves

$$\begin{aligned}&\Delta _{\xi } {\tilde{v}}=0, \quad \; \xi \in \Omega _0,\\&\frac{\partial {\tilde{v}}}{\partial \nu }=0, \quad \; \xi \in \partial \Omega _0, \end{aligned}$$

for any fixed $\sigma \in (0,\Sigma ).$ Due to the imposed Neumann boundary condition then ${\tilde{v}}$ is a spatial homogeneous (independent of $\xi$) solution, and thus averaging (1.19) over $\Omega _0$ we get that ${-}\!\int _{\Omega _0}{\tilde{v}}(\xi ,\sigma )\, d\xi :=\frac{1}{|\Omega _0|}\int _{\Omega _0} {\tilde{v}}(\xi ,\sigma )\, \mathrm{d}\xi :=\eta (\sigma )$. It follows then that $\eta (\sigma )$ satisfies the following partial differential equation

$$\begin{aligned} \tau \frac{\mathrm{d}\eta }{\mathrm{d} \sigma }=-\Phi (\sigma )\eta +\phi ^2(\sigma )\displaystyle \frac{{-}\!\int _{\Omega _0}{\tilde{u}}^r}{\eta ^s}, \quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.22)

where

$$\begin{aligned} \Phi (\sigma )=:\left( \phi ^2(\sigma )+N\frac{{\dot{\phi }}(\sigma )}{\phi (\sigma )}\right) \end{aligned}$$

(1.23)

and ${-}\!\int _{\Omega _0}{\tilde{u}}^r\,d\xi :=\frac{1}{|\Omega _0|}\int _{\Omega _0} {\tilde{u}}^r\,\mathrm{d}\xi .$

Finally, we can infer that the pair $({\tilde{u}}, \eta )$ satisfies the shadow system

$$\begin{aligned}&{\tilde{u}}_{\sigma }= D_1 \Delta _{\xi } {\tilde{u}}-\Phi (\sigma ){\tilde{u}}+\phi ^2(\sigma )\displaystyle \frac{{\tilde{u}}^p}{\eta ^q}, \qquad \xi \in \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.24)

$$\begin{aligned}&\tau \frac{d\eta }{d\sigma }=-\Phi (\sigma )\eta +\phi ^2(\sigma )\displaystyle \frac{{-}\!\int _{\Omega _0}{\tilde{u}}^r\,d\xi }{\eta ^s}, \quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.25)

$$\begin{aligned}&\frac{\partial {\tilde{u}}}{\partial \nu }=0, \qquad \; \xi \in \partial \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.26)

$$\begin{aligned}&{\tilde{u}}(\xi ,0)={\hat{u}}_0(\xi )>0,\quad \eta (0)=\eta _0:={-}\!\int _{\Omega _0}{\tilde{v}}(\xi ,0)\, d\xi >0, \quad \xi \in \Omega _0. \end{aligned}$$

(1.27)

In the limit case $\tau \rightarrow 0,$ i.e. when the inhibitor’s response to the growth of the activator is quite small, then the shadow system is reduced to a single, though, non-local equation. Indeed, when $\tau =0$, (1.25) entails that $\eta (\sigma )=\left( \frac{\phi ^2(\sigma )}{\Phi (\sigma )}{-}\!\int _{\Omega _0}{\tilde{u}}^r\,d\xi \right) ^{\frac{1}{s+1}},$ and thus (1.24)–(1.27) reduce to

$$\begin{aligned}&{\tilde{u}}_{\sigma }= D_1 \Delta _{\xi } {\tilde{u}}-\Phi (\sigma ){\tilde{u}}+\displaystyle \frac{\Psi (\sigma ){\tilde{u}}^p}{\left( {-}\!\int _{\Omega _0}{\tilde{u}}^r\,d\xi \right) ^{\gamma }}, \qquad \xi \in \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.28)

$$\begin{aligned}&\frac{\partial {\tilde{u}}}{\partial \nu }=0, \qquad \; \xi \in \partial \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.29)

$$\begin{aligned}&{\tilde{u}}(\xi ,0)={\hat{u}}_0(\xi )>0,\quad \xi \in \Omega _0, \end{aligned}$$

(1.30)

recalling $\gamma =\frac{q}{s+1}$ and

$$\begin{aligned} \Psi (\sigma )=\phi ^{2(1-\gamma )}(\sigma )\Phi ^{\gamma }(\sigma ). \end{aligned}$$

(1.31)

Recovering the t variable entails that the following partial differential equation holds

$$\begin{aligned}&{\hat{u}}_{t}= \frac{D_1}{\rho ^2(t)} \Delta _{\xi } {\hat{u}}-L(t){\hat{u}} +\,L^{-\gamma }(t)\displaystyle \frac{{\hat{u}}^p}{\left( {-}\!\int _{\Omega _0}{\hat{u}}^r\,\mathrm{d}\xi \right) ^{\gamma }}, \qquad \xi \in \Omega _0,\quad t\in (0,T),\qquad \quad \end{aligned}$$

(1.32)

$$\begin{aligned}&\frac{\partial {\hat{u}}}{\partial \nu }=0, \qquad \xi \in \partial \Omega _0,\quad t\in (0,T), \end{aligned}$$

(1.33)

$$\begin{aligned}&{\hat{u}}(\xi ,0)={\hat{u}}_0(\xi )>0,\quad \xi \in \Omega _0, \end{aligned}$$

(1.34)

where $L(t):=\left( 1+N\frac{{\dot{\rho }}(t)}{\rho (t)}\right) .$ We note that formulation (1.28)–(1.30) is more appropriate for the demonstrated mathematical analysis; however, all of our theoretical results can be directly interpreted in terms of the equivalent formulation (1.32)–(1.34). Besides, formulation (1.32)–(1.34) is more appropriate for our numerical experiments since the calculation of the functions $\Phi (\sigma )$ and $\Psi (\sigma )$ is not always possible.

The primary aim of the current work is to investigate the long-time dynamics of the non-local problem (1.28)–(1.30). Then, it is also examined under which circumstances the dynamics of (1.28)–(1.30) resembles that of the reaction–diffusion system (1.18)–(1.21), which is not always the case, as it has been pointed out in Jiang (2006), Karali et al. (2013), Li and Ni (2009), and Li and Yip (2014). The latter study is performed by using analytical methods, but when these methods fail, then a numerical approach will be implemented. We also use the numerical approach to verify the derived analytical results.

Biologically speaking, we will investigate whether it is necessary to study the dynamics of both reactants or only the study of the activator’s dynamics is sufficient. This is done under the assumption that the inhibitor’s response to the growth of the activator is quite small and that it also diffuses much faster than the activator. From here onwards, we take $D_1=1$, revert to the initial variables x, u instead of $\xi , {\widetilde{u}}$ and we drop the index $\xi$ from the Laplacian $\Delta$ without any loss of generality. Hence, we will focus our study on the following single nonlocal partial differential equation

$$\begin{aligned}&u_{\sigma }= \Delta u-\Phi (\sigma )u+\displaystyle \frac{\Psi (\sigma )u^p}{\left( {-}\!\int _{\Omega _0}u^r\,dx\right) ^{\gamma }}, \qquad x\in \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.35)

$$\begin{aligned}&\frac{\partial u}{\partial \nu }=0, \qquad \; x\in \partial \Omega _0,\quad \sigma \in (0,\Sigma ), \end{aligned}$$

(1.36)

$$\begin{aligned}&u(x,0)=u_0(x)>0,\quad x\in \Omega _0. \end{aligned}$$

(1.37)

Hence, the layout of the current work is as follows. Section 2 deals with the derivation of various blow-up results, induced by the non-local reaction term (ODE blow-up results), together with some global-time existence results for problem (1.35)–(1.37). The notion of finite-time blow-up should be understood biologically as an overcrowding of activator’s population, which mathematically means that $\lim _{\sigma \rightarrow \Sigma _b} ||u(\cdot ,\sigma )||_{\infty }=\infty .$ The impact of domain growth on the finite-time blow-up of a reaction-diffusion equation was first discussed in Labadie (2008); however, the novelty of our approach, both demonstrated in Sects. 2 and 3, consists of the fact that we investigate both growing and shrinking domains but for a non-local reaction-diffusion equation whose dynamics are more complicated. Following the approach developed in Kavallaris and Suzuki (2017, 2018), in Sect. 3 we present and prove a Turing instability result associated with (1.35)–(1.37). This Turing instability occurs under the Turing condition (1.5) and is exhibited in the form of a diffusion-driven blow-up (DDBU). It is not the first time that DDBU phenomenon is observed, see for example Fila and Ninomiya (2005), Hu and Yin (1995), Kavallaris and Suzuki (2017, 2018) and Mizoguchi et al. (1998), nevertheless according to our knowledge it is the first time that such a result is proven for non-local reaction-diffusion equations defined on evolving domains. Finally, in Sect. 4 we appeal to various numerical experiments in order to confirm some of the theoretical results presented in Sects. 2 and 3. More importantly, the numerical approach is also used to compare the long-time dynamics of the non-local problem (1.35)–(1.37) with that of the reaction–diffusion system (1.24)–(1.27). It is also applied to study the dynamics of the shadow system (1.35)–(1.37) when analytical methods fail to do so.

2 ODE Blow-up and Global Existence

The current section is devoted to the presentation of some ODE blow-up results for problem (1.35)–(1.37), i.e. blow-up results induced by the kinetic (non-local) term in (1.35). Here, by blow-up of the solution u of (1.35)–(1.37) we mean the occurrence of a finite time $\Sigma _b$ so that $\lim _{\sigma \rightarrow \Sigma _b} ||u(\cdot , \sigma )||=\infty .$ Next, some global-in-time existence results for the solution u of (1.35)–(1.37) are also presented, that is u exists for any $\sigma >0$ and it is bounded. It should be pointed out that local-in-time existence of non-local problem (1.35)–(1.37) is easily obtained by using ideas in Quittner and Souplet (2007).

Throughout the manuscript, we use the notation C and c to denote positive constants with big and small values, respectively. Our first observation is that the concentration of the activator cannot become zero in finite time. Indeed, the following proposition holds.

Proposition 2.1

Assume that

$$\begin{aligned} \inf _{(0,\Sigma )} \Psi (\sigma ):=m_{\Psi }>0,\quad \inf _{(0,\Sigma )} \Phi (\sigma ):=m_{\Phi }>0\quad \mathrm{and}\quad \sup _{(0,\Sigma )} \Phi (\sigma ):=M_{\Phi }<+\infty ,\nonumber \\ \end{aligned}$$

(2.1)

then for each $\Sigma >0$ there exists $C_{\Sigma }>0$ such that for the solution $u(x,\sigma )$ of (1.35)–(1.37) the following inequality holds

$$\begin{aligned} u(x,\sigma )\ge C_{\Sigma }\quad \mathrm{in}\quad \Omega _0\times [0,\Sigma ). \end{aligned}$$

(2.2)

Proof

Owing to the maximum principle and by using (2.1), we derive that $u=u(x,\sigma )>0.$ By virtue of the comparison principle, we also deduce that $u(x,\sigma )\ge {{\tilde{u}}}(\sigma )$, where ${\tilde{u}}={\tilde{u}}(\sigma )$ is the solution to $\frac{d{\tilde{u}}}{d\sigma }=-M_{\Phi }{\tilde{u}}\, \text{ in } (0, \Sigma ),\quad {\tilde{u}}(0)={\tilde{u}}_0\equiv \inf _{\Omega _0} u_0(x)>0,$ and thus (2.2) is satisfied with $C_{}={\tilde{u}}_0\mathrm{e}^{-M_{\Phi } \Sigma }$. $\square$

Remark 2.1

It is easily checked that condition (2.1) is satisfied for any decreasing function $\phi (\sigma )$ satisfying

$$\begin{aligned} \phi (\sigma )> \frac{1}{\sqrt{2N\sigma +1}},\; 0<\sigma <\Sigma , \end{aligned}$$

(2.3)

since then by virtue of (1.23)

$$\begin{aligned} 0<\Phi (\sigma )=\left( \phi ^2(\sigma )+N\frac{{\dot{\phi }}(\sigma )}{\phi (\sigma )}\right)< \phi ^2(\sigma )< \phi ^2(0)=1,\quad 0<\sigma <\Sigma . \end{aligned}$$

(2.4)

Then, (2.4) via (1.31) implies that

$$\begin{aligned} 0<\Psi (\sigma )=\left( \phi (\sigma )\right) ^{2(1-\gamma )} \Phi ^{\gamma }(\sigma )<1 ,\quad \text{ for }\quad 0<\gamma<1,\quad 0<\sigma <\Sigma \end{aligned}$$

(2.5)

and

$$\begin{aligned} 0<\Psi (\sigma )=\left( \phi (\sigma )\right) ^{2(1-\gamma )} \Phi ^{\gamma }(\sigma )<m^{2(1-\gamma )}_{\Phi } ,\quad \text{ for }\quad \gamma >1,\quad 0<\sigma <\Sigma , \end{aligned}$$

(2.6)

when $m_{\Phi }=\inf _{(0,\Sigma )}\Phi (\sigma )>0.$

A key estimate for obtaining some blow-up results presented throughout is the following proposition.

Proposition 2.2

Let $\Psi (\sigma )$ and $\Phi (\sigma )$ satisfy (2.1), then there exists $\delta _0>0$ such for any $0<\delta \le \delta _0$ the following estimate is fulfilled

$$\begin{aligned} {-}\!\int _{\Omega _0}u^{-\delta }\le C\quad \mathrm{for\, any}\quad 0<\sigma <\Sigma , \end{aligned}$$

(2.7)

where the positive constant C is independent of time $\sigma .$

Proof

Define $\chi =u^{\frac{1}{\alpha }}$ for $\alpha \ne 0$, then we can easily check that $\chi$ satisfies

$$\begin{aligned} \alpha \chi _{\sigma }= & {} \alpha \left( \Delta \chi +4 (\alpha -1) \vert \nabla \chi ^{\frac{1}{2}}\vert ^2\right) \nonumber \\&-\,\Phi \chi +\frac{\Psi u^{p-1+\frac{1}{\alpha }}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }} \quad \text{ in }\quad \Omega _0\times (0,\Sigma ), \end{aligned}$$

(2.8)

$$\begin{aligned} \frac{\partial \chi }{\partial \nu }= & {} 0, \quad \text{ on }\quad \partial \Omega _0\times (0,\Sigma ), \end{aligned}$$

(2.9)

$$\begin{aligned} \chi (x,0)= & {} u_0^{\frac{1}{\alpha }}(x),\quad \text{ in }\quad \Omega _0. \end{aligned}$$

(2.10)

Averaging (2.8) over $\Omega _0$, we obtain

$$\begin{aligned} \alpha \frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +4\alpha (1-\alpha ){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2+\Phi {-}\!\int _{\Omega _0}\chi = \frac{{-}\!\int _{\Omega _0}\Psi u^{p-1+\frac{1}{\alpha }}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}, \end{aligned}$$

(2.11)

and hence

$$\begin{aligned} \frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +4(1-\alpha ){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2+\frac{\Phi }{\alpha }{-}\!\int _{\Omega _0}\chi \le 0, \end{aligned}$$

(2.12)

for $\alpha <0.$ Setting $\delta =-\frac{1}{\alpha }$, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\sigma }{-}\!\int _{\Omega _0}\chi +4(1+\delta ^{-1}){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2 \le M_{\Phi }\delta {-}\!\int _{\Omega _0}\chi . \end{aligned}$$

Now, recall the Sobolev’s inequality, Brezis (2011), that reads

$$\begin{aligned} \Vert \nabla w\Vert _2^2\ge C_1(N,\Omega _0)\Vert w\Vert _2^2,\quad \text{ for } \text{ any }\quad w\in H^1(\Omega _0)\quad \text{ and }\quad N\ge 2, \end{aligned}$$

(2.13)

where $C_1(N,\Omega _0)$ is a positive constant depending only on dimension N and domain $\Omega _0.$

Then, by choosing $0<\delta \ll 1,$ (2.12) in conjunction with (2.13) and for $w=\chi ^{\frac{1}{2}}$ it follows that $\frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +C_2{-}\!\int _{\Omega _0}\chi \le 0$, for some positive constant $C_2.$ Consequently, Gröwnwall’s lemma yields that $\chi (\sigma )\le C<\infty$ for any $0<\sigma <\Sigma$ and thus (2.7) follows due to the fact that $\chi =u^{-\delta }.$ $\square$

Remark 2.2

Note that Proposition 2.2 guarantees that the non-local term of problem (1.35)–(1.37) stays away from zero and hence solution u can never decay to zero. In fact, inequality (2.7) implies $\displaystyle { {-}\!\int _{\Omega _0}u^{\delta }\ge c=C^{-1}}$ and then

$$\begin{aligned} {-}\!\int _{\Omega _0}u^r\ge \left( {-}\!\int _{\Omega _0}u^\delta \right) ^{r/\delta }\ge c^{r/\delta }>0\quad \text{ for } \text{ any }\quad 0<\sigma <\Sigma , \end{aligned}$$

(2.14)

follows by Jensen’s inequality, Evans (2010), and taking $\delta \le r,$ where again c is independent of time $\sigma .$ The latter estimate rules out the possibility of (finite time or infinite time) quenching, i.e. $\lim _{\sigma \rightarrow \Sigma } ||u(\cdot ,\sigma )||_{\infty }=0\quad \text{ for }\quad \Sigma <\infty \quad \text{ or }\quad \Sigma =\infty ,$ cannot happen, and thus activator’s extinction in the long run is not possible.

Remark 2.3

In case $\Phi (\sigma )$ is not bounded from above, as it happens for $\rho (t)=\mathrm{e}^{\beta t},\beta >0,$ when $\Phi (\sigma )=(1+N\beta )(1-2\beta \sigma )^{-1}, 0<\sigma <\frac{1}{2\beta },$ then both of the estimates (2.7) and (2.14) still hold true; however, the involved constants depend on time $\sigma$ and thus (finite or infinite time) quenching cannot be ruled out.

Next, we present our first ODE-type blow-up result for problem (1.35)–(1.37) when an anti-Turing condition, the reverse of (1.5) is satisfied.

Theorem 2.1

Take $p \ge r, 0<\gamma <1$ and $\omega =p-r\gamma >1.$ Assume also $\Psi (\sigma )>0$ and consider initial data $u_0(x)$ such that

$$\begin{aligned} {\bar{u}}_0:={-}\!\int _{\Omega _0}u_0\,dx>(\omega -1)^{\frac{1}{1-\omega }}\,I^{\frac{1}{1-\omega }}(\Sigma )>0, \end{aligned}$$

(2.15)

provided that

$$\begin{aligned} I(\Sigma ):=\int _0^{\Sigma } \Psi (\theta ) \mathrm{e}^{(1-\omega )\int ^{\theta } \Phi (\eta )\,\mathrm{d}\eta }\,\mathrm{d} \theta <\infty , \end{aligned}$$

(2.16)

then the solution of (1.35)–(1.37) blows up in finite time $\Sigma _b<\Sigma$, i.e. $\lim _{\sigma \rightarrow \Sigma _b}\Vert u(\cdot ,\sigma )\Vert _\infty =+\infty .$

Proof

Since $p>1$ and $p\ge r$, then by virtue of the Hölder’s inequality ${-}\!\int _{\Omega _0}u^p\ge \left( {-}\!\int _{\Omega _0}u\right) ^{p}$ and $\left( {-}\!\int _{\Omega _0}u^r\right) ^\gamma \le \left( {-}\!\int _{\Omega _0}u^p \right) ^{\frac{\gamma r}{p}}.$ Then, ${\bar{u}}(\sigma )={-}\!\int _{\Omega _0}u(x,\sigma )\,dx$ satisfies

$$\begin{aligned} \displaystyle \frac{\mathrm{d} {\bar{u}}}{\mathrm{d}\sigma }= -\Phi (\sigma ){\bar{u}}+\Psi (\sigma )\displaystyle {\frac{{-}\!\int _{\Omega _0}u^p}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}}\ge -\Phi (\sigma ){\bar{u}}+\Psi (\sigma ){\bar{u}}^{p-r\gamma }\quad \text{ for }\quad 0<\sigma <\Sigma . \end{aligned}$$

(2.17)

Set now $F(\sigma )$ to be the solution of the following Bernoulli’s type initial value problem $\frac{d F}{d\sigma }=-\Phi (\sigma )F(\sigma )+\Psi (\sigma )F^{\omega }(\sigma ),\; 0<\sigma <\Sigma ,\quad F(0)={\bar{u}}_0>0,$ then via the comparison principle $F(\sigma )\le {\bar{u}}(\sigma )$ for $0<\sigma <\Sigma$ and $F(\sigma )$ is given by $F(\sigma )=\mathrm{e}^{(\omega -1)\int ^{\sigma } \Phi (\eta )\,d\eta }(G(\sigma ))^{\frac{1}{1-\omega }},$ where $G(\sigma ):=\left[ {\bar{u}}_0^{1-\omega }-(\omega -1)\int _0^{\sigma } \Psi (\theta )\mathrm{e}^{(1-\omega )\int ^{\theta } \Phi (\eta )\,\mathrm{d}\eta }\,\mathrm{d} \theta \right] .$ Note that $F(\sigma )$ blows up in finite-time if there exists $\sigma ^*<\Sigma$ such that $G(\sigma ^*)=0.$ First note that $G(0)>0;$ furthermore, under the assumption (2.15) we have $\lim _{\sigma \rightarrow \Sigma }G(\sigma )<0$ and thus by virtue of the intermediate value theorem there exists $\sigma ^*<\Sigma$ such that $G(\sigma ^*)=0.$ The latter implies that $\lim _{\sigma \rightarrow \sigma ^*} F(\sigma )=+\infty$ and therefore, $\lim _{s\rightarrow \Sigma _b} {\bar{u}}(\sigma )=+\infty$ for some $\Sigma _b\le \sigma ^*,$ which completes the proof. $\square$

Remark 2.4

Note that for an exponentially growing domain, i.e. when $\rho (t)=\mathrm{e}^{\beta t}, \beta >0,$ condition (2.16) is satisfied since then $1<\Phi (\sigma )=\left( 1+N\beta \right) (1-2\beta \sigma )^{-1}$ and $1<\Psi (\sigma )=\left( 1+N\beta \right) ^{\gamma }(1-2\beta \sigma )^{-1}$ for all $\sigma \in \left( 0,\frac{1}{2\beta }\right)$. Thus,

$$\begin{aligned} I(\Sigma )=\left( 1+N\beta \right) ^{\gamma }\int _0^{\frac{1}{2 \beta }} \left( 1-2\beta \theta \right) ^{\frac{(\omega -1)(1+N\beta )}{2\beta }-1}\,\mathrm{d} \theta =\frac{\left( 1+N\beta \right) ^{\gamma -1}}{(\omega -1)}<+\infty , \end{aligned}$$

and according to Theorem 2.1 finite-time blow-up takes place at time

$$\begin{aligned} \Sigma _g\le \sigma _{g}:=\frac{1}{2\beta }\left\{ 1-\left[ 1-(1+N\beta )^{1-\gamma }{\bar{u}}_0^{1-\omega }\right] ^{\frac{2 \beta }{(\omega -1)(1+N\beta )}}\right\} , \end{aligned}$$

(2.18)

and for initial data $u_0$ satisfying ${\bar{u}}_0>\left( 1+N\beta \right) ^{\frac{1-\gamma }{\omega -1}}.$ Notably the bigger the exponent $\beta >0$ is, the faster the evolving domain grows, then a rather large initial condition $u_0$ is needed in order to get blow-up according to Theorem 2.1.

Conversely, for an exponentially shrinking domain, i.e. when $\rho (t)=\mathrm{e}^{-\beta t}, 0<\beta <\frac{1}{N},$ then again condition (2.16) is valid since then

$$\begin{aligned} 0<\Phi (\sigma )=\left( 1-N\beta \right) (1+2\beta \sigma )^{-1}<1,\quad \sigma \in (0,\infty ), \end{aligned}$$

(2.19)

and

$$\begin{aligned} 0<\Psi (\sigma )=\left( 1-N\beta \right) ^{\gamma }(1+2\beta \sigma )^{-1}<1,\quad \sigma \in (0,\infty ). \end{aligned}$$

(2.20)

In that case

$$\begin{aligned} I(\Sigma )=\left( 1-N\beta \right) ^{\gamma }\int _0^{+\infty } \left( 1+2\beta \theta \right) ^{\frac{(\omega -1)(1-N\beta )}{2\beta }-1}\,\mathrm{d} \theta =\frac{\left( 1-N\beta \right) ^{\gamma -1}}{(\omega -1)}<+\infty , \end{aligned}$$

and again finite-time blow-up occurs at

$$\begin{aligned} \Sigma _d\le \sigma _{d}:=\frac{1}{2\beta }\left\{ \left[ 1-(1-N\beta )^{1-\gamma }{\bar{u}}_0^{1-\omega }\right] ^{\frac{2 \beta }{(1-\omega )(1-N\beta )}}-1\right\} , \end{aligned}$$

(2.21)

provided that the initial data satisfy ${\bar{u}}_0>\left( 1-N\beta \right) ^{\frac{1-\gamma }{\omega -1}}.$ Therefore, the smallest $0<\beta <\frac{1}{N}$ is chosen, the fastest the evolving domain shrinks, then the smaller initial data $u_0$ are required for the occurrence of finite-time blow-up predicted by Theorem 2.1.

For a stationary domain, i.e. when $\rho (t)=\phi (\sigma )=1,$ we have $\Phi (\sigma )=\Psi (\sigma )=1$ and thus finite-time blow-up occurs at

$$\begin{aligned} \Sigma _s\le \sigma _s:=\frac{1}{1-\omega }\ln \left( 1-{\bar{u}}_0^{1-\omega }\right) , \end{aligned}$$

(2.22)

provided that ${\bar{u}}_0>1,$ cf. Kavallaris and Suzuki (2017, 2018).

In conclusion, conditions (2.15) and (2.16) imply, since $\omega >1$ and $\Psi (s)>0,$ that the faster the evolving domain expands, then the bigger initial data are required to obtain finite-time blow-up. On the other hand, the faster the evolving domain shrinks, then the smaller initial data $u_0$ are needed for finite-time blow-up to occur.

Note also that by relations (2.18), (2.21) and (2.22), we cannot really obtain an ordering of blowing-up times $\Sigma _g, \Sigma _d$ and $\Sigma _s$ since there is not a clear ordering of the corresponding upper bounds $\sigma _g, \sigma _d, \sigma _s$ However, we conjecture that $\Sigma _g>\Sigma _s>\Sigma _d,$ a conjecture which is verified by numerical Experiment 1 in Sect. 4; see in particular Fig. 1.

Remark 2.5

When the domain evolves logistically, which is a feasible choice in the context of biology, cf. Plaza et al. (2004), i.e. when $\rho (t)=\frac{\mathrm{e}^{\beta t}}{1+\frac{1}{m}\left( \mathrm{e}^{\beta t}-1\right) }\quad \text{ for }\quad m\ne 1,$ then equation (1.17) cannot be solved for t and it is more convenient to deal with problem (1.32)–(1.34) instead. Then, following the same approach as in Theorem 2.1 it can be shown that the solution of (1.32)–(1.34) exhibits finite-time blow-up under the same conditions for parameters p, $\gamma$, r provided that the initial condition satisfies

$$\begin{aligned} {\bar{u}}_0:={-}\!\int _{\Omega _0}u_0\,\mathrm{d}x>(\omega -1)^{\frac{1}{1-\omega }}\,\int _0^{\infty } L^{-\gamma }(\theta ) \mathrm{e}^{(1-\omega )\int ^{\theta }L(\eta )\,\mathrm{d}\eta }\,\mathrm{d} \theta , \end{aligned}$$

(2.23)

where now the quantity $L(t)=1+\frac{N\beta \left( 1-\frac{1}{m}\right) }{1+\frac{1}{m}\left( \mathrm{e}^{\beta t}-1\right) }.$

Remark 2.6

Assume now that

$$\begin{aligned} 0<{\bar{u}}_0<(\omega -1)^{\frac{1}{1-\omega }}\,I^{\frac{1}{1-\omega }}(\Sigma ), \end{aligned}$$

(2.24)

then $G(\Sigma )>0$ and since $G(\sigma )$ is strictly decreasing we get that $G(\sigma )>0$ for any $0<\sigma <\Sigma$ which implies that $F(\sigma )$ never blows up. Therefore, since $F(\sigma )\le {\bar{u}}(\sigma ),$ there is still a possibility that ${\bar{u}}(\sigma )$ does not blow up either; however, we cannot be sure and it remains to be verified numerically; more precisely see Fig.  2 of Experiment 1 in Sect. 4.

Next, we investigate the dynamics of some $L^\ell$-norms $||u(\cdot ,\sigma )||_{\ell },$ which identify some invariant regions in the phase space. We first define $\zeta (\sigma )={-}\!\int _{\Omega _0}u^r\,dx$, $y(\sigma )={-}\!\int _{\Omega _0}u^{-p+1+r}\,dx$ and $w(\sigma )={-}\!\int _{\Omega _0}u^{p-1+r}\,dx$, then Hölder’s inequality implies

$$\begin{aligned} w(\sigma )y(\sigma )\ge \zeta ^2(\sigma ), \quad 0\le \sigma <\Sigma . \end{aligned}$$

(2.25)

Our first result in this direction provides some conditions under which a finite-time blow-up takes place, when an anti-Turing condition is in place and is stated as follows.

Theorem 2.2

Take $0<\gamma <1$ and $r\le 1<\frac{p-1}{r}.$ Assume that $\Phi (\sigma )$, $\Psi (\sigma )$ satisfy (2.1), then if one of the following conditions holds:

• (1) $w(0)<\frac{m_{\Psi }}{M_{\Phi }}\zeta (0)^{1-\gamma },$

• (2) $\frac{p-1}{r}\ge 2$ and $w(0)<1,$

then finite-time blow-up occurs.

Proof

Set $\chi =u^{\frac{1}{\alpha }}$ with $\alpha \ne 0$, then following the same steps as in Proposition 2.2 we derive

$$\begin{aligned}&\quad \alpha \chi _{\sigma }=\alpha \left( \Delta \chi +4 (\alpha -1) \vert \nabla \chi ^{\frac{1}{2}}\vert ^2\right) -\Phi \chi +\Psi \frac{u^{p-1+\frac{1}{\alpha }}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}, \end{aligned}$$

(2.26)

$$\begin{aligned}& \text {for} \;x\in \Omega _0,\; \sigma \in (0,\Sigma ),\nonumber \\&\quad \frac{\partial \chi }{\partial \nu }=0, \quad \; x\in \partial \Omega _0,\; \sigma \in (0,\Sigma ), \end{aligned}$$

(2.27)

$$\begin{aligned}&\quad \chi (x,0)=u_0^{\frac{1}{\alpha }}(x), \quad x\in \Omega _0. \end{aligned}$$

(2.28)

Averaging (2.26) over $\Omega _0$ and using zero-flux boundary condition (2.27), we obtain

$$\begin{aligned} \alpha \frac{\mathrm{d}}{\mathrm{d}\sigma }{-}\!\int _{\Omega _0}\chi =-4\alpha (1-\alpha ){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2-\Phi (\sigma ){-}\!\int _{\Omega _0}\chi +\Psi (\sigma )\frac{{-}\!\int _{\Omega _0}u^{p-1+\frac{1}{\alpha }}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}.\qquad \quad \end{aligned}$$

(2.29)

Relation (2.29) for $\alpha =\frac{1}{r},$ since also $r\le 1,$ implies that

$$\begin{aligned} \frac{1}{r}\frac{\mathrm{d} \zeta }{\mathrm{d}\sigma }&=-\frac{4}{r}\left( 1-\frac{1}{r}\right) {-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2-\Phi (\sigma )\zeta (\sigma )+\Psi (\sigma )\frac{{-}\!\int _{\Omega _0}u^{p-1+r}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}\nonumber \\&\ge -M_{\Phi } \zeta (\sigma )+m_{\Psi }\frac{\zeta ^{2-\gamma }(\sigma )}{w(\sigma )}\ge \frac{\zeta (\sigma )}{w(\sigma )}\left( -M_{\Phi } w(\sigma )+m_{\Psi }\zeta ^{1-\gamma }(\sigma )\right) ,\nonumber \\ \end{aligned}$$

(2.30)

which suffices by using (2.25) together with (2.1). Furthermore, since $\frac{p-1}{r}>1$, then (2.29) for $\alpha =\frac{1}{-p+1+r}$ leads to

$$\begin{aligned} \alpha \frac{\mathrm{d}w}{\mathrm{d}\sigma }=4\alpha (\alpha -1){-}\!\int _{\Omega _0}\vert \nabla u^{\frac{1}{2\alpha }}\vert ^2-\Phi (\sigma )w+\Psi (\sigma )\zeta ^{1-\gamma }, \end{aligned}$$

(2.31)

which, owing to (2.1) and using the fact that $\alpha =\frac{1}{-p+1+r}<0$ ensures that

$$\begin{aligned} \frac{1}{p-1-r} \frac{\mathrm{d}w}{\mathrm{d}\sigma }\le M_{\Phi } w(\sigma )-m_{\Psi }\zeta ^{1-\gamma }(\sigma ). \end{aligned}$$

(2.32)

Note that since $0<\gamma <1$, we have that the curve $\Gamma _1: w=\frac{m_{\Psi }\zeta ^{1-\gamma }}{M_{\Phi }}$, $\zeta >0,$ is concave in $w\zeta -$plane, with its endpoint at the origin (0, 0). Furthermore, relations (2.30) and (2.32) imply that the region ${\mathcal {R}}=\{ (\zeta ,w) \mid w<\frac{m_{\Psi }\zeta ^{1-\gamma }}{M_{\Phi }}\}$ is invariant, and $\zeta (\sigma )$ and $w(\sigma )$ are increasing and decreasing on ${\mathcal {R}},$ respectively. Under the assumption that $w(0)<\frac{m_{\Psi }\zeta ^{1-\gamma }(0)}{M_{\Phi }}$, we have $\frac{\mathrm{d}w}{\mathrm{d}\sigma }<0$, $\frac{\mathrm{d}\zeta }{\mathrm{d}\sigma }>0$, for $0\le \sigma <\Sigma$, and thus,

$$\begin{aligned} \frac{m_{\Psi }}{w(\sigma )}-\frac{M_{\Phi }}{\zeta ^{1-\gamma }(\sigma )}\ge \frac{m_{\Psi }}{w(0)}-\frac{M_{\Phi }}{\zeta ^{1-\gamma }(0)}\equiv c_0>0, \quad \text{ for }\quad 0\le \sigma <\Sigma . \end{aligned}$$

Therefore, by virtue of (2.30), we derive the differential inequality

$$\begin{aligned} \frac{1}{r}\frac{\mathrm{d}\zeta }{\mathrm{d}\sigma }\ge -M_{\Phi }\zeta (\sigma )+m_{\Psi }\frac{\zeta ^{2-\gamma }(\sigma )}{w(\sigma )}= & {} \zeta ^{2-\gamma }(\sigma )\left( \frac{1}{w(\sigma )}-\frac{M_{\Phi }}{m_{\Psi }\zeta ^{1-\gamma }(\sigma )}\right) \nonumber \\\ge & {} c_0\zeta ^{2-\gamma }(\sigma ), \quad 0\le \sigma <\Sigma . \end{aligned}$$

(2.33)

Since $2-\gamma >1$, inequality (2.33) implies that $\zeta (\sigma )$ blows up in finite time $\sigma _1\le {\hat{\sigma }}_1\equiv \frac{\zeta ^{\gamma -1}(0)}{(1-\gamma )c_0 r}<\infty ,$ and since $\zeta (\sigma )={-}\!\int _{\Omega _0}u^r\,\mathrm{d}x\le \Vert u(\cdot ,\sigma )\Vert ^r_{\infty }$ we conclude that $u(x,\sigma )$ blows up in finite time $\Sigma _b\le {\hat{\sigma }}_1.$

We now consider the latter case when $\frac{p-1}{r}\ge 2$, then $q=\frac{p-1-r}{r}\ge 1$, and thus by virtue of Jensen’s inequality, Evans (2010), we obtain ${-}\!\int _{\Omega _0}u^r \left( {-}\!\int _{\Omega _0}(u^{-r})^q\right) ^{\frac{1}{q}}\ge {-}\!\int _{\Omega _0}u^{r} {-}\!\int _{\Omega _0}u^{-r}\ge 1,$ which entails $\zeta ^{\frac{1}{r}}(\sigma )\ge w^{-\frac{1}{p-1-r}}(\sigma )$, and thus by virtue of (2.1)

$$\begin{aligned} w(\sigma )\ge \zeta ^{-\frac{p-1-r}{r}(\sigma )}=\zeta ^{1-\frac{p-1}{r}}(\sigma )> \frac{1}{\Phi (\sigma )}\zeta ^{1-\frac{p-1}{r}}(\sigma )\ge \frac{m_{\Psi }}{M_{\Phi }}\zeta ^{1-\frac{p-1}{r}}(\sigma ),\quad \end{aligned}$$

(2.34)

for any $\sigma \in [0,\Sigma )$. Since $\frac{p-1}{r}\ge 2$, the curve $\Gamma _2: w=\frac{m_{\Psi }\zeta ^{1-\frac{p-1}{r}}}{M_{\Phi }}, \ \zeta >0,$ is convex and approaches $+\infty$ and 0 as $\zeta \downarrow 0^+$ and $\zeta \uparrow +\infty$, respectively. Moreover, the curves $\Gamma _1$ and $\Gamma _2$ intersect at the point $(\zeta ,w)=(1,1),$ and therefore, $w(0)<1$ combined with (2.34) implies that $w(0)<\frac{m_{\Psi }\zeta ^{1-\gamma }(0)}{M_{\Phi }}$. Thus, the latter case is reduced to the former case and once again finite-time blow-up for the solution $u(x,\sigma )$ is established.

$\square$

Remark 2.7

Note that in the case of a stationary domain then $\zeta (\sigma )$ blows up, see Kavallaris and Suzuki (2017, 2018), in finite time $\sigma _2\le {\hat{\sigma }}_2\equiv \frac{\zeta ^{\gamma -1}(0)}{(1-\gamma )c_1 r},$ where $c_1\equiv \frac{1}{w(0)}-\frac{1}{\zeta ^{1-\gamma }(0)},$ and thus $u(x,\sigma )$ blows in finite time $\Sigma _1\le {\hat{\sigma }}_2$ under the condition $w(0)<\zeta (0)^{1-\gamma }.$

Remark 2.8

For a logistically growing or shrinking domain problem, (1.32)–(1.34) exhibit finite-time blow-up under the assumptions of Theorem 2.2 whenever $w(0)<M_{L}^{-(\gamma +1)}\zeta (0)^{1-\gamma },$ where $M_{L}:=\sup _{(0,\infty )} L(t)=\sup _{(0,\infty )}\left( 1+\frac{N\beta \left( 1-\frac{1}{m}\right) }{1+\frac{1}{m}\left( \mathrm{e}^{\beta t}-1\right) }\right) .$ In particular, for a logistically growing domain, when $m>1,$ then $M_L=L(0)=1+N\beta \left( 1-\frac{1}{m}\right) ,$ whilst for logistically decaying domain, when $0<m<1$ we have $M_L=\lim _{t\rightarrow +\infty }L(t)=1$ and hence in that case blow-up conditions (1) and (2) of Theorem 2.2 coincide with the ones of Kavallaris and Suzuki (2017, Theorem 3.5), see also Remark 2.7.

Now, we present a global-in-time existence result stated as follows.

Theorem 2.3

Assume that $\frac{p-1}{r}<\min \{1, \frac{2}{N}, \frac{1}{2}(1-\frac{1}{r})\}$ and $0<\gamma <1.$ Consider functions $\Phi (\sigma )$, $\Psi (\sigma )>0$ with

$$\begin{aligned} \inf _{(0,\Sigma )} \Phi (\sigma ):=m_{\Phi }>0\quad \text{ and }\quad \sup _{(0,\Sigma )} \Psi (\sigma ):=M_{\Psi }<+\infty , \end{aligned}$$

(2.35)

then problem (1.35)–(1.37) has a global-in-time solution.

Proof

We assume $\frac{p-1}{r}<\min \{ 1, \frac{2}{N}, \frac{1}{2}(1-\frac{1}{r})\}$ and $0<\gamma <1$. We also assume $N\ge 2$ since the complementary case $N=1$ is simpler.

Note that for $p>1$, we have $\frac{p-1}{r}<\frac{2}{N}$ and $r>p$. Therefore, we have $0<\frac{1}{r-p+1}<\min \left\{ 1, \left( \frac{1}{p-1}\right) \left( \frac{2}{N-2}\right) , \frac{1}{1-p+r\gamma }\right\} ,$ since $0<\gamma <1$. Choosing $\frac{1}{r-p+1}<\alpha <\min \{ 1, \frac{1}{p-1}\cdot \frac{2}{N-2}, \frac{1}{1-p+r\gamma } \}$, it follows that the $\max \left\{ \frac{N-2}{N}, \frac{1}{\alpha r}\right\} <\frac{1}{-\alpha +1+\alpha p},$ and then we can find $\beta >0$ such that

$$\begin{aligned} \max \left\{ \frac{N-2}{N}, \frac{1}{\alpha r}\right\}<\frac{1}{\beta }<\frac{1}{-\alpha +1+\alpha p}<2, \end{aligned}$$

(2.36)

which satisfies

$$\begin{aligned} \frac{\beta }{\alpha r}<1<\frac{\beta }{-\alpha +1+\alpha p}. \end{aligned}$$

(2.37)

Recalling that $\chi =u^{\frac{1}{\alpha }}$ satisfies (2.26)–(2.28) with $\displaystyle {{-}\!\int _{\Omega _0}\frac{u^{p-1+\frac{1}{\alpha }}}{\left( {-}\!\int _{\Omega _0}u^r\right) ^\gamma }= \frac{{-}\!\int _{\Omega _0}\chi ^{-\alpha +1+\alpha p}}{\left( {-}\!\int _{\Omega _0}\chi ^{\alpha r}\right) ^\gamma }},$ then by virtue of (2.37)

$$\begin{aligned} {-}\!\int _{\Omega _0}\chi ^{-\alpha +1+\alpha p}\le \left( {-}\!\int _{\Omega _0}\chi ^\beta \right) ^{\frac{-\alpha +1+\alpha p}{\beta }}\quad \text{ and }\quad \left( {-}\!\int _{\Omega _0}\chi ^{\alpha r}\right) ^\gamma \ge \left( {-}\!\int _{\Omega _0}\chi ^\beta \right) ^{\frac{\alpha r}{\beta }\cdot \gamma }, \end{aligned}$$

thus we obtain the following estimate

$$\begin{aligned} \frac{{-}\!\int _{\Omega _0}\chi ^{-\alpha +1+\alpha p}}{\left( {-}\!\int _{\Omega _0}\chi ^{\alpha r}\right) ^\gamma }\le \left( {-}\!\int _{\Omega _0}\chi ^\beta \right) ^{\frac{-\alpha +1+\alpha p-\alpha r\gamma }{\beta }}=\Vert \chi ^{\frac{1}{2}}\Vert _{2\beta }^{2(1-\lambda )}, \end{aligned}$$

(2.38)

with $0<\lambda =\alpha \{1-p+r\gamma \}<1$, recalling that $\frac{p-1}{r}<\gamma$ and $\alpha <\frac{1}{1-p+r\gamma }$. Averaging (2.26) over $\Omega _0$ leads to the following,

$$\begin{aligned} \alpha \frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +4\alpha (1-\alpha ){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2+\Phi (\sigma ){-}\!\int _{\Omega _0}\chi =\Psi (\sigma )\frac{{-}\!\int _{\Omega _0}\chi ^{-\alpha +1+\alpha p}}{\left( {-}\!\int _{\Omega _0}\chi ^{\alpha r}\right) ^\gamma }, \end{aligned}$$

(2.39)

and hence

$$\begin{aligned} \frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +4(1-\alpha ){-}\!\int _{\Omega _0}\vert \nabla \chi ^{\frac{1}{2}}\vert ^2+\frac{m_{\Phi }}{\alpha }{-}\!\int _{\Omega _0}\chi \le \frac{M_{\Psi }}{\alpha }\Vert \chi ^{\frac{1}{2}}\Vert _{2\beta }^{2(1-\lambda )}, \end{aligned}$$

by virtue of (2.35), (2.36) and (2.38). Now since $1<2\beta <\frac{2N}{N-2}$ holds due to (2.36) and applying first the Sobolev’s and then Young’s inequalities we obtain

$$\begin{aligned} \frac{d}{d\sigma }{-}\!\int _{\Omega _0}\chi +c\Vert \chi ^{\frac{1}{2}}\Vert _{H^1}^2+\frac{M_{\Psi }}{\alpha }{-}\!\int _{\Omega _0}\chi \le C, \end{aligned}$$

which implies ${-}\!\int _{\Omega _0}\chi \le C.$ Since $\frac{1}{\alpha }$ can be chosen to be close to $r-p+1$, the above estimate gives

$$\begin{aligned} \Vert u(\cdot ,\sigma )\Vert _q\le C_q, \quad \text{ for } \text{ any }\quad 1\le q<r-p+1, \end{aligned}$$

(2.40)

recalling that $\chi =u^{\frac{1}{\alpha }}.$ Note that $\frac{p-1}{r}<\frac{1}{2}(1-\frac{1}{r})$ implies $\frac{r-p+1}{p}>1$ and thus we obtain global-in-time existence by using the same bootstrap argument as in Kavallaris and Suzuki (2017, Theorem 3.4). $\square$

Remark 2.9

Note that condition (2.35) is satisfied in the case of an exponentially shrinking domain as indicated in Remark 2.4, see in particular (2.19) and (2.20).

3 Turing Instability and Pattern Formation

In this section, we state and prove a Turing-instability, that is a diffusion-driven instability, result for problem (1.35)–(1.37). Due to technical restrictions, we focus on the radial case $\Omega _0=B_1(0):=\{x\in \mathbb {R}^N \mid \vert x\vert <1\}$ and for dimensions $N\ge 3;$ however, in Sect. 4, we treat numerically the two-dimensional case $N=2$ as well, see Fig. 6 Next, we consider a radially decreasing and spiky initial datum of the form, (Hu and Yin 1995),

$$\begin{aligned} u_0(R)=\lambda \psi _\delta (R), \end{aligned}$$

(3.1)

with $0<\lambda \ll 1$ and

$$\begin{aligned} \psi _\delta (R)={\left\{ \begin{array}{ll} R^{-a},&{} \delta \le R\le 1, \\ \delta ^{-a}\left( 1+\frac{a}{2}\right) -\frac{a}{2}\delta ^{-(a+2)}R^2, &{} 0\le R<\delta , \end{array}\right. } \end{aligned}$$

(3.2)

where $a=\frac{2}{p-1}$ and $0<\delta <1$. Notably $u_0(R)\in L^{\infty }(0,1),$ which is compatible with assumption (1.10).

Then, the solution u (1.35)–(1.37) is radially symmetric and decreasing , i.e. $u(x,\sigma )=u(R,\sigma )$ for $R=|x|$ and $u_R(R,\sigma )\le 0$ and thus, it satisfies the following

$$\begin{aligned} u_{\sigma }= & {} \Delta _R u-\Phi (\sigma )u+\displaystyle \frac{\Psi (\sigma )u^p}{\left( {-}\!\int _{\Omega _0}u^r\right) ^{\gamma }}, \quad R\in (0,1),\; \sigma \in (0,\Sigma ), \end{aligned}$$

(3.3)

$$\begin{aligned} u_R(0,\sigma )= & {} u(1,\sigma )=0, \quad \sigma \in (0,\Sigma ), \end{aligned}$$

(3.4)

$$\begin{aligned} u(R,0)= & {} u_0(R),\quad 0<R<1, \end{aligned}$$

(3.5)

where $\Delta _R u:=u_{RR}+\frac{N-1}{R}u_R.$

Remarkably, under the Turing condition (1.5), the spatial homogeneous solutions of (3.3)–(3.5), i.e. the solution of the problem

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}\sigma }= & {} -\Phi (\sigma )u+\Psi (\sigma )u^{p-r\gamma },\\ \left. u\right| _{\sigma =0}= & {} {\bar{u}}_0>0, \end{aligned}$$

never exhibit blow-up, as long as $\Phi (\sigma ), \Psi (\sigma )$ are both bounded, since the nonlinearity $f(u)=u^{p-r\gamma }$ is sub-linear [see also Kavallaris and Suzuki (2017, 2018)]. Otherwise, considering spatial inhomogeneous solutions of (3.3)–(3.5), the following diffusion-driven blow-up (Turing instability ) result holds true.

Theorem 3.1

Consider $N\ge 3,\;1\le r\le p$, $p>\frac{N}{N-2}E$, $\frac{2}{N}<\frac{p-1}{r}<\gamma$ and $\gamma >1.$ Assume that both $\Phi (\sigma )$ and $\Psi (\sigma )$ are positive and bounded. Then, there exists $\lambda _0>0$ such that for any $0<\lambda \le \lambda _0$ there exists $0<\delta _0=\delta _0(\lambda )<1$, then any solution of problem (3.3)–(3.5) with spiky initial data of the form (3.1) and $0<\delta \le \delta _0$ blows up in finite time.

Note that the maximum principle is not applicable for the non-local problem (3.3)–(3.5) and hence comparison techniques fail, see for example Quittner and Souplet (2007, Proposition 52.24). Therefore, to overcome this obstacle, and finally prove Theorem 3.1, we derive a lower estimate of the non-local term

$$\begin{aligned} {\widetilde{K}}(\sigma ):=\frac{\Psi (\sigma )}{\Big (\displaystyle {-}\!\int _{\Omega _0}u^r \Big )^{\gamma }}. \end{aligned}$$

and then deal with a local problem for which comparison techniques become applicable. To that end, following an approach used in Hu and Yin (1995) and Kavallaris and Suzuki (2017, 2018), we need to prove first some auxiliary results.

First, it is easily seen that for $\psi _{\delta }$ given by (3.2) the following lemma holds (Kavallaris and Suzuki 2017, 2018).

Lemma 3.1

For the function $\psi _{\delta }$ defined by (3.2), we have:

• (i) For any $0<\delta <1,$ there holds in a weak sense

  $$\begin{aligned} \displaystyle { \Delta _R \psi _\delta \ge -Na\psi _\delta ^p}. \end{aligned}$$

  (3.6)

• (ii) If $m>0$ and $N>ma$, we have

  $$\begin{aligned} {-}\!\int _{\Omega _0}\psi _\delta ^m=\frac{N}{N-ma}+O\left( \delta ^{N-ma}\right) , \quad \delta \downarrow 0. \end{aligned}$$

  (3.7)

Now, if we consider

$$\begin{aligned} \mu >1+r\gamma \end{aligned}$$

(3.8)

and set $\alpha _1=\sup _{0<\delta <1}\frac{1}{{\bar{\psi }}_\delta ^\mu }{-}\!\int _{\Omega _0}\psi _\delta ^p$, and $\alpha _2=\inf _{0<\delta <1}\frac{1}{{\bar{\psi }}_\delta ^\mu }{-}\!\int _{\Omega _0}\psi _\delta ^p$,

then since $p>\frac{N}{N-2},$ relation (3.7) is applicable for $m=p$ and $m=1$, and thus owing to (3.8) we obtain

$$\begin{aligned} 0<\alpha _1, \alpha _2<\infty . \end{aligned}$$

(3.9)

Furthermore, it follows that

$$\begin{aligned} d\equiv \inf _{0<\delta <1}\left( \frac{1}{2\alpha _1}\right) ^{\frac{r\gamma }{p}}\left( \frac{1}{2{\bar{\varphi }}_\delta }\right) ^{\frac{r\gamma }{p}\mu }>0, \end{aligned}$$

(3.10)

and the initial data $u_0$ defined by (3.1) and (3.2) also satisfy the following lemma, for the proof see Kavallaris and Suzuki (2017, 2018).

Lemma 3.2

If $p>\frac{N}{N-2}$ and $\frac{p-1}{r}<\gamma$, there exists $\lambda _0=\lambda _0(d)>0$ such that for any $0<\lambda \le \lambda _0$ there holds

$$\begin{aligned} \Delta _R u_0+d\lambda ^{-r\gamma }u_0^p\ge 2u_0^p. \end{aligned}$$

(3.11)

Hereafter, we fix $0<\lambda \le \lambda _0=\lambda _0(d)$ such that (3.11) is satisfied. Given $0<\delta <1$, let $\Sigma _\delta >0$ be the maximal existence time of the solution to (3.3)–(3.5) with initial data of the form (3.1)-(3.2). Next, we introduce the new variable $z:=\mathrm{e}^{\int ^{\sigma }\Phi (s)\,ds }u,$ such that the linear dissipative term $-\Phi (\sigma ) u$ in (3.3) is eliminated and z satisfies

$$\begin{aligned} z_\sigma= & {} \Delta _R z \nonumber \\&+\,K(\sigma )z^p,\quad R\in (0,1),\; \sigma \in (0,\Sigma _\delta ), \end{aligned}$$

(3.12)

$$\begin{aligned} z_R(0,\sigma )= & {} z_R(1,\sigma )=0,\quad \sigma \in (0,\Sigma _\delta ), \end{aligned}$$

(3.13)

$$\begin{aligned} z(R,0)= & {} u_0(R),\quad 0<R<1, \end{aligned}$$

(3.14)

where

$$\begin{aligned} K(\sigma ):=\frac{\Psi (\sigma )\mathrm{e}^{(1+r\gamma -p)\int ^\sigma \Phi (s)\,ds}}{\Big (\displaystyle {-}\!\int _{\Omega _0}z^r \Big )^{\gamma }}. \end{aligned}$$

(3.15)

It is clear that as long as $\Phi (\sigma )$ is bounded then u blows-up in finite time if and only if z does so. Assuming now that both $\Phi (\sigma )$ and $\Psi (\sigma )$ are positive and bounded, which is the case for the evolution provided by $\psi (\sigma )$ satisfying (2.3) or for an exponentially shrinking domain as indicated in Remarks 2.1 and 2.4, then by virtue of (2.14) we have

$$\begin{aligned} 0< K(\sigma )=\frac{\Psi (\sigma )\mathrm{e}^{(1-p)\int ^\sigma \Phi (s)\,ds}}{\Big (\displaystyle {-}\!\int _{\Omega _0}u^r \Big )^{\gamma }}\le C<\infty . \end{aligned}$$

(3.16)

Averaging of (3.12) entails

$$\begin{aligned} \frac{d{\bar{z}}}{d\sigma }=K(\sigma ){-}\!\int _{\Omega _0}z^p, \end{aligned}$$

(3.17)

and thus (3.16) yields

$$\begin{aligned} {\bar{z}}(\sigma )\ge {\bar{z}}(0)={\bar{u}}_0:={-}\!\int _{\Omega _0}u_0. \end{aligned}$$

(3.18)

Henceforth, the positivity and the boundedness of $\Phi (\sigma )$, and $\Psi (\sigma )$ as well as the Turing condition (1.5) are imposed.

Next, we provide a useful estimate of z that will be frequently used throughout the sequel.

Lemma 3.3

The solution z of problem (3.12)–(3.14) satisfies

$$\begin{aligned}&R^N z(R,\sigma )\le {\bar{z}}(\sigma ) \quad \mathrm{in}\quad (0,1)\times (0,\Sigma _{\delta }), \end{aligned}$$

(3.19)

$$\begin{aligned}&\mathrm{and} \nonumber \\&z_R\left( \frac{3}{4},\sigma \right) \le -c, \ \ 0\le \sigma <\Sigma _\delta , \end{aligned}$$

(3.20)

for any $0<\delta <1$ and some positive constant c.

Proof

Let us define $w=R^{N-1}z_{R}$, then it follows that w satisfies ${\mathcal {H}}[w]=0, \quad \text{ for }\quad (R,\sigma )\in (0,1)\times (0,\Sigma _{\delta })$, with $w(0,\sigma )=w(1,\sigma )=0$, for $\sigma \in (0,\Sigma _{\delta })$, and $w(R,0)<0$, for $0<R<1$, where ${\mathcal {H}}[w]\equiv w_\sigma -w_{RR}+\frac{N-1}{\rho }w_R-p K(\sigma )z^{p-1}w$. Owing to the maximum principle, and recalling that $K(\sigma )$ is bounded by (3.16), we get that $w\le 0$, which implies $z_{R}\le 0$ in $(0,1)\times (0,\Sigma _{\delta })$. Accordingly, inequality (3.19) follows since

$$\begin{aligned} R^N z(R,\sigma )= & {} z(R,\sigma )\int _0^R N s^{N-1}\,\mathrm{d}s\le \int _0^R Nz(s,\sigma )s^{N-1}\,\mathrm{d}s \\\le & {} \int _0^1Nz(s,\sigma )s^{N-1}\,\mathrm{d}s={-}\!\int _{\Omega _0}z={\bar{z}}(\sigma ). \end{aligned}$$

Now, given that $w\le 0$ together with (3.16), we have

$$\begin{aligned} w_\sigma -w_{RR}+\frac{N-1}{\rho }w_R=pK(\sigma )z^{p-1}w\le 0 \quad \text{ in }\quad (0,1)\times (0,\Sigma _{\delta }), \end{aligned}$$

with $w\left( \frac{1}{2},\sigma \right) \le 0,\quad w\left( 1,\sigma \right) \le 0$, for $\sigma \in (0,\Sigma _{\delta })$, and $w(R,0)=\rho ^{N-1}u'_{0}(R)\le -c$, for $\frac{1}{2}<\rho <1$, which implies $w\le -c$ in $(\frac{1}{2},1)\times (0, \Sigma _\delta )$, and thus (3.20) holds. $\square$

The next result is vital for proving the key estimate provided by Proposition 3.1 below.

Lemma 3.4

Take $\varepsilon >0$ and $1<q<p$ then $\vartheta$ defined as

$$\begin{aligned} \vartheta :=R^{N-1}z_R+\varepsilon \cdot \frac{R^Nz^q}{{\bar{z}}^{\gamma +1}}, \end{aligned}$$

(3.21)

satisfies

$$\begin{aligned} {\mathcal {H}}[\vartheta ]\le & {} -\frac{2q\varepsilon }{{\bar{z}}^{\gamma +1}}z^{q-1}\vartheta +\frac{\varepsilon R^Nz^q}{{\bar{z}}^{2(\gamma +1)}}\left\{ 2q\varepsilon z^{q-1}\right. \nonumber \\&\quad \left. -\,m_{\Psi }(\gamma +1){\bar{z}}^{\gamma -r\gamma }{-}\!\int _{\Omega _0}z^p-m_{\Psi }(p-q)z^{p-1}{\bar{z}}^{\gamma +1-r\gamma }\right\} , \qquad \end{aligned}$$

(3.22)

for $(R,\sigma )\in (0,1)\times (0,\Sigma _{\delta }),$ where $m_{\Psi }=\inf _{\sigma \in (0,\Sigma _\delta )}\Psi (\sigma )>0.$

Proof

It is readily checked that ${\mathcal {H}}\left[ R^{N-1} z_{R}\right] =0,$ while by straightforward calculations we derive

$$\begin{aligned} {\mathcal {H}}\left[ \varepsilon R^{N} \frac{z^q}{\overline{z}^{\gamma +1}}\right]= & {} \frac{2q(N-1)\varepsilon R^{N-1}z^{q-1}}{\overline{z}^{\gamma +1}}z_{R}\nonumber \\&+\,\frac{q\varepsilon R^N z^{p-1+q}}{\overline{z}^{\gamma +1}} K(\sigma )\nonumber \\&-\,\frac{(\gamma +1) \varepsilon R^N z^q}{\overline{z}^{\gamma +2}}\,K(\sigma )\,{-}\!\int _{\Omega _0}z^p\,\mathrm{d}x \nonumber \\&-\,\frac{2 q N\varepsilon R^{N-1} z^{q-1}}{\overline{z}^{\gamma +1}}z_R\nonumber \\&-\,\frac{q(q-1)\varepsilon R^N z^{q-2}}{\overline{z}^{\gamma +1}}z_R^2-\frac{p\varepsilon R^N z^{p-1+q}}{\overline{z}^{\gamma +1}} K(\sigma )\nonumber \\\le & {} -\frac{2 q \varepsilon z^{q-1}}{\overline{z}^{\gamma +1}}\vartheta \nonumber \\&+\,\frac{\varepsilon R^N z^q}{\overline{z}^{2(\gamma +1)}} \left[ 2q\varepsilon z^{q-1}-\frac{\Psi (\sigma )(\gamma +1) \overline{z}^\gamma \mathrm{e}^{(1+r\gamma -p) \int ^\sigma \Phi (s)\,ds}}{\left( {-}\!\int _{\Omega _0}z^r\,\mathrm{d}x\right) ^\gamma }{-}\!\int _{\Omega _0}z^p\,\mathrm{d}x\right. \nonumber \\&\left. \quad - \,\frac{\Psi (\sigma )(p-q) z^{p-1} \overline{z}^{\gamma +1}}{\left( {-}\!\int _{\Omega _0}z^r\,\mathrm{d}x\right) ^\gamma } \mathrm{e}^{(1+r\gamma -p)\int ^\sigma \Phi (s)\,\mathrm{d}s} \right] . \end{aligned}$$

(3.23)

Then, by virtue of the Hölder’s inequality, and since $1\le r\le p,$ (3.23) entails the desired estimate (3.22). $\square$

Next, note that when $p>\frac{N}{N-2}$, there is $1<q<p$ such that $N>\frac{2p}{q-1}$ and thus the following quantities

$$\begin{aligned} A_1\equiv \sup _{0<\delta<1}\frac{1}{{\bar{u}}_0^{\mu }} {-}\!\int _{\Omega _0}u_0^p=\lambda ^{\mu -p}\alpha _1\quad \text{ and }\quad A_2\equiv \inf _{0<\delta <1}\frac{1}{{\bar{u}}_0^{\mu }} {-}\!\int _{\Omega _0}u_0^p=\lambda ^{\mu -p}\alpha _2, \end{aligned}$$

(3.24)

are finite due to (3.9).

An essential ingredient for the proof of Theorem 3.1 is the following key estimate of the $L^p-$norm of z in terms of $A_1$ and $A_2.$

Proposition 3.1

There exist $0<\delta _0<1$ and $0<\sigma _0\le 1$ independent of any $0<\delta \le \delta _0,$ such that the following estimate is satisfied

$$\begin{aligned} \frac{1}{2} A_2\overline{z}^{\mu }\le {-}\!\int _{\Omega _0}z^p\,\mathrm{d}x\le 2 A_1 \overline{z}^{\mu }, \end{aligned}$$

(3.25)

for any $0<\sigma <\min \{\sigma _0,\Sigma _{\delta }\}.$

The proof of Proposition 3.1 requires some further auxiliary results provided below. Let us define $0<\sigma _0(\delta )<\Sigma _\delta$ to be the maximal time for which inequality (3.25) is valid in $0<\sigma <\sigma _0(\delta ),$ then we have

$$\begin{aligned} \frac{1}{2}A_2{\bar{z}}^\mu \le {-}\!\int _{\Omega _0}z^p\le 2A_1{\bar{z}}^\mu . \end{aligned}$$

(3.26)

We only regard the case $\sigma _0(\delta )\le 1,$ since otherwise there is nothing to prove. Then, the following lemma holds true.

Lemma 3.5

There exists $0<\sigma _1<1$ such that

$$\begin{aligned} {\bar{z}}(\sigma )\le 2{\bar{u}}_0, \quad 0<\sigma <\min \{ \sigma _1, \sigma _0(\delta )\}, \end{aligned}$$

(3.27)

for any $0<\delta <1$.

Proof

Since $r\ge 1$ and $\sigma _0(\delta )\le 1$, then by virtue of (3.15) and (3.17)

$$\begin{aligned} \frac{\mathrm{d}{\bar{z}}}{\mathrm{d}\sigma }\le 2A_1 M_{\Psi } \mathrm{e}^{(1+r\gamma -p)M_{\Phi }}{\bar{z}}^{\mu -r\gamma },\quad \text{ for } \;0<\sigma <\sigma _0(\delta ), \end{aligned}$$

recalling that $M_{\Phi }=\sup _{\sigma \in (0,\Sigma _{\delta })}\Phi (\sigma )<+\infty$ and $M_{\Psi }=\sup _{\sigma \in (0,\Sigma _{\delta })}\Psi (\sigma )<+\infty .$

Setting $C_1=2A_1 M_{\Psi } \mathrm{e}^{(1+r\gamma -p)M_{\Phi }}$ and taking into account (3.8), we then derive

$$\begin{aligned} \overline{z}(\sigma )\le \left[ {\bar{u}}_0^{1+r\gamma -\mu }-C_1(\mu -r\gamma -1)\sigma \right] ^{-\frac{1}{\mu -r\gamma -1}}. \end{aligned}$$

Accordingly, (3.27) holds for any $0<\sigma <\min \{ \sigma _1, \sigma _0(\delta )\}$ where $\sigma _1$ is independent of any $0<\delta <1$ and is estimated as $\sigma _1\le \min \left\{ \frac{1-2^{1+r\gamma -\mu }}{C_1(\mu -r\gamma -1)}\overline{u}_0^{1+r\gamma -\mu },1\right\} .$ $\square$

Another fruitful estimate is provided by the next lemma.

Lemma 3.6

There exist $0<\delta _0<1$ and $0<R_0<\frac{3}{4}$ such that for any $0<\delta \le \delta _0$ the following estimate is valid

$$\begin{aligned} \frac{1}{\vert \Omega \vert }\int _{B_{R_0}(0)}z^p\le \frac{A_2}{8}{\bar{z}}^\mu , \quad \text{ for }\quad 0<\sigma <\min \{ \sigma _1, \sigma _0(\delta )\}, \end{aligned}$$

(3.28)

where $B_{R_0}(0)=\{x\in \mathbb {R}^N \mid \vert x\vert <R_0\}.$

Proof

By virtue of (3.18) and (3.27), it follows that

$$\begin{aligned} {\bar{u}}_0\le {\bar{z}}(\sigma )\le 2{\bar{u}}_0, \quad \text{ for }\quad 0<\sigma <\min \{\sigma _1, \sigma _0(\delta )\}. \end{aligned}$$

(3.29)

Furthermore, we note that the growth of ${-}\!\int _{\Omega _0}z^p$ is controlled by the estimate (3.25) for $0<\min \{ \sigma _1, \sigma _0(\delta )\}$ and since $p>q$ then Young’s inequality ensures that the second term of the right-hand side in (3.22) is negative for $0<\sigma <\min \{ \sigma _1, \sigma _0(\delta )\}$, uniformly in $0<\delta <1$, provided that $0<\varepsilon \le \varepsilon _0$ for some $0<\varepsilon _0\ll 1.$ Therefore,

$$\begin{aligned} {\mathcal {H}}[\vartheta ]\le -\frac{2q\varepsilon z^{q-1}}{{\bar{z}}^{\gamma +1}} \vartheta \quad \text{ in }\quad (0,1)\times (0,\min \{ \sigma _1, \sigma _0(\delta )\}). \end{aligned}$$

(3.30)

Moreover, (3.19) and (3.29) imply

$$\begin{aligned} \vartheta (R,\sigma )= & {} R^{N-1}z_R+\varepsilon \frac{R^Nz^q}{{\bar{z}}^{\gamma +1}} \le R^{N-1}z_R+\varepsilon R^{N(1-q)}{\bar{z}}^{q-\gamma -1} \\\le & {} R^{N-1}z_R+C \varepsilon R^{N(1-q)}\quad \text{ in }\quad (0,1)\times (0,\min \{ \sigma _1, \sigma _0(\delta )\}), \end{aligned}$$

which, for $0<\varepsilon \le \varepsilon _0,$ entails

$$\begin{aligned} \vartheta \left( \frac{3}{4},\sigma \right)<0, \quad 0<\sigma <\min \{ \sigma _1, \sigma _0(\delta )\}, \end{aligned}$$

(3.31)

owing to (3.20) and provided that $0<\varepsilon _0\ll 1.$ Additionally, (3.21) for $t=0$ gives

$$\begin{aligned} \vartheta (R,0) = R^{N-1}\left( \lambda \psi _{\delta }'(R)+\varepsilon \lambda ^{q-\gamma -1}R \frac{\psi _\delta ^q}{{\bar{\psi }}_\delta ^{\gamma +1}}\right) . \end{aligned}$$

(3.32)

For $0\le R<\delta$ and $\varepsilon$ small enough and independent of $0<\delta <\delta _0,$ then the right-hand side of (3.32) can be estimated as

$$\begin{aligned} R^{N}\lambda \left( -a\delta ^{-a-2}+\varepsilon \lambda ^{q-\gamma -2} \frac{\psi _\delta ^q}{{\bar{\psi }}_\delta ^{\gamma +1}} \right) \lesssim R^{N}\lambda \left( -a\delta ^{-a-2}+\varepsilon \lambda ^{q-\gamma -2} \delta ^{-aq} \right) \lesssim 0, \end{aligned}$$

since by virtue of (3.2) and (3.7) and for $m=1,$ there holds $\displaystyle \frac{\psi _\delta ^q}{{\bar{\psi }}_\delta ^{\gamma +1}}\lesssim \delta ^{-aq},\; \delta \downarrow 0,$ uniformly in $0\le R<\delta ,$ taking also into account that $a+2=ap>ak.$

On the other hand, for $\delta \le R\le 1$ and by using (3.7) for $m=1$ we take

$$\begin{aligned} \vartheta (R,0)=R^N\lambda \left( -a R^{-a-1}+\varepsilon \lambda ^{q-\gamma -1}\frac{R^{1-aq}}{{\bar{\psi }}_R^{\gamma +1}}\right) , \end{aligned}$$

(3.33)

which, since $a+2=ap>aq$ implies $-a-1<-aq+1$, finally yields $\vartheta (R,0)<0, \quad \delta \le R\le \frac{3}{4},$ for any $0<\delta \le \delta _0$ and $0<\varepsilon \le \varepsilon _0$, provided $\varepsilon _0$ is chosen sufficiently small. Accordingly, it follows that

$$\begin{aligned} \vartheta (R,0)<0, \quad \text {and} \quad 0\le R\le \frac{3}{4}, \end{aligned}$$

(3.34)

for any $0<\delta \le \delta _0$ and $0<\varepsilon \le \varepsilon _0$, provided $0<\varepsilon _0\ll 1.$

In conjunction of (3.30), (3.31) and (3.34), we deduce $\vartheta (R,\sigma )=R^{N-1}z_R+\varepsilon \frac{R^Nz^q}{{\bar{z}}^{\gamma +1}}\le 0$ in $(0,\frac{3}{4})\times (0,\min \{ \sigma _1, \sigma _0(\delta )\})$, and finally

$$\begin{aligned} z(R,\sigma )\le \left( \frac{\varepsilon }{2}(q-1)\right) ^{-\frac{1}{q-1}} R^{-\frac{2}{q-1}} {\bar{z}}^{\frac{\gamma -1}{q-1}}(\sigma ) \quad \text{ in } (0,\frac{3}{4})\times (0,\min \{ \sigma _1, \sigma _0(\delta )\}). \end{aligned}$$

(3.35)

Note that owing to $N>\frac{2p}{q-1}$ there holds $-\left( \frac{2}{q-1}\right) p+N-1>-1$ and thus (3.28) is valid for some $0<R_0<\frac{3}{4}.$ $\square$

Remark 3.1

Estimate (3.35) entails that $z(R,\sigma )$ can only blow-up in the origin $R=0;$ that is, only a single-point blow-up is feasible.

Next, we prove the key estimate (3.25) using essentially Lemmas 3.5 and 3.6.

Proof of Proposition 3.1

By virtue of (3.8) and since $\frac{p-1}{r}<\delta$, there holds that $\ell =\frac{\mu }{p}>1.$ We can easily check that $\theta =\displaystyle {\frac{z}{\overline{z}^{\ell }}}$ satisfies (Kavallaris and Suzuki 2017, 2018)

$$\begin{aligned} \theta _{\sigma }=\Delta _R \theta + \Psi (\sigma )\mathrm{e}^{(r\gamma +1-p)\int ^\sigma \Phi (s)\,\mathrm{d}s}\left[ \frac{z^p}{\overline{z}^{\ell }\left( {-}\!\int _{\Omega _0}z^r\right) ^{\gamma }}-\frac{\ell z{-}\!\int _{\Omega _0}z^p}{\overline{z}^{\ell +1}\left( {-}\!\int _{\Omega _0}z^r\right) ^{\gamma }} \right] , \end{aligned}$$

in $\Omega _0\times (0, \min \{ \sigma _0, \Sigma _\delta \})$, with $\frac{\partial \theta }{\partial \nu }=0$, on $\partial \Omega _0\times (0, \min \{ \sigma _0, \Sigma _\delta \})$, and $\theta (x,0)=\frac{z(x,0)}{{\bar{z}}_0^{\ell }}$, on $\Omega _0$. In conjunction with (2.14), (3.18), (3.19), (3.26), and (3.27), we deduce that

$$\begin{aligned} \left\| \theta ,\ \frac{z^p}{\overline{z}^{\ell }\left( {-}\!\int _{\Omega _0}z^r\right) ^{\gamma }}, \ \frac{\ell z{-}\!\int _{\Omega _0}z^p}{\overline{z}^{\ell +1}\left( {-}\!\int _{\Omega _0}z^r\right) ^{\gamma }}\right\| _{L^\infty ((\Omega _0\setminus B_{R_0}(0))\times \min \{ \sigma _1, \sigma _0(\delta )\})} <+\infty ,\qquad \end{aligned}$$

(3.36)

uniformly in $0<\delta \le \delta _0,$ and using the fact that $\Psi (\sigma )$ and $\Psi (\sigma )$ are both bounded and positive. Estimate (3.36) according to the standard parabolic regularity condition, see DeGiorgi–Nash–Moser estimates in Lieberman (1996, pp. 144–145), entails the existence of $0<\sigma _2\le \sigma _1$ independent of $0<\delta \le \delta _0$: $\sup _{0< \sigma <\min \{\sigma _2, \sigma _0(\delta )\}}\left\| \theta ^p(\cdot ,\sigma )-\theta ^p(\cdot ,0)\right\| _{L^1(\Omega _0\setminus B_{R_0}(0)}\le \frac{A_2}{8}\vert \Omega _0\vert ,$ which yields

$$\begin{aligned} \left| \frac{1}{\vert \Omega _0\vert }\int _{\Omega _0\setminus B_{R_0}(0)}\frac{z^p}{{\bar{z}}^\mu }-\frac{1}{\vert \Omega _0\vert }\int _{\Omega _0\setminus B_{R_0}(0)}\frac{z_0^p}{{\bar{z}}_0^\mu } \right| \le \frac{A_2}{8}, \end{aligned}$$

(3.37)

with $0<\sigma <\min \{ \sigma _2, \sigma _0(\delta )$ for any $0<\delta \le \delta _0$. Combining (3.28) and (3.37) we deduce $\left| {-}\!\int _{\Omega _0}\frac{z^p}{\overline{z}^\mu }-{-}\!\int _{\Omega _0}\frac{z_0^p}{\overline{z}_0^\mu }\right| \le \frac{3 A_2}{8}, \;\text{ for }\; 0<\sigma<\min \{ \sigma _2, \sigma _0(\delta )\}\;\text{ and }\; 0<\delta \le \delta _0,$ and thus, we finally obtain

$$\begin{aligned} \frac{5A_2}{8}\le {-}\!\int _{\Omega _0}\frac{z^p}{\overline{z}^{\mu }} \le \frac{11 A_1}{8}, \quad \text {for} \quad 0<\sigma<\min \{ \sigma _2, \sigma _0(\delta )\}, \ 0<\delta \le \delta _0, \end{aligned}$$

(3.38)

taking also into consideration $A_2\le {-}\!\int _{\Omega _0}\frac{z_0^p}{{\bar{z}}_0^\mu }\le A_1.$ Consequently, if we consider $\sigma _0(\delta )\le \sigma _2$, then it follows that $\frac{1}{2}A_2{\bar{z}}^\mu<\frac{5}{8}A_2{\bar{z}}^\mu \le {-}\!\int _{\Omega _0}z^p\le \frac{11}{8}A_1{\bar{z}}^\mu <2A_1{\bar{z}}^\mu$, for $0<\sigma <\sigma _0(\delta )$, and thus a continuity argument implies that $\frac{1}{2}A_2{\bar{z}}^\mu \le {-}\!\int _{\Omega _0}z^p\le 2A_1{\bar{z}}^\mu$, with $0<\sigma <\sigma _0(\delta )+\eta$, for some $\eta >0,$ which contradicts the definition of $\sigma _0(\delta )$. Accordingly, we derive that $\sigma _2<\sigma _0(\delta )$ for any $0<\delta \le \delta _0$, and the proof of Proposition 3.1 is complete for $\sigma _0=\sigma _2.$ $\square$

Now, we are ready to proceed with the proof of Theorem 3.1.

Proof of Theorem 3.1

First, note that $\sigma _0\le \sigma _1$ in (3.27), then from (3.10) and (3.24), we have

$$\begin{aligned} \quad K(\sigma ) \ge \frac{m_{\Psi }}{\left( {-}\!\int _{\Omega _0}z^p\right) ^{\frac{r\gamma }{p}}}\ge m_{\Psi }\left( \frac{1}{2\alpha _1}\right) ^{\frac{r\gamma }{p}}\cdot \left( \frac{1}{2{\bar{\psi }}_\delta }\right) ^{\frac{r\gamma }{p}\mu }\lambda ^{-r\gamma }\ge m_{\Psi }\mathrm{d}\lambda ^{-r\gamma }\equiv D, \nonumber \\ \end{aligned}$$

(3.39)

for $0<\sigma <\min \{ \sigma _0, \Sigma _\delta \}.$ Note also that for $0<\lambda \le \lambda _0(d)$, then inequality (3.11) entails

$$\begin{aligned} \Delta u_0+Du_0^p\ge 2u_0^p \end{aligned}$$

(3.40)

for any $0<\delta \le \delta _0$. The comparison principle in conjunction with (3.39) and (3.40) then yields

$$\begin{aligned} z\ge {\tilde{z}} \quad \text{ in }\quad Q_0\equiv \Omega _0\times (0, \min \{ \sigma _0, \Sigma _\delta \}), \end{aligned}$$

(3.41)

where ${{\tilde{z}}}={{\tilde{z}}}(x,t)$ solves the following partial differential equation

$$\begin{aligned}&{\tilde{z}}_{\sigma }=\Delta {\tilde{z}}+D{\tilde{z}}^p, \quad \text{ in }\quad Q_0, \end{aligned}$$

(3.42)

$$\begin{aligned}&\frac{\partial {\tilde{z}}}{\partial \nu }=0,\quad \text{ on }\quad \partial \Omega _0\times (0, \min \{ \sigma _0, \Sigma _\delta \}), \end{aligned}$$

(3.43)

$$\begin{aligned}&{\tilde{z}}(|x|,\sigma )=u_0(\vert x\vert )\quad \text{ in }\quad \Omega _0. \end{aligned}$$

(3.44)

Setting $h(x,\sigma ):={\tilde{z}}_{\sigma }(x,\sigma )-{\tilde{z}}^p(x,\sigma ),$ then

$$\begin{aligned} h_{\sigma } = \Delta h+p(p-1) {\tilde{z}}^{p-2} |\nabla {\tilde{z}}|^2+ D p {\tilde{z}}^{p-1}\,h \ge \Delta h+ D p {\tilde{z}}^{p-1}\,h, \quad \text{ in }\quad Q_0, \end{aligned}$$

with

$$\begin{aligned} h(x,0)= & {} \Delta {\tilde{z}}(x,0)+D{\tilde{z}}^p(x,0)\\&-{\tilde{z}}^p(x,0)=\Delta u_0+(D-1)u_0^p\ge u_0^p>0,\quad \text{ in }\quad \Omega _0, \end{aligned}$$

whilst $\frac{\partial h}{\partial \nu }=0\;\text{ on }\;\partial \Omega _0\times (0, \min \{ \sigma _0, \Sigma _\delta \}).$ Therefore, owing to the maximum principle, we derive ${\tilde{z}}_{\sigma }>{\tilde{z}}^p\quad \text{ in }\quad Q_0,$ and thus via integration we obtain

$$\begin{aligned} {\tilde{z}}(0,\sigma )\ge \left( \frac{1}{z_0^{p-1}(0)}-(p-1)\sigma \right) ^{-\frac{1}{p-1}}=\left\{ \left( \frac{\delta ^{a}}{\lambda (1+\frac{a}{2})}\right) ^{p-1}-(p-1)\sigma \right\} ^{-\frac{1}{p-1}} \end{aligned}$$

for $0<\sigma <\min \{\sigma _0, \Sigma _\delta \}$, and therefore,

$$\begin{aligned} \min \{\sigma _0, \Sigma _\delta \}<\frac{1}{p-1}\cdot \left( \frac{\delta ^a}{\lambda (1+\frac{a}{2})}\right) ^{p-1}. \end{aligned}$$

(3.45)

Note that for $0<\delta \ll 1$, the right-hand side on (3.45) is less than $\sigma _0$, so $\Sigma _\delta<\frac{1}{p-1} \left( \frac{\delta ^a}{\lambda (1+\frac{a}{2})}\right) ^{p-1}<+\infty$. $\square$

Remark 3.2

Recalling that $z=\mathrm{e}^{\int ^{\sigma }\Phi (s)\,ds} u$, we also obtain the occurrence of a single-point blow-up for the solution u of problem (3.3)–(3.5).

Remark 3.3

Notably, by (3.45) we conclude that $\Sigma _\delta \rightarrow 0$ as $\delta \rightarrow 0,$ i.e. the more spiky initial data we consider then the faster the diffusion-driven blow-up for z and consequently for u as well.

A diffusion-driven instability (Turing instability) phenomenon, as was first indicated in the seminal paper (Turing 1952), is often followed by pattern formation. A similar situation is observed as a consequence of the driven-diffusion finite-time blow-up provided by Theorem 3.1, and it is described below. The blow-up rate of the solution u of (3.3)–(3.5) and the blow-up pattern (profile) identifying the formed pattern are given.

Theorem 3.2

Take $N\ge 3,\;\max \{r, \frac{N}{N-2}\}<p<\frac{N+2}{N-2}$ and $\frac{2}{N}<\frac{p-1}{r}<\gamma .$ Assume that both $\Phi (\sigma )$ and $\Psi (\sigma )$ are positive and bounded. Then, the blow-up rate of the solution of (3.3)–(3.5) can be characterized as follows

$$\begin{aligned} \Vert u(\cdot , \sigma )\Vert _\infty \ \approx \ (\Sigma _{\max }-\sigma )^{-\frac{1}{p-1}}, \quad t\uparrow \Sigma _{\max },\quad \end{aligned}$$

(3.46)

where $\Sigma _{\max }$ stands for the blow-up time.

Proof

We first perceive that by virtue of (3.16) and in view of the Hölder’s inequality, since $p>r,$ the following inequality holds

$$\begin{aligned} 0<K(\sigma )=\frac{\Psi (\sigma )\mathrm{e}^{(1+r\gamma -p)\int ^\sigma \Phi (s)\,\mathrm{d}s}}{\Big (\displaystyle {-}\!\int _{\Omega _0}z^r \Big )^{\gamma }}\le C_1<+\infty . \end{aligned}$$

(3.47)

Define now $\Theta$ satisfying the partial differential equation

$$\begin{aligned}\Theta _{\sigma }=\Delta \Theta +C_1\Theta ^{p},\quad \text{ in }\quad \Omega _0\times (0,\Sigma _{\max }),\end{aligned}$$

with $\frac{\partial \Theta }{\partial \nu }=0,$ on $\partial \Omega _0\times (0,\Sigma _{\max })$, and $\Theta (x,0)=z_0(x)$, in $\Omega _0$, then via comparison $z\le \Theta$ in $\Omega _0\times (0,\Sigma _{\max }).$ Yet it is known, see Quittner and Souplet (2007, Theorem 44.6), that $|\Theta (x,\sigma )|\le C_{\eta }|x|^{-\frac{2}{p-1}-\eta }\quad \text{ for }\quad \eta >0,$ and thus

$$\begin{aligned} |z(x,\sigma )|\le C_{\eta }|x|^{-\frac{2}{p-1}-\eta }, \quad \text{ for }\quad (x,\sigma )\in \Omega _0\times (0, \Sigma _{\max }). \end{aligned}$$

(3.48)

Following the same steps as in the proof of Kavallaris and Suzuki (2017, Theorem 9.1), we derive

$$\begin{aligned} \lim _{\sigma \rightarrow \Sigma _{\max }} K(\sigma )=\omega \in (0,+\infty ). \end{aligned}$$

(3.49)

By virtue of (3.49) and applying Quittner and Souplet (2007, Theorem 44.3(ii)), we can find a constant $C_{U}>0$ such that

$$\begin{aligned} \left| \left| z(\cdot ,\sigma )\right| \right| _{\infty }\le C_{U}\left( \Sigma _{\max }-\sigma \right) ^{-\frac{1}{(p-1)}}, \quad \text{ in }\quad (0, \Sigma _{\max }). \end{aligned}$$

(3.50)

Setting $N(\sigma ):=\left| \left| z(\cdot ,\sigma )\right| \right| _{\infty }=z(0,\sigma ),$ then $N(\sigma )$ is differentiable for almost every $\sigma \in (0,\Sigma _{\delta }),$ in view of Friedman and McLeod (1985), and $\frac{\mathrm{d}N}{\mathrm{d}\sigma }\le K(\sigma ) N^p(\sigma ).$ Notably, $K(\sigma )\in C([0,\Sigma _{\max }))$ and owing to (3.47) it is bounded in any time interval $[0,\sigma ],\; \sigma <\Sigma _{\max };$ then, upon integration we obtain

$$\begin{aligned} \left| \left| z(\cdot ,\sigma )\right| \right| _{\infty }\ge C_L\left( \Sigma _{\max }-\sigma \right) ^{-\frac{1}{(p-1)}}, \quad \text{ in }\quad (0, \Sigma _{\max }), \end{aligned}$$

(3.51)

for some positive constant $C_L.$

Recalling that $z(x,\sigma )=\mathrm{e}^{\int ^{\sigma }\Phi (s)\,ds} u(x,\sigma )$ then (3.50) and (3.51) entail

$$\begin{aligned} {\widetilde{C}}_L\left( \Sigma _{\max }-\sigma \right) ^{-\frac{1}{(p-1)}}\le \left| \left| u(\cdot ,\sigma )\right| \right| _{\infty }\le {\widetilde{C}}_U\left( \Sigma _{\max }-t\right) ^{-\frac{1}{(p-1)}}, \quad \text{ for }\quad \sigma \in (0, \Sigma _{\max }), \end{aligned}$$

where now ${\widetilde{C}}_L$, and ${\widetilde{C}}_U$ depend on $\Sigma _{\max },$ and thus (3.46) is proved. $\square$

Remark 3.4

We first note that (3.48) provides a rough form of the blow-up pattern for z and thus for u as well. Additionally, owing to (3.47) the non-local problem (3.12)–(3.14) can be treated as a local one for which the more accurate asymptotic blow-up profile, Duong et al. (Duong et al. 2020) and Merle and Zaag (Merle and Zaag 1998), is known and is given by $\lim _{\sigma \rightarrow \Sigma _{\max }}z(|x|,\sigma )\sim C\left[ \frac{|\log |x||}{|x|^2}\right] , \;\text{ for }\; |x|\ll 1,\;\text{ and }\; C>0.$ Using again the relation between z and u, we end up with a similar asymptotic blow-up profile for the diffusion-driven-induced blow-up solution u of problem (3.3)–(3.5). This blow-up profile actually determines the form of the developed patterns which are induced as a result of the diffusion-driven instability, and it is numerically investigated in the next section.

4 Numerical Experiments

To confirm and illustrate some of the theoretical results of the previous sections, we perform a series of numerical experiments for which we solve the involved PDE problems using the finite element method (Johnson 1987), using piecewise linear basis functions and implemented using the adaptive finite-element toolbox ALBERTA (Schmidt and Siebert 2005). In all our simulations (unless stated otherwise), the domain was triangulated using 16384 elements, the discretisation in time was done using the forward Euler method taking $5\times 10^{-4}$ as time-step and the resulting linear system solved using Generalized Minimal Residual iterative solver (Saad 2003).

4.1 Experiment 1

We take an initial condition $u_0$ and a set of parameters satisfying the assumptions of Theorem 2.1. Then, solve (1.32)–(1.34) on $\Omega _0=\left[ -1, 1 \right] ^2$ with initial condition of the form

$$\begin{aligned} u_0(\mathbf{x},0)={\left\{ \begin{array}{ll} -8 ||\mathbf{x}||^2+3, &{} ||\mathbf{x}|| <\frac{1}{2},\\ 1, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(4.1)

As for the domain evolution, we consider four different cases:

• $\rho (t)=\mathrm{e}^{\beta t}$ (exponentially growing domain);

• $\rho (t)=\mathrm{e}^{-\beta t}$ (exponentially decaying domain);

• $\rho (t)=\frac{\mathrm{e}^{\beta t}}{1+\frac{1}{m}\left( \mathrm{e}^{\beta t}-1\right) }$ (logistically growing domain);

• $\rho (t)=1$ (static domain).

We summarise all parameters used in Table 1. In Fig. 1, we demonstrate the $||u(x,t)||_\infty$ for each of the domain evolutions, so we can monitor their respective blow-up times.

Table 1 Set of parameters used in Experiment 1

Fig. 1

Plots representing $||u(x,t)||_\infty$, where u(x, t) is the numerical solution of (1.32)–(1.34) for different domain evolutions: static, exponentially decaying and growing, and logistically growing domains, starting from the initial condition (4.1) in $\Omega _0=\left[ -1, 1\right] ^2$. Parameters are shown in Table 1 and satisfy conditions of Theorem 2.1. (Colour version online)

If we denote by $\Sigma _g$, $\Sigma _d$, $\Sigma _{lg}$ and $\Sigma _s$ the blow-up times for the case of exponentially growing and decaying, the logistically growing domains and the static domain, respectively, we observe from Fig. 1 that we have the following ordering $\Sigma _g>\Sigma _{lg}>\Sigma _s>\Sigma _d,$ which is in agreement with the mathematical intuition, but it cannot be derived by our analytical results cf. Remark 2.4.

We now take the same initial condition, $u_0$ and the same initial domain which we assume is evolving exponentially and consider parameters $D_1=1$, $p=1.4$, $q=1$, $r=1$ and $s=2$ for which inequality (2.24) of Remark 2.6 holds. As we can see in Fig. 2, we have an example of a solution u for which its mean value ${\bar{u}}$ does not blow up, as already conjectured in the aforementioned remark. Hence, this numerical experiment predicts a very interesting phenomenon both mathematically and biologically which has been conjectured but not proven by Theorem 2.1. It predicts the infinite-time quenching of the solution of problem (1.32)–(1.34), and thus, the extinction of the activator in the long run, see also Remark 2.3. It must also be noted that this result is not in contradiction with Proposition 2.2, where infinite-time quenching is ruled out since condition (2.1) is not satisfied for an exponentially growing domain where $\Phi (\sigma )$ is an unbounded function as indicated in Remark 2.4.

Fig. 2

The plot of $||{\bar{u}}||_\infty$ resulting from the numerical solution of (1.32)–(1.34) starting from the initial condition (4.1) in $\Omega _0=\left[ -1, 1\right] ^2$, evolving exponentially, considering parameters $D_1=1$, $\beta =0.1$, $p=1.4$, $q=1$, $r=1$ and $s=2$. (Colour version online)

4.2 Experiment 2

This experiment is meant to illustrate Theorem 2.3, and we take as initial data $u_0=\cos (\pi y)+2$ and take $\Omega _0$ as the unit square when numerically solving equations (1.32)–(1.34). As for domain evolution, we consider $\rho (t)=\mathrm{e}^{\beta t}$, with $\beta =0.1$. To proceed, we consider two sets of parameters, one for which assumptions of Theorem 2.3 are satisfied and another for which those assumptions are not fulfilled. See Table 2 for model parameters.

Table 2 Set of parameters used in Experiment 2

Fig. 3

The plot of $||u(x,t)||_\infty$, where u(x, t) is the numerical solution of (1.32)–(1.34). Initial condition is $u_0=\cos (\pi y)+2$, and $\Omega _0$ is the unit square evolving according to exponential growth ($\beta =0.1$). (Colour version online)

Results shown in Fig. 3 are in agreement with theoretical predictions of Theorem 2.3 since the solutions exist for all times when the assumption of the theorem is met (Fig. 3a), otherwise, a finite-time blow-up is exhibited to occur (Fig. 3b).

4.3 Experiment 3

In this experiment, we intend to illustrate Theorem 3.1, so we numerically solve (1.32)–(1.34) in $\mathbb {R}^3$, taking $\Omega _0$ as the unit sphere and initial condition $u_0$ given by (3.1), considering $\delta =0.8$ and $\lambda =0.1.$ As for other parameters, we choose $D_1=1$, $p=4$, $q=4$, $r=2$ and $s=1$, which satisfy the conditions of the theorem. In Fig. 4, we display the $L^\infty -$norm of the solution u for three types of evolution laws implemented, namely: exponential decay, logistic decay and no evolution. For the exponential and logistic decay, we select the same set of parameters as used in Experiment 1. As we can observe, for all the cases the solution blows up, as theoretically predicted by Theorem 3.1. Again the blow-up times satisfy the ordering $\Sigma _s>\Sigma _{ls}>\Sigma _d,$ where $\Sigma _{ls}$ stands for the blow-up time for the logistic decay evolution, being in agreement with the mathematical intuition. Such an ordering, again, cannot be obtained via the theoretical result of Theorem  3.1.

Fig. 4

Plots for $||u(x,t)||_\infty$, where u(x, t) is the numerical solution of (1.32)–(1.34), in $\mathbb {R}^3$, considering $\Omega _0$ the unit sphere. Three evolution laws considered: exponential decay, logistic decay and no evolution (static domain). Parameters used: $p=4$, $q=4$, $r=2$, $s=1$ and initial condition given by (3.1) taking $\delta =0.8$ and $\lambda =0.1$. (Colour version online)

In Fig. 5a, b, we compare the initial solution with the solution at $t=0.03$, respectively, for the logistic decay, close to the blow-up time $t=0.03$, by looking at a cross section of the three-dimensional unit sphere $\Omega _0.$ Besides, in Fig. 5c, d again the solution at section cross of $\Omega _0$ is depicted but now for the stationary and exponential decaying case, respectively. Through this experiment, we can observe the formation of blow-up (Turing-instability) patterns around the origin $R=0.$ We actually conclude that the evolution of the domain has no impact on the form of blow-up patterns; however, it certainly affects the spreading of Turing-instability patterns as it is obvious from Fig. 5b, c, d.

Fig. 5

Numerical solution of experiment 3. The above figures show the blow-up (Turing-instability) patterns on a cross section of the three-dimensional sphere $\Omega _0$: a initial profile of solution for logistic growth; b blow-up pattern for logistic growth at $t=0.03$; c blow-up pattern for stationary case at $t=0.07$; and, d blow-up pattern for exponential decay at $t=0.03$.(Colour version online)

Notably Theorem 3.1 holds only to $N\ge 3$; however, we have numerically tested the occurrence of blow-up predicted by that theorem also for $N=2$, taking $\Omega$ as the unit circle and the same parameters used in Experiment 3. It is then numerically verified the exhibition of finite-time blow-up. The numerical results are displayed, in Fig. 6 where the $L^\infty$-norm of the solution u for the same three types of evolution laws is depicted for the $N=3$ case. The initial condition used is displayed in Fig. 7.

Fig. 6

Plots for $||u(x,t)||_\infty$, where u(x, t) is the numerical solution of (1.32)–(1.34), in $\mathbb {R}^2$, considering $\Omega _0$ the unit circle. Three domain evolution laws considered: exponential decay, logistic decay and no evolution (static domain). Parameters used: $p=4$, $q=4$, $r=2$, $s=1$ and initial condition given by (3.1) taking $\delta =0.8$ and $\lambda =0.1$. (Colour version online)

Fig. 7

Initial condition used for Experiment 3 when $N=2$, $\Omega _0$ is the unit circle and initial condition is given by (3.1) taking $\delta =0.8$ and $\lambda =0.1$. (Colour version online)

Fig. 8

Plots of the $L_\infty$ norm for the numerical solutions of (1.18)–(1.19) and (1.32)–(1.34). The initial condition is prescribed as $u_0={\hat{u}}_0=\cos (\pi y)+2$ where $\Omega _0$ is an exponentially decaying unit square with $\beta =0.1$. (Colour version online)

4.4 Experiment 4

Next, we design a numerical experiment to compare the dynamics of the reaction–diffusion system (1.18)–(1.19) with that of the non-local problem (1.32)–(1.34) under the assumptions of Theorem 2.1. To this end, we perform an experiment considering $u_0={\hat{u}}_0=\cos (\pi y)+2$, $\Omega _0=\left[ 0, 1\right] ^2$, $p=3$, $q=2$, $r=1$ and $s=2$. For the reaction–diffusion system (1.18)–(1.19), we also take in addition $D_1=0.01$, $D_2=1$, $\tau =0.01$ and $v_0=2$ whilst for (1.32)–(1.34) we only choose $D_1=0.01.$ For both cases, we consider an exponential decaying evolution of the domain, with $\beta =0.1$. Unlike previous numerical examples, here the domain was triangulated using 786432 elements and a timestep $10^{-4}$ was taken.

The obtained results are displayed in Fig. 8, and they demonstrate that the reaction–diffusion system (1.18)–(1.19), and the non-local problem (1.32)–(1.34) share the same long time dynamics. In particular, the solutions of both problems exhibit blow-up which takes place in finite time. The latter, biologically speaking, means that in the examined case we just need to monitor only the dynamics of the activator, whose dynamics are governed by the non-local problem (1.32)–(1.34). Therefore, we can get an insight regarding the interaction between both of the chemical reactants (activator and inhibitor) provided by the reaction–diffusion system (1.18)–(1.19).