1 Introduction

Understanding chemotaxis-driven dynamics of population distributions far from spatial homogeneity is forming a recurrent theme in several disciplines from biology and especially in corresponding theoretical studies [30]. Stimulated by findings on spontaneous formation of aggregates and, more generally, on various facets of complex collective behavior in chemotactically migrating populations when distributed in a strongly heterogeneous manner, substantial efforts in mathematical analysis have led to considerable insight into numerous qualitative aspects related to the destabilizing potential of taxis-type cross-diffusion, as constituting the apparently most characteristic common ingredient not only in the celebrated Keller–Segel model for such processes [18], in its simplest form reducing to the parabolic system

$$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta u - \nabla \cdot (u\nabla v), \\ v_t=\Delta v-v+u, \end{array} \right. \end{aligned}$$
(1.1)

for the population density \(u=u(x,t)\) and the chemoattractant concentration \(v=v(x,t)\), but beyond this also in a large class of relatives thereof [15, 30]. Here the knowledge seems most comprehensive with regard to the question under which circumstances solutions may develop singularities either in finite or infinite time. In the context of (1.1), for instance, it is known that such unboundedness phenomena do occur in spatially two- or higher-dimensional settings [14, 42], but that any such explosion is ruled out in the associated one-dimensional version [29]; in fact, a large literature has revealed remarkably detailed information on corresponding dichotomies in several refinements and derivates of (1.1) [8, 12, 23, 27, 36, 40, 46], in its most delicate part, namely arguments concerned with blowup detection, typically relying on certain simplifications in which the chemoattractant evolution is governed by an elliptic equation [3, 16, 22, 26, 37, 47].

In comparison to this, much less seems known with regard to the question how chemotactic evolution copes with locally extreme population concentrations in the sense of singularities which are supposedly present at some instant. Here adaptation and refinement of straightforward functional analytical approaches, e.g., based on transformation to integral equations via Duhamel formulae, and on design of suitable fixed point frameworks, have made it possible to construct certain generalized local-in-time solutions, instantaneously becoming smooth and classical, for initial data with quite low integrability properties. In the prototypical setup of the Neumann problem for (1.1) in bounded domains \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 2\), for instance, immediate regularization in this sense has been shown to occurs for all initial data \((u_0,v_0)\) from \(L^1(\varOmega )\times W^{1,n}(\varOmega )\), and actually even for slightly larger classes involving either Radon measure-type distributions in the first component when \(n=2\), or suitable Morrey spaces in both components when \(n\ge 3\) [4, 5, 33]; requirements of this flavor apparently cannot be substantially relaxed even when beyond this, more subtle use is made of a priori estimates implied from the well-known gradient-flow structure of the particular cross-diffusive interaction in (1.1) [6, 7]. This is further supported by observations indicating absence of smoothing in presence of initial data close to the above spaces: In some two-dimensional Keller–Segel systems, namely, some finite-mass but measure-valued cell distributions are known to evolve into persistent measure-type singularities ([13, 24]; cf. also [2, 34]); in the case \(n\ge 3\), the stationary singularities becoming manifest in the so-called Chandrasekhar solutions to a simplified parabolic–elliptic variant of (1.1), as determined by \(u(x,t):=\frac{2(n-2)}{|x|^2}\), \(x\ne 0\), even provide explicit examples that strongly indicate criticality of spatial \(L^\frac{n}{2}\) spaces as regards the possibility of instantaneous regularization.

1.1 Main results: detecting additional regularization effects of logistic dampening

The purpose of the present study is to investigate how far the evolution of singular structures may be affected by the presence of logistic-type growth restrictions. In fact, a large literature suggests that accordingly obtained logistic Keller–Segel systems can be viewed as a natural first step to adapt the prototypical and hence quite simple model (1.1) so as to account for mechanisms of competition-induced overcrowding prevention which seem virtually ubiquitous in numerous situations of biological relevance (see, e.g., [11, 35, 39] or [32] for some examples, or also [15, 30] for a broader overview). In fact, several precedents have been indicating that on the way from (1.1) to fully realistic models, despite their increased complexity and especially their apparent lack of appropriate energy structures, such logistic chemotaxis systems remain accessible to a considerably large variety of mathematical tools; accordingly, not only quite comprehensive conclusions are available in the fields of global classical solvability for smooth data [28, 38, 41, 48], of constructing attractors [1, 28, 29], and of proving asymptotic homogenization in cases of strong dampening [9, 43] (cf. also [21]), but beyond this even some facets of the rich dynamics of such systems, as reported by numerical findings [31], could be captured by some rigorous results on emergence of extremely large cell densities at possibly intermediate time scales [17, 20, 44, 45].

In contract to this, the knowledge with regard to the smoothing potential of logistic death terms when confronted with locally large distributions seems quite thin; after all, the apparently furthest reaching explicitly formulated results in this respect, addressing cases of \(L^2\) initial values in two-dimensional domains [28], seem to allow extension, through adaptation of the arguments, e.g., in [5], at least to \(L^\frac{n}{2}\) data in general n-dimensional settings. To our impression, however, covering classes of initial data substantially larger than those known to be admissible already for (1.1) has nowhere been achieved so far. Our main results in this direction now identify a genuine supplementary smoothing action of quadratic degradation, as forming the essential additional ingredient in logistic Keller–Segel systems when compared to (1.1). Indeed, we shall see that in such systems instantaneous smoothing may in fact occur for initial data exhibiting singular behavior much more extreme than power-type, even up to some exponential strength, and thus far below levels of integrability.

To make this more precise, let us resort to a setting as simple as possible but yet potentially capturing the main characteristics of the problem context under consideration, and hence consider radially symmetric solutions to the parabolic–elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} u_t = \Delta u - \chi \nabla \cdot (u\nabla v) + \kappa u - \mu u^2, &{}\quad x\in \varOmega , \quad t>0, \\ 0 = \Delta v - m(t) + u, &{}\quad x\in \varOmega , \quad t>0, \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0, &{} \quad x\in \partial \varOmega , \quad t>0, \\ u(\cdot ,0)=u_0(x), &{}\quad x\in \varOmega , \end{array} \right. \end{aligned}$$
(1.2)

in the ball \(\varOmega =B_R(0)\subset {\mathbb {R}}^n\) with \(n\ge 1\) and \(R>0\), where \(\chi >0\), \(\kappa \in {\mathbb {R}}\), \(\mu >0\) and

$$\begin{aligned} m(t) := \frac{1}{|\varOmega |} \int _\varOmega u(x,t) \mathrm{d}x, \qquad t>0. \end{aligned}$$
(1.3)

Our standing assumption on the initial data will be that

$$\begin{aligned} \left\{ \begin{array}{ll} u_0\in C^1(\overline{\varOmega }{\setminus } \{0\}) &{}\quad \text{ is } \text{ nonnegative } \text{ and } \text{ radially } \text{ symmetric } \\ &{}\quad \text{ and } \text{ nonincreasing } \text{ with } \text{ respect } \text{ to } |x|\in (0,R], \end{array} \right. \end{aligned}$$
(1.4)

and in most places we will furthermore suppose that

$$\begin{aligned} u_0(r) \le K\phi (r) \qquad \text{ for } \text{ all } r\in (0,R), \end{aligned}$$
(1.5)

where

$$\begin{aligned} \phi \in C^2((0,\infty )) \text{ is } \text{ such } \text{ that }\,\phi \ge 1\quad \hbox {and}\quad \phi '<0,\,\hbox {and that}\,\phi (r)\nearrow +\infty \,\hbox {as}\, r\searrow 0. \end{aligned}$$
(1.6)

Here and throughout the sequel, whenever this appears convenient and without any risk to cause confusion, we shall tacitly switch to the usual notation using radial variables, thus ad lib replacing, e.g., \(u_0(x)\) with \(u_0(r)\) for \(r=|x|\).

Now our main results will require that the singular behavior of \(\phi \), and hence of \(u_0\), can be controlled in such a way that besides the technical but otherwise comparatively mild hypotheses that

$$\begin{aligned} \phi ''(r) \le K_1 \phi ^2(r) \qquad \text{ for } \text{ all } r\in (0,1) \end{aligned}$$
(1.7)

and that

$$\begin{aligned} \phi (2r) \le K_2 r^n \phi (r) \qquad \text{ for } \text{ all } r\in (0,1), \end{aligned}$$
(1.8)

the crucial logarithmic integrability condition

$$\begin{aligned} \int _0^1 r^{n-1} \ln \phi (r) \mathrm{d}r < \infty \end{aligned}$$
(1.9)

is satisfied. In this general framework, we shall obtain the following statement on global existence of solutions which immediately become smooth and classical, and which attain the prescribed and possibly singular initial trace locally uniformly outside the spatial origin:

Theorem 1.1

Let \(\varOmega =B_R(0)\subset {\mathbb {R}}^n\) with some \(n\ge 1\) and \(R>0\), and let \(\kappa \in {\mathbb {R}}, \chi >0\) and \(\mu >0\) be such that

$$\begin{aligned} \mu >\chi . \end{aligned}$$
(1.10)

Moreover, assume that \(u_0\) satisfies (1.4) and (1.5) with some \(K>0\) and some function \(\phi :(0,\infty )\rightarrow {\mathbb {R}}\) for which there exist \(K_1>0\) and \(K_2>0\) such that (1.6)–(1.9) hold. Then one can find a pair (u, v) of nonnegative radially symmetric functions

$$\begin{aligned} \left\{ \begin{array}{l} u\in C^0((\overline{\varOmega }{\setminus } \{0\})\times [0,\infty )) \cap C^{2,1}(\overline{\varOmega }\times (0,\infty )) \qquad \text{ and } \\ v\in C^{2,0}(\overline{\varOmega }\times (0,\infty )) \end{array} \right. \end{aligned}$$
(1.11)

which solves the boundary value problem in (1.2) in the classical sense in \(\overline{\varOmega }\times (0,\infty )\), for which the total mass functional enjoys the integrability property

$$\begin{aligned} \int _0^{\mathrm{T}} \int _\varOmega u(x,t) \mathrm{d}x < \infty \qquad \text{ for } \text{ all } T>0, \end{aligned}$$
(1.12)

and which is such that furthermore

$$\begin{aligned} u(\cdot ,t) \rightarrow u_0 \quad \text{ in } C^0_{\mathrm{loc}}(\overline{\varOmega }{\setminus } \{0\}) \qquad \text{ as } t\searrow 0. \end{aligned}$$
(1.13)

When specified to the particular choice of functions \(\phi \) representing exponential singularities, this directly implies the following concrete consequence.

Corollary 1.2

Let \(n\ge 1, R>0\) and \(\varOmega =B_R(0)\subset {\mathbb {R}}^n\), let \(\kappa \in {\mathbb {R}}, \chi >0\) and \(\mu >\chi \), and suppose that \(u_0\) satisfies (1.4) as well as

$$\begin{aligned} u_0(r) \le K e^{\lambda r^{-\alpha }} \qquad \text{ for } \text{ all } r\in (0,R) \end{aligned}$$
(1.14)

with some positive constants K, \(\lambda \) and \(\alpha \) such that

$$\begin{aligned} \alpha <n. \end{aligned}$$
(1.15)

Then there exists a pair (u, v) of functions for which the conclusion from Theorem 1.1 holds.

1.2 Main ideas

As thanks to the comparatively simple parabolic–elliptic structure of (1.2), assumption (1.10) can be seen to entail a pointwise upper bound for \(\frac{t}{t+1} u\) by means of a comparison argument (Lemma 2.2), constructing a smooth limit in \(\overline{\varOmega }\times (0,\infty )\) of solutions to an appropriately regularized problem can be achieved by quite well-established bootstrap and compactness reasonings (Lemmas 2.6, 2.7). The main challenge will thus consist in asserting the claimed behavior of this solution to the boundary value problem in (1.2), and it turns out that for this it is not only sufficient but also essentially necessary (see Proposition 3.1) to control the asymptotic behavior of the nonlocal ingredient m(t) in (1.2), and hence of the total mass functional \(\int _\varOmega u(x,t)\mathrm{d}x\), in the sense of time integrability near \(t=0\) as in (1.12). This will be achieved on the basis of a second comparison argument, which will rely on the fact that thanks to (1.7) large multiples of \(\phi \) become stationary supersolutions to an equivalent version of the first equation in (1.2) when restricted to radially nonincreasing functions u and v [see Lemma 3.2 and (2.8)], which forms the major motivation for the initial monotonicity, as assumed in (1.4) and fortunately inherited not only by \(u(\cdot ,t)\) but also by \(v(\cdot ,t)\) for \(t>0\) (Lemmas 2.4, 2.5).

Since (1.8) and (1.9) guarantee that a combination of the above two pointwise inequalities for u entails an estimate of the mass functional against an integrable function of time (Lemmas 3.3, 3.4), through the second equation in (1.2) thus having at hand suitable bounds for \(\nabla v\), yet \(L^1\) with respect to time, we will firstly conclude that the initial trace is attained in the topology of \(W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \) for each \(\delta \in (0,R)\) (Lemma 3.7). Secondly, this \(L^1\) information on \(\nabla v\) will enable us to derive precompactness in \(C^0_{\mathrm{loc}}(\overline{\varOmega }{\setminus }\{0\})\) of \((u(\cdot ,t))_{t\in (0,1)}\) and hence imply the desired convergence property (1.13), through a two-step regularity argument (Lemmas 3.8, 3.9).

2 Approximate problems and their limit behavior for \(t>0\)

2.1 A family of approximate problems and a fundamental pointwise estimate

Let us introduce a convenient regularization of (1.2) through approximation of a given and possibly nonsmooth initial function \(u_0\) by a family \((u_{0\varepsilon })_{\varepsilon \in (0,1)}\) of suitably smooth functions on \(\overline{\varOmega }\) in the sense that

$$\begin{aligned} 0\not \equiv u_{0\varepsilon } \in C^2(\overline{\varOmega }) \quad \text{ is } \text{ radially } \text{ symmetric } \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$
(2.1)

that, when expressed in the variable \(r=|x|\in [0,R]\), their gradients \(u_{0\varepsilon r}\) satisfy

$$\begin{aligned} u_{0\varepsilon r} (r) \le 0 \qquad \text{ for } \text{ all }\,r\in [0,R]\,\hbox {and} \varepsilon \in (0,1) \end{aligned}$$
(2.2)

as well as the compatibility condition

$$\begin{aligned} u_{0\varepsilon r} (R)=0 \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$
(2.3)

and that \(u_{0\varepsilon }\) approaches \(u_0\) in the sense that

$$\begin{aligned} u_{0\varepsilon } \le u_0+1 \qquad \text{ in }\,\overline{\varOmega }{\setminus } \{0\}\,\hbox {for all}\, \varepsilon \in (0,1) \end{aligned}$$
(2.4)

and

$$\begin{aligned} u_{0\varepsilon } \rightarrow u_0 \quad \text{ in } C^1_{\mathrm{loc}}(\overline{\varOmega }{\setminus } \{0\}) \qquad \text{ as } \varepsilon \searrow 0. \end{aligned}$$
(2.5)

Then for each \(\varepsilon \in (0,1)\), well-established arguments [25, 47] assert local solvability of the approximate version of (1.2) given by

$$\begin{aligned} \left\{ \begin{array}{ll} u_{\varepsilon t} = \Delta u_\varepsilon - \chi \nabla \cdot (u_\varepsilon \nabla v_\varepsilon ) + \kappa u_\varepsilon - \mu u_\varepsilon ^2 &{}\quad x\in \varOmega , \quad t>0, \\ 0 = \Delta v_\varepsilon - m_\varepsilon (t) + u_\varepsilon , \quad m_\varepsilon (t):=\frac{1}{|\varOmega |} \int _\varOmega u_\varepsilon (\cdot ,t), &{}\quad x\in \varOmega , \quad t>0, \\ \frac{\partial u_\varepsilon }{\partial \nu }=\frac{\partial v_\varepsilon }{\partial \nu }=0, &{}\quad x\in \partial \varOmega , \quad t>0, \\ u_\varepsilon (x,0)=u_{0\varepsilon }(x), &{}\quad x\in \varOmega , \end{array} \right. \end{aligned}$$
(2.6)

in the following sense:

Lemma 2.1

Let \(\chi >0, \kappa \in {\mathbb {R}}\) and \(\mu >0\), and suppose that \(u_0\) and \(u_{0\varepsilon }\) satisfy (1.4) and (2.1)–(2.5). Then for all \(\varepsilon \in (0,1)\) there exist \(T_{\mathrm{max},\varepsilon }\in (0,\infty ]\) and a uniquely determined classical solution \((u_\varepsilon ,v_\varepsilon ) \in (C^0(\overline{\varOmega }\times [0,\infty )) \cap C^{2,1}(\overline{\varOmega }\times (0,\infty )))^2\) such that \(u_\varepsilon (\cdot ,t)\) and \(v_\varepsilon (\cdot ,t)\) are positive and radially symmetric in \(\overline{\varOmega }\) for all \(t>0\), and that

$$\begin{aligned} \text{ if } T_{\mathrm{max},\varepsilon }<\infty \text{, } \text{ then } \quad \limsup _{t\nearrow T} \Vert u_\varepsilon (\cdot ,t)\Vert _{L^\infty (\varOmega )}=\infty . \end{aligned}$$
(2.7)

For later reference in several places, let us observe on combining the first two equations in (2.6) that \(u_\varepsilon \) actually solves the Neumann problem for the scalar parabolic equation

$$\begin{aligned} u_{\varepsilon t} = \Delta u_\varepsilon - \chi \nabla u_\varepsilon \cdot \nabla v_\varepsilon - \chi m_\varepsilon (t) u_\varepsilon + \kappa u_\varepsilon - (\mu -\chi ) u_\varepsilon ^2, \qquad x\in \varOmega , \quad t\in (0,T_{\mathrm{max},\varepsilon }), \end{aligned}$$
(2.8)

with the coefficient function \(\nabla v_\varepsilon \) in fact forming a nonlocal ingredient due to its dependence on \(u_\varepsilon \) through the second equation in (2.6).

Here with regard to our overall goal of achieving regularization, the rightmost summand appears to be favorable if \(\chi \) and \(\mu \) comply with the hypotheses from Theorem 1.1. Throughout the sequel, we shall accordingly assume that

$$\begin{aligned} \mu >\chi , \end{aligned}$$

and then may draw a first but substantial conclusion thereof through parabolic comparison with spatially flat functions as follows.

Lemma 2.2

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that (1.4) holds. Then for each \(\varepsilon \in (0,1)\), the solution \((u_\varepsilon ,v_\varepsilon )\) of (2.6) is global in time and satisfies

$$\begin{aligned} u_\varepsilon (x,t) \le \frac{1}{(\mu -\chi )t} + \frac{\kappa _+}{\mu -\chi } \qquad \text{ for } \text{ all }\,x\in \varOmega \,\hbox {and}\,t>0. \end{aligned}$$
(2.9)

Proof

For fixed \(\varepsilon \in (0,1)\) we may use the regularity statement in Lemma 2.1 to pick \(c_1(\varepsilon )>0\) such that

$$\begin{aligned} u_\varepsilon (x,t) \le c_1(\varepsilon ) \qquad \text{ for } \text{ all }\,x\in \varOmega \,\hbox {and}\, t\in (0,T_\varepsilon ), \end{aligned}$$
(2.10)

where \(T_\varepsilon :=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\). Then given \(\tau _0>0\), for any such \(\varepsilon \) we can fix \(\tau =\tau (\varepsilon ,\tau _0)>0\) such that \(\tau \le \tau _0\) and \(\tau <T_\varepsilon \) as well as

$$\begin{aligned} \tau \le \frac{1}{(\mu -\chi ) c_1(\varepsilon )}, \end{aligned}$$
(2.11)

because \(\mu >\chi \). Introducing

$$\begin{aligned} \overline{u}(x,t):=\frac{1}{(\mu -\chi )t} + \frac{\kappa _+}{\mu -\chi }, \qquad x\in \overline{\varOmega }, \quad t\ge \tau , \end{aligned}$$
(2.12)

from (2.10) and (2.11) we therefore obtain that

$$\begin{aligned} \overline{u}(x,\tau ) \ge \frac{1}{(\mu -\chi )\tau } \ge c_1(\varepsilon ) \ge u_\varepsilon (x,\tau ) \qquad \text{ for } \text{ all } x\in \varOmega , \end{aligned}$$

and clearly \(\frac{\partial \overline{u}(x,t)}{\partial \nu }=0=\frac{\partial u_\varepsilon (x,t)}{\partial \nu }\) for all \(x\in \partial \varOmega \) and \(t\in (\tau ,T_{\mathrm{max},\varepsilon })\). Furthermore, differentiating in (2.12) we see that

$$\begin{aligned}&\overline{u}_t - \Delta \overline{u}+ \chi \nabla v_\varepsilon \cdot \nabla \overline{u}+ \chi m_\varepsilon (t) \overline{u}- \kappa \overline{u}+ (\mu -\chi ) \overline{u}^2 \\&\quad = \overline{u}_t + \chi m_\varepsilon (t) \overline{u}- \kappa \overline{u}+ (\mu -\chi )\overline{u}^2 \\&\quad \ge \overline{u}_t - \kappa _+ \overline{u}+ (\mu -\chi )\overline{u}^2 \\&\quad = - \frac{1}{(\mu -\chi )t^2} - \kappa _+ \cdot \Big \{ \frac{1}{(\mu -\chi )t} + \frac{\kappa _+}{\mu -\chi } \Big \} + (\mu -\chi ) \cdot \Big \{ \frac{1}{(\mu -\chi )t} + \frac{\kappa _+}{\mu -\chi } \Big \}^2 \\&\quad = \frac{\kappa _+}{(\mu -\chi )t} \\&\quad \ge 0 \qquad \text{ for } \text{ all } x\in \varOmega \text{ and } t\in (\tau ,T_{\mathrm{max},\varepsilon }), \end{aligned}$$

whence in view of (2.8) the comparison principle applies so as to assert the inequality \(\overline{u}\ge u_\varepsilon \) throughout \(\overline{\varOmega }\times [\tau ,T_{\mathrm{max},\varepsilon }) \supset \overline{\varOmega }\times [\tau _0,T_{\mathrm{max},\varepsilon })\). Firstly fixing, e.g., \(\tau _0:=T_\varepsilon \) here, from the extensibility criterion in Lemma 2.1 we thereby infer that indeed \(T_{\mathrm{max},\varepsilon }=\infty \) for all \(\varepsilon \in (0,1)\), whereupon we secondly achieve (2.9) upon taking \(\tau _0\searrow 0\). \(\square \)

2.2 Downward radial monotonicity of both solution components

Besides the mere radial symmetry, a further indispensable prerequisite for the most essential among our subsequent arguments will be provided by the observation that the initially assumed monotonicity property expressed in (1.4) and (2.2) is inherited by \(u_\varepsilon \) and hence in fact also carries over to \(v_\varepsilon \). Both these statements rely on the following elementary fact.

Lemma 2.3

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and assume (1.4). Then for all \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} r^{n-1} v_{\varepsilon r}(r,t) = \frac{1}{n} m_\varepsilon (t) r^n - \int _0^r \rho ^{n-1} u_\varepsilon (\rho ,t) \mathrm{d}\rho \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0. \end{aligned}$$
(2.13)

Proof

Since when written in radial variables the second equation in (2.6) becomes

$$\begin{aligned} (r^{n-1} v_{\varepsilon r})_r = r^{n-1} m_\varepsilon (t) - r^{n-1} u_\varepsilon (r,t) \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0, \end{aligned}$$

the claim directly results upon integration over (0, r) for \(r\in (0,R)\). \(\square \)

Indeed, this enables us to suitably reduce the differentiated version of (2.8) so as to conclude, again by comparison and again making use of the assumption \(\mu >\chi \), that \(u_\varepsilon \) remains radially nonincreasing.

Lemma 2.4

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and that (1.4) holds. Then for each \(\varepsilon \in (0,1)\),

$$\begin{aligned} u_{\varepsilon r}(r,t) \le 0 \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0. \end{aligned}$$
(2.14)

Proof

We fix \(\varepsilon \in (0,1)\) and \(T>0\) and then choose \(\lambda _\varepsilon >0\) large enough fulfilling

$$\begin{aligned} \lambda _\varepsilon >\kappa _+ + 3\chi \Vert u_\varepsilon \Vert _{L^\infty (\varOmega \times (0,T))}, \end{aligned}$$
(2.15)

and for \(\eta >0\) we let

$$\begin{aligned} z_{\varepsilon \eta }(r,t):=u_{\varepsilon r}(r,t) - \eta e^{\lambda _\varepsilon t}, \qquad r\in [0,R], \quad t\in [0,T]. \end{aligned}$$

Then since due to (2.1) and (2.3) classical parabolic theory [19] ensures that \(u_{\varepsilon r} \in C^0([0,R]\times [0,T]) \cap C^{2,1}([0,R] \times (0,T])\) and that \(u_{\varepsilon r}(0,t)=u_{\varepsilon r}(R,t)=0\) for all \(t\in [0,T]\), it follows that \(z_{\varepsilon \eta }\) belongs to the same space and satisfies

$$\begin{aligned} z_{\varepsilon \eta }(0,t) = z_{\varepsilon \eta }(R,t)=- \eta e^{\lambda _\varepsilon t} < 0 \qquad \text{ for } \text{ all } t\in [0,T] \end{aligned}$$
(2.16)

as well as

$$\begin{aligned} z_{\varepsilon \eta }(r,0)= u_{0\varepsilon r}(r)-\eta \le -\eta <0 \qquad \text{ for } \text{ all } r\in [0,R] \end{aligned}$$
(2.17)

according to (2.2). Moreover, differentiating the identity (2.8) with respect to \(r=|x|\), we see that

$$\begin{aligned} u_{\varepsilon rt}= u_{\varepsilon rrr} + a_\varepsilon (r,t) u_{\varepsilon rr} + b_\varepsilon (r,t) u_{\varepsilon r} \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T), \end{aligned}$$

with

$$\begin{aligned} a_\varepsilon (r,t):=\frac{n-1}{r} - \chi v_{\varepsilon r}, \qquad r\in (0,R), \quad t\in (0,T), \end{aligned}$$

and

$$\begin{aligned} b_\varepsilon (r,t):=-\frac{n-1}{r^2} - \chi v_{\varepsilon rr} - \chi m_\varepsilon (t) + \kappa - 2(\mu -\chi )u_\varepsilon , \qquad r\in (0,R), \quad t\in (0,T), \end{aligned}$$

and that hence

$$\begin{aligned} z_{\varepsilon \eta t}&= z_{\varepsilon \eta rr} + a_\varepsilon (r,t) z_{\varepsilon \eta r} + b_\varepsilon (r,t) z_{\varepsilon \eta } + \Big (b_\varepsilon (r,t)-\lambda _\varepsilon \Big ) \cdot \eta e^{\lambda _\varepsilon t}\nonumber \\&\quad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T). \end{aligned}$$
(2.18)

Here from the second equation in (2.6), we know that

$$\begin{aligned} v_{\varepsilon rr} = m_\varepsilon (t) - u_\varepsilon - \frac{n-1}{r} v_{\varepsilon r} \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T), \end{aligned}$$

and that, by Lemma 2.3,

$$\begin{aligned} v_{\varepsilon r}(r,t)&= \frac{1}{n} m_\varepsilon (t) r - r^{1-n} \int _0^r \rho ^{n-1} u_\varepsilon (\rho ,t) \mathrm{d}\rho \le \frac{1}{n} m_\varepsilon (t) r \\&\quad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T), \end{aligned}$$

whence

$$\begin{aligned} -\chi v_{\varepsilon rr} \le -\chi m_\varepsilon (t) + \chi u_\varepsilon + \frac{(n-1)\chi }{r} \cdot \frac{1}{n} m_\varepsilon (t) r = - \frac{\chi }{n} m_\varepsilon (t) + \chi u_\varepsilon \le \chi u_\varepsilon \end{aligned}$$

for all \(r\in (0,R)\) and \(t\in (0,T)\). Therefore, (2.15) ensures that

$$\begin{aligned}&b_\varepsilon (r,t) \le \chi u_\varepsilon - \chi m_\varepsilon (t) + \kappa - 2(\mu -\chi )u_\varepsilon \le \kappa _+ + 3\chi u_\varepsilon \le \lambda _\varepsilon \\&\quad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T), \end{aligned}$$

and that thus

$$\begin{aligned} z_{\varepsilon \eta t} \le z_{\varepsilon \eta rr} + a_\varepsilon (r,t) z_{\varepsilon \eta r} + b_\varepsilon (r,t) z_{\varepsilon \eta } \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t\in (0,T) \end{aligned}$$

according to (2.18). As a consequence of (2.16) and (2.17), it thus follows from a standard maximum principle argument that in none of the domains \([0,R]\times [0,T_0]\), \(T_0\in (0,T]\), \(z_{\varepsilon \eta }\) can attain its maximal value 0, and that hence \(z_{\varepsilon \eta }\) actually must remain negative throughout \([0,R]\times [0,T]\), which on taking \(\eta \searrow 0\) implies (2.14). \(\square \)

Fortunately, the structure of the equation in (2.6) governing \(v_\varepsilon \), actually reducing to (2.13), is simple enough so as to allow for a similar conclusion concerning the second solution component:

Lemma 2.5

If \(\kappa \in {\mathbb {R}}\), \(0<\chi <\mu \) and (1.4) holds, then for arbitrary \(\varepsilon \in (0,1)\),

$$\begin{aligned} v_{\varepsilon r}(r,t) \le 0 \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0. \end{aligned}$$
(2.19)

Proof

We fix \(t>0\) and then obtain from the downward monotonicity of \([0,R] \ni r \mapsto u_\varepsilon (r,t)\), as asserted by Lemma 2.4, that for all \(r\in (0,R)\),

$$\begin{aligned} \frac{n}{r^n} \int _0^r \rho ^{n-1} u_\varepsilon (\rho ,t)\mathrm{d}\rho= & {} \frac{1}{|B_r(0)|} \int _{B_r(0)} u_\varepsilon (x,t) \mathrm{d}x \\\ge & {} \frac{1}{|B_R(0)|} \int _{B_R(0)} u_\varepsilon (x,t) \mathrm{d}x \\= & {} m_\varepsilon (t). \end{aligned}$$

In view of (2.13), this precisely means that \(v_{\varepsilon r}(r,t) \le 0\) for all \(r\in (0,R)\). \(\square \)

2.3 Constructing a smooth limit in \(\overline{\varOmega }\times (0,\infty )\)

Returning to the outcome of Lemma 2.2, we can now proceed to the construction of a radially decreasing limit couple (u, v) which is smooth for \(t>0\) and solves the corresponding identities in (1.2) classically in this region. Indeed, by quite a straightforward reasoning we obtain the following result on higher regularity away from the temporal origin.

Lemma 2.6

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and assume (1.4). Then for all \(\tau \in (0,1)\) and \(T>1\) there exist \(\theta =\theta (\tau ,T)\in (0,1)\) and \(C(\tau ,T)>0\) such that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{2+\theta ,1+\frac{\theta }{2}}(\overline{\varOmega }\times [\tau ,T])} + \Vert v_\varepsilon \Vert _{C^{2+\theta ,\frac{\theta }{2}}(\overline{\varOmega }\times [\tau ,T])} \le C(\tau ,T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(2.20)

Proof

On the basis of the uniform bound for \(u_\varepsilon \) in \(L^\infty (\varOmega \times (\tau ,\infty ))\) provided by Lemma 2.2 for each \(\tau \in (0,1)\), this can be seen by adapting a standard bootstrap procedure to the present setting in a straightforward manner (cf., e.g., [10, Section 5]). \(\square \)

The compactness features thereby provided directly entail the announced existence result.

Lemma 2.7

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and that (1.4) holds. Then there exists \((\varepsilon _j)_{j\in {\mathbb {N}}} \subset (0,1)\) such that \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \), and such that as \(\varepsilon =\varepsilon _j\searrow 0\) we have

$$\begin{aligned} u_\varepsilon \rightarrow u \quad \text{ in }\quad C^{2,1}_{\mathrm{loc}}(\overline{\varOmega }\times (0,\infty )) \end{aligned}$$
(2.21)

and

$$\begin{aligned} v_\varepsilon \rightarrow v \quad \text{ in }\quad C^{2,0}_{\mathrm{loc}}(\overline{\varOmega }\times (0,\infty )) \end{aligned}$$
(2.22)

with some nonnegative functions u and v which form a classical solution of the boundary value problem in (1.2) in \(\overline{\varOmega }\times (0,\infty )\). Moreover, \(u(\cdot ,t)\) and \(v(\cdot ,t)\) are radially symmetric and nonincreasing with respect to |x| for all \(t>0\).

Proof

According to Lemma 2.6, this can readily be derived by using the Arzelà–Ascoli theorem and an appropriate limit passage in (2.6), Lemmas 2.4 and 2.5. \(\square \)

3 Linking regularity at \(t=0\) to time integrability of total mass

We next approach the core of our analysis by focusing on the initial behavior of the solution (u, v) obtained in Lemma 2.7, and it will turn out that a crucial role in this regard is played by the behavior of the total mass functional \(\int _\varOmega u(x,t)\mathrm{d}x\) near \(t=0\), which through the nonlocal coupling expressed in (2.13) is closely linked to regularity of the cross-diffusive gradient \(v_{\varepsilon r}\). Here bearing in mind our ultimate goal of treating initial data merely fulfilling, e.g., (1.14), we may by far not expect boundedness of this functional, while on the other hand the bound therefore trivially implied by the upper estimate for \(u_\varepsilon \) from Lemma 2.2 appears to be too rough by only yielding a multiple of \(\frac{1}{t}\) for \(t<1\) as a majorant not integrable near \(t=0\). Driven by the observation that, as we shall see by an independent reasoning in Sect. 3.1, such an integrability feature seems essential for any nontrivial solution behavior near \(t=0\), in Sect. 3.2 we will employ another comparison argument to derive a further pointwise upper bound for \(u_\varepsilon \) which is now singular at the spatial origin but uniform in time. In Sect. 3.3 this will be seen to imply an integrable control for the mass functional, which will thereafter entail attainment of the initial trace firstly with respect to some quite rough topology (Sect. 3.4), and finally, upon another spatially local regularity reasoning in Sect. 3.5, also in the desired locally uniform sense (Sect. 3.6).

3.1 Necessity of temporal mass integrability for nontrivial initial traces

Our further considerations can be motivated by the following observation indicating a crucial role that integrability of the total mass functional plays with regard to nontrivial attainment of initial traces.

Proposition 3.1

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that for some \(T>0\), \((u,v)\in C^{2,1}(\overline{\varOmega }\times (0,T]) \times C^{2,0}(\overline{\varOmega }\times (0,T])\) is a classical solution of the boundary value problem in (1.2) in \(\overline{\varOmega }\times (0,T]\) which is such that \(u(\cdot ,t)\) and \(v(\cdot ,t)\) are radially symmetric and nonincreasing with respect to \(|x|\in (0,R)\) for all \(t\in (0,T)\), and such that

$$\begin{aligned} \int _0^{\mathrm{T}} \int _\varOmega u(x,t) \mathrm{d}x\mathrm{d}t = \infty . \end{aligned}$$
(3.1)

If, apart from that,

$$\begin{aligned} \liminf _{t\searrow 0} u(x,t) <\infty \qquad \text{ for } \text{ some } x\in \varOmega , \end{aligned}$$
(3.2)

then for all \(r_0\in (|x|,R)\) we have

$$\begin{aligned} \liminf _{t\searrow 0} \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega {\setminus } B_{r_0}(0))} =0. \end{aligned}$$
(3.3)

In particular, if there exists \(\widehat{u}_0 :\varOmega \rightarrow {\mathbb {R}}\) such that

$$\begin{aligned} u(\cdot ,t) \rightarrow \widehat{u}_0 \quad \text{ a.e. } \text{ in } \varOmega \qquad \text{ as } t\searrow 0, \end{aligned}$$
(3.4)

then

$$\begin{aligned} \widehat{u}_0=0 \qquad \text{ a.e. } \text{ in } \varOmega . \end{aligned}$$
(3.5)

Proof

Fixing \(x\in \varOmega \) such that (3.2) holds, this hypothesis states the existence of \(c_1>0\) and \((t_k)_{k\in {\mathbb {N}}} \subset (0,T)\) such that \(t_k\searrow 0\) as \(k\rightarrow \infty \), and such that writing \(r_\star :=|x| \in [0,R)\) and again using radial variables we have \(u(r_\star ,t_k) \le c_1\) for all \(k\in {\mathbb {N}}\). As also by assumption we know that \(u_r(\cdot ,t_k)\le 0\) in (0, R), this in particular entails that

$$\begin{aligned} y(t) := \int _{r^\star }^R (r-r_\star )^4 u(r,t) \mathrm{d}r, \qquad t\in (0,T), \end{aligned}$$
(3.6)

satisfies

$$\begin{aligned} y(t_k) \le c_2:=c_1 \int _{r_\star }^R (r-r_\star )^4 \mathrm{d}r = \frac{c_1(R-r_\star )^5}{5} \qquad \text{ for } \text{ all } k\in {\mathbb {N}}. \end{aligned}$$
(3.7)

Now if (3.3) was false for some \(r_0\in (r_\star ,R)\), then we could find \(c_3>0\) and \(\tau \in (0,T)\) with the property that \(u(r_0,t)\ge c_3\) for all \(t\in (0,\tau )\), again by monotonicity of \((0,R) \ni r\mapsto u(r,t)\) meaning that

$$\begin{aligned} y(t) \ge c_4:= c_3\cdot \int _{r_\star }^{r_0} (r-r_\star )^4 \mathrm{d}r = \frac{c_3(r_0-r_\star )^5}{5} \qquad \text{ for } \text{ all } t\in (0,\tau ). \end{aligned}$$
(3.8)

To derive a contradiction from this, in full analogy to (2.8), we combine the first two equations in (1.2) to see that

$$\begin{aligned} u_t = u_{rr} + \frac{n-1}{r} u_r - \chi u_r v_r -\chi m(t) u + \kappa u - (\mu -\chi ) u^2 \qquad \text{ for }\,r\in (0,R)\,\hbox {and}\, t\in (0,T), \end{aligned}$$
(3.9)

with \(m(t)=\frac{n}{R^n} \int _0^R r^{n-1} u(r,t) \mathrm{d}r\), \(t\in (0,T)\), fulfilling

$$\begin{aligned} \int _0^\tau m(t) \mathrm{d}t=\infty \end{aligned}$$
(3.10)

as a consequence of (3.1). Moreover, since our assumptions ensure that \(u_r\) and \(v_r\) are nonnegative, (3.9) implies that

$$\begin{aligned} u_t \le u_{rr} - \chi m(t) u + \kappa u - (\mu -\chi )u^2 \qquad \text{ in } (0,R)\times (0,T), \end{aligned}$$

and that thus the function y from (3.6) satisfies

$$\begin{aligned} y'(t)\le & {} \int _{r_\star }^R (r-r_\star )^4 u_{rr}(r,t) \mathrm{d}r - \chi m(t) y(t) + \kappa y(t) \nonumber \\&- (\mu -\chi ) \int _{r_\star }^R (r-r_\star )^4 u^2(r,t) \mathrm{d}r \qquad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$
(3.11)

Here two integrations by parts show that since \(u_r(R,t)=0\) for all \(t\in (0,T)\),

$$\begin{aligned} \int _{r_\star }^R (r-r_\star )^4 u_{rr}(r,t) \mathrm{d}r= & {} 12 \int _{r_\star }^R (r-r_\star )^2 u(r,t) \mathrm{d}r \\ \qquad+ & {} \Big [ (r-r_\star )^4 u_r(r,t) - 4(r-r_\star )^3 u(r,t)\Big ]_{r=r_\star }^{r=R} \\= & {} 12 \int _{r_\star }^R (r-r_\star )^2 u(r,t) \mathrm{d}r - 4(R-r_\star )^3 u(R,t) \\\le & {} 12 \int _{r_\star }^R (r-r_\star )^2 u(r,t) \mathrm{d}r \qquad \text{ for } \text{ all } t\in (0,T), \end{aligned}$$

so that using Young’s inequality along with our assumption that \(\mu >\chi \) we obtain that

$$\begin{aligned} \int _{r_\star }^R (r-r_\star )^4 u_{rr}(r,t) \mathrm{d}r \le \frac{\mu -\chi }{2} \int _{r_\star }^R (r-r_\star )^4 u^2(r,t) \mathrm{d}r + c_5 \qquad \text{ for } \text{ all } t\in (0,T) \end{aligned}$$

with \(c_5:=\frac{72(R-r_\star )}{\mu -\chi }>0\). As, by the same token, writing \(c_6:=\frac{\kappa _+^2}{2(\mu -\chi )} \int _{r_\star }^R (r-r_\star )^4 \mathrm{d}r = \frac{\kappa _+^2 (r-r_\star )^5}{10(\mu -\chi )}\), we have

$$\begin{aligned} \kappa y(t) \le \frac{\mu -\chi }{2} \int _{r_\star }^R (r-r_\star )^4 u^2(r,t) \mathrm{d}r + c_6 \qquad \text{ for } \text{ all } t\in (0,T), \end{aligned}$$

from (3.11) we infer that

$$\begin{aligned} y'(t) + \chi m(t) y(t) \le c_5 +c_6 \qquad \text{ for } \text{ all } t\in (0,T), \end{aligned}$$

and that hence, after integration,

$$\begin{aligned} y(\tau ) + \chi \int _{t_k}^\tau m(t)y(t) \mathrm{d}t \le y(t_k) + (c_5+c_6) \cdot (\tau -t_k) \qquad \text{ for } \text{ all } k \ge k_0 \end{aligned}$$

if we let \(k_0\in {\mathbb {N}}\) be large enough such that \(t_k<\tau \) for all \(k \ge k_0\). In view of (3.8) and (3.7), however, this particularly entails that

$$\begin{aligned} c_4\chi \int _{t_k}^\tau m(t) \mathrm{d}t \le c_2 + (c_5+c_6)\tau \qquad \text{ for } \text{ all } k\ge k_0, \end{aligned}$$

which, due to the positivity of \(\chi \), in the limit \(k\rightarrow \infty \) contradicts (3.10) and thereby completes the proof. \(\square \)

3.2 A spatially singular but temporally uniform pointwise upper bound for \(u_\varepsilon \)

Relying on the fact that (1.7) guarantees a favorable supersolution property enjoyed by large multiples of \(\phi \) for solutions of (2.8) with nonincreasing derivatives \(u_{\varepsilon r}\) and \(v_{\varepsilon r}\), we can turn our hypothesis (1.5) into a second pointwise bound for \(u_\varepsilon \) of the announced flavor.

Lemma 3.2

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that beyond (1.4), \(u_0\) satisfies (1.5) with some \(K>0\) and some \(\phi \) fulfilling (1.6) and (1.7) with a certain \(K_1>0\). Then there exists \(C>0\) such that for each \(\varepsilon \in (0,1)\),

$$\begin{aligned} u_\varepsilon (r,t) \le C\phi (r) \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0. \end{aligned}$$
(3.12)

Proof

Using that \(\mu >\chi \) and that \(\phi (R)\) is positive, we choose a positive constant \(c_1\) large enough such that

$$\begin{aligned} c_1\ge \frac{2K_1}{\mu -\chi } \end{aligned}$$
(3.13)

and

$$\begin{aligned} c_1 \ge \frac{2\kappa _+}{(\mu -\chi )\phi (R)} \end{aligned}$$
(3.14)

as well as

$$\begin{aligned} c_1 \ge K + \frac{1}{\phi (R)}. \end{aligned}$$
(3.15)

For fixed \(\varepsilon \in (0,1)\) and \(\delta _0\in (0,R)\) we may then rely on the boundedness of \(u_\varepsilon \) in \(\varOmega \times (0,\infty )\), as asserted by Lemma 2.1 when combined with Lemma 2.2, to find \(c_2(\varepsilon )>0\) such that

$$\begin{aligned} u_\varepsilon (r,t) \le c_2(\varepsilon ) \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0, \end{aligned}$$
(3.16)

and recall that by (1.6) we have \(\phi (r)\nearrow + \infty \) as \(r\searrow 0\), so that we can pick \(\delta =\delta (\varepsilon ,\delta _0)\in (0,R)\) such that \(\delta \le \delta _0\) and

$$\begin{aligned} \phi (\delta ) \ge \frac{c_2(\varepsilon )}{c_1}. \end{aligned}$$
(3.17)

Here the latter together with (3.16) ensures that

$$\begin{aligned} \overline{u}(r,t):=c_1\phi (r), \qquad r\in [0,R], \quad t\ge 0, \end{aligned}$$

satisfies

$$\begin{aligned} \overline{u}(\delta ,t) = c_1\phi (\delta ) \ge c_2(\varepsilon ) \ge u_\varepsilon (\delta ,t) \qquad \text{ for } \text{ all } t>0. \end{aligned}$$
(3.18)

Moreover, by monotonicity of \(\phi \),

$$\begin{aligned} \overline{u}_r(R,t)=c_1\phi '(R) \le 0 = u_{\varepsilon r}(R,t) \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
(3.19)

and at the initial time, our assumption (1.5) along with (2.4) warrants that due to (3.15) we have

$$\begin{aligned} \overline{u}(r,0)= & {} K\phi (r) + (c_1-K)\phi (r) \nonumber \\\ge & {} K\phi (r) + (c_1-K) \phi (R) \nonumber \\\ge & {} u_0(r) + 1 \nonumber \\\ge & {} u_{0\varepsilon }(r) \qquad \text{ for } \text{ all } r\in (\delta ,R). \end{aligned}$$
(3.20)

Now computing

$$\begin{aligned}&\overline{u}_t - \overline{u}_{rr} - \frac{n-1}{r} \overline{u}_r - \kappa \overline{u}+ (\mu -\chi )\overline{u}^2 = c_1 \phi ''(r) - c_1 \cdot \frac{n-1}{r} \phi '(r) \nonumber \\&\quad - c_1\kappa \phi (r) + c_1^2 (\mu -\chi ) \phi ^2(r), \qquad r\in (\delta ,R), \quad t>0, \end{aligned}$$
(3.21)

we see that again due to the monotonicity of \(\phi \), by using (3.14) we may estimate

$$\begin{aligned} - c_1\cdot \frac{n-1}{r} \phi '(r) \ge 0 \qquad \text{ for } \text{ all } r\in (\delta ,R) \end{aligned}$$

and

$$\begin{aligned} \frac{c_1 \kappa \phi (r)}{\frac{1}{2} c_1^2 (\mu -\chi )\phi ^2(r)} = \frac{2\kappa }{c_1(\mu -\chi ) \phi (r)} \le \frac{2\kappa _+}{c_1(\mu -\chi )\phi (R)} \le 1 \qquad \text{ for } \text{ all } r\in (\delta ,R). \end{aligned}$$

As finally (1.7) in view of (3.13) implies that also

$$\begin{aligned} \frac{c_1 \phi ''(r)}{\frac{1}{2} c_1^2 (\mu -\chi )\phi ^2(r)} = \frac{2\phi ''(r)}{(\mu -\chi ) c_1\phi ^2(r)} \le \frac{2K_1}{(\mu -\chi ) c_1} \le 1 \qquad \text{ for } \text{ all } r\in (\delta ,R), \end{aligned}$$

from (3.21) it follows that

$$\begin{aligned} \overline{u}_t \ge \overline{u}_{rr} + \frac{n-1}{r} \overline{u}_r + \kappa \overline{u}- (\mu -\chi ) \overline{u}^2 \qquad \text{ for } \text{ all }\,r\in (\delta ,R)\,\hbox {and any} \, t>0, \end{aligned}$$

while thanks to the inequalities \(u_{\varepsilon r} \le 0\) and \(v_{\varepsilon r} \le 0\) asserted by Lemmas 2.4 and 2.5, respectively, from (2.8) we obtain that

$$\begin{aligned} u_{\varepsilon t}= & {} u_{\varepsilon rr} + \frac{n-1}{r} u_{\varepsilon r} -\chi u_{\varepsilon r} v_{\varepsilon r} - \chi m_\varepsilon (t) u_\varepsilon + \kappa u_\varepsilon - (\mu -\chi ) u_\varepsilon ^2 \\\le & {} u_{\varepsilon rr} + \frac{n-1}{r} u_{\varepsilon r} + \kappa u_\varepsilon - (\mu -\chi ) u_\varepsilon ^2 \qquad \text{ for } \text{ all }\,r\in (\delta ,R)\,\hbox {and} \, t>0. \end{aligned}$$

On the basis of (3.18)–(3.20), we may therefore conclude by a comparison argument that \(\overline{u}(r,t)\ge u_\varepsilon (r,t)\) for all \(r\in (\delta (\varepsilon ,\delta _0),R)\) and \(t>0\), which since \(\delta (\varepsilon ,\delta _0)\le \delta _0\) entails that

$$\begin{aligned} u_\varepsilon (r,t) \le c_1\phi (r) \qquad \text{ for } \text{ all }\,r\in (\delta _0,R)\, \hbox {and}\, t>0 \end{aligned}$$

and therefore establishes (3.12) in the limit \(\delta _0\searrow 0\). \(\square \)

3.3 Uniform integrability of the mass functional

Now having at hand the upper inequalities for \(u_\varepsilon \) both from Lemma 2.2 and from Lemma 3.2, on combining these and relying on the technical assumption (1.8), we can construct a majorant for the mass functional in the following sense.

Lemma 3.3

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and assume that (1.4) and (1.5) hold with some \(K>0\) and some \(\phi \) satisfying (1.6) and (1.7) as well as (1.8) with some \(K_1>0\) and \(K_2>0\). Then there exist \(t_0\in (0,1)\) and \(C>0\) such that for any \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \int _\varOmega u_\varepsilon (x,t) \mathrm{d}x \le C \cdot \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \qquad \text{ for } \text{ all } t\in (0,t_0). \end{aligned}$$
(3.22)

Proof

According to Lemma 2.2, we can find \(c_1>0\) such that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned} u_\varepsilon (r,t) \le \frac{c_1}{t} \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and each} \,t\in (0,1), \end{aligned}$$
(3.23)

while Lemma 3.2 provides \(c_2>0\) fulfilling

$$\begin{aligned} u_\varepsilon (r,t) \le c_2 \phi (r) \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and}\, t>0 \end{aligned}$$
(3.24)

whenever \(\varepsilon \in (0,1)\). Thus, if we let \(t_0\in (0,1)\) be such that \(t_0 \le \frac{1}{\phi (\frac{R}{2})}\), then for \(t\in (0,t_0)\) we have \(2\phi ^{-1}(\frac{1}{t}) < 2\phi ^{-1}(\phi (\frac{R}{2}))=R\), and hence splitting

$$\begin{aligned} \int _0^R r^{n-1} u_\varepsilon (r,t) \mathrm{d}r = \int _0^{2\phi ^{-1}\left( \frac{1}{t}\right) } r^{n-1} u_\varepsilon (r,t) \mathrm{d}r + \int _{2\phi ^{-1}\left( \frac{1}{t}\right) }^R r^{n-1} u_\varepsilon (r,t) \mathrm{d}r, \end{aligned}$$
(3.25)

we can combine (3.23) with (3.24) to estimate

$$\begin{aligned} \int _0^{2\phi ^{-1}\left( \frac{1}{t}\right) } r^{n-1} u_\varepsilon (r,t) \mathrm{d}r\le & {} \frac{c_1}{t} \cdot \int _0^{2\phi ^{-1}\left( \frac{1}{t}\right) } r^{n-1} \mathrm{d}r \\= & {} \frac{2^n c_1}{n} \cdot \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \end{aligned}$$

and

$$\begin{aligned} \int _{2\phi ^{-1}\left( \frac{1}{t}\right) }^R r^{n-1} u_\varepsilon (r,t) \mathrm{d}r\le & {} c_2\phi \Big (2\phi ^{-1}\Big (\frac{1}{t}\Big )\Big ) \cdot \int _{2\phi ^{-1}\left( \frac{1}{t}\right) }^R r^{n-1} \mathrm{d}r \\\le & {} \frac{c_2 R^n}{n} \cdot \phi \Big (2\phi ^{-1}\Big (\frac{1}{t}\Big )\Big ) \end{aligned}$$

for all \(t\in (0,t_0)\). Now since (1.8) ensures that herein

$$\begin{aligned} \phi \Big (2\phi ^{-1}\Big (\frac{1}{t}\Big )\Big )\le & {} K_2 \cdot \Big [ 2\phi ^{-1} \Big (\frac{1}{t}\Big ) \Big ]^n \cdot \phi \Big (\phi ^{-1}\Big (\frac{1}{t}\Big )\Big ) \\= & {} 2^n K_2 \cdot \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \qquad \text{ for } \text{ all } t\in (0,1), \end{aligned}$$

from (3.25) it follows that if we abbreviate \(c_3:=\frac{2^n c_1}{n} + \frac{c_2 R^n}{n} \cdot 2^n K_2\), then

$$\begin{aligned} \int _0^R r^{n-1} u_\varepsilon (r,t) \mathrm{d}r \le c_2 \cdot \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \qquad \text{ for } \text{ all } t\in (0,t_0), \end{aligned}$$

which precisely yields (3.22). \(\square \)

Now our final and most crucial hypothesis (1.9) ensures integrability of the right-hand side in (3.22) near \(t=0\):

Lemma 3.4

Suppose that \(\phi \) satisfies (1.6) as well as (1.9). Then

$$\begin{aligned} \int _0^{t_0} \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \mathrm{d}t < \infty \qquad \text{ for } \text{ all } t_0\in (0,1). \end{aligned}$$
(3.26)

Proof

Substituting \(r=\phi ^{-1}(\frac{1}{t})\) and integrating by parts, for \(\delta \in (0,t_0)\) we can rewrite

$$\begin{aligned} \int _\delta ^{t_0} \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \mathrm{d}t= & {} - \int _{\phi ^{-1}\left( \frac{1}{\delta }\right) }^{\phi ^{-1}\left( \frac{1}{t_0}\right) } \frac{r^n}{\frac{1}{\phi (r)}} \cdot \frac{\phi '(r)}{\phi ^2(r)} \mathrm{d}r \nonumber \\= & {} - \int _{\phi ^{-1}\left( \frac{1}{\delta }\right) }^{\phi ^{-1}\left( \frac{1}{t_0}\right) } r^n \cdot \frac{d}{\mathrm{d}r} \ln \phi (r) \mathrm{d}r \nonumber \\= & {} n \int _{\phi ^{-1}\left( \frac{1}{\delta }\right) }^{\phi ^{-1}\left( \frac{1}{t_0}\right) } r^{n-1} \ln \phi (r) \mathrm{d}r \nonumber \\&- \Big [\phi ^{-1}\Big (\frac{1}{t_0}\Big )\Big ]^n \cdot \ln \frac{1}{t_0} + \Big [\phi ^{-1}\Big (\frac{1}{\delta }\Big )\Big ]^n \cdot \ln \frac{1}{\delta }. \end{aligned}$$
(3.27)

Here, clearly,

$$\begin{aligned} \int _{\phi ^{-1}\left( \frac{1}{\delta }\right) }^{\phi ^{-1}\left( \frac{1}{t_0}\right) } r^{n-1} \ln \phi (r) \mathrm{d}r\le & {} n \int _0^1 r^{n-1} \ln \phi (r) \mathrm{d}r + n \int _1^{\phi ^{-1}\left( \frac{1}{t_0}\right) } r^{n-1} \ln \phi (1) \mathrm{d}r \nonumber \\\le & {} c_1:= n\int _0^1 r^{n-1} \ln \phi (r) \mathrm{d}r + \Big [\phi ^{-1}\Big (\frac{1}{t_0}\Big )\Big ]^n \!\! \cdot \ln \phi (1) \ \nonumber \\&\text{ for } \text{ all } \delta \in (0,t_0), \end{aligned}$$
(3.28)

with \(c_1\) being finite and positive by (1.9) and (1.6). Apart from that, (1.9) ensures that there must exist \((r_j)_{j\in {\mathbb {N}}}\subset (0,\phi ^{-1}(\frac{1}{t_0}))\) such that \(r_j\searrow 0\) as \(j\rightarrow \infty \) and \(r_j^{n-1} \ln \phi (r_j) \le \frac{1}{r_j}\) for all \(j\in {\mathbb {N}}\), for otherwise \(\int _0^1 r^{n-1} \ln \phi (r)\mathrm{d}r \ge \int _0^1 \frac{\mathrm{d}r}{r}=\infty \). The numbers \(\delta _j:=\frac{1}{\phi (r_j)}\), \(j\in {\mathbb {N}}\) thus satisfy \(\delta _j<t_0\) as well as

$$\begin{aligned} \Big [\phi ^{-1}\Big (\frac{1}{\delta _j}\Big )\Big ]^n \cdot \ln \frac{1}{\delta _j} = r_j^n \ln \phi (r_j) \le 1 \end{aligned}$$

for all \(j\in {\mathbb {N}}\), whence (3.27) along with (3.28) entails that

$$\begin{aligned} \int _{\delta _j}^{t_0} \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \mathrm{d}t \le c_1+1 \qquad \text{ for } \text{ all } j\in {\mathbb {N}}, \end{aligned}$$

because \(\ln \frac{1}{t_0}\) is positive. Since the assumed convergence property of \((r_j)_{j\in {\mathbb {N}}}\) together with (1.6) warrants that \(\delta _j\searrow 0\) as \(j\rightarrow \infty \), this establishes (3.26). \(\square \)

A further and quite immediate consequence of Lemma 3.3 on \(v_{\varepsilon r}\) will be of substantial importance both in Lemma 3.6 and in Lemma 3.8.

Lemma 3.5

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and assume that (1.4), (1.5) and (1.6)–(1.8) hold with some \(K>0\), \(K_1>0\) and \(K_2>0\). Then there exist \(t_0\in (0,1)\) and \(C>0\) such that for any choice of \(\varepsilon \in (0,1)\),

$$\begin{aligned} |v_{\varepsilon r}(r,t)| \le C r^{1-n} \cdot \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t} \qquad \text{ for } \text{ all }\,r\in (0,R)\,\hbox {and each}\, t\in (0,t_0). \end{aligned}$$
(3.29)

Proof

As on the right-hand side of (2.13), we can estimate

$$\begin{aligned} \frac{1}{n} m_\varepsilon (t) r^n -\int _0^r \rho ^{n-1} u_\varepsilon (\rho ,t) \mathrm{d}\rho\ge & {} - \int _0^R \rho ^{n-1} u_\varepsilon (\rho ,t) \mathrm{d}\rho \\= & {} - \frac{1}{n|B_1(0)|} \cdot \int _\varOmega u_\varepsilon (\cdot ,t)\\&\qquad \text{ for } \text{ all }\,r\in (0,R), t>0\,\hbox {and}\,\varepsilon \in (0,1), \end{aligned}$$

thanks to Lemma 2.5 this directly results from Lemma 3.3. \(\square \)

3.4 Approaching initial traces in \((W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \)

Through (2.13), the information gained in Lemmas 3.3 and 3.4 now provides regularity properties of the gradient \(\nabla v_\varepsilon \) that are sufficient for the derivation of the following statement on compactness in appropriately large dual spaces.

Lemma 3.6

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that (1.4) and (1.5) hold as well as (1.6)–(1.9) with some positive constants \(K, K_1\) and \(K_2\). Then for each \(\delta \in (0,R)\) and any \(T>1\),

$$\begin{aligned} (u_\varepsilon )_{\varepsilon \in (0,1)} \quad \text{ is } \text{ relatively } \text{ compact } \text{ in } \quad C^0\Big ( [0,T]; (W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \Big ). \end{aligned}$$
(3.30)

Proof

For fixed \(\delta \in (0,R)\), Lemma 3.2 together with (1.6) allows us to fix \(c_1=c_1(\delta )>0\) such that

$$\begin{aligned} u_\varepsilon \le c_1 \quad \text{ in } (\varOmega {\setminus } \overline{B}_\delta (0)) \times (0,\infty ) \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$
(3.31)

and Lemma 3.5 in conjunction with Lemma 2.6 states that if we let \(t_0\in (0,1)\) be as given by Lemma 3.5, and define

$$\begin{aligned} f(t):=\left\{ \begin{array}{ll} \frac{\left[ \phi ^{-1}\left( \frac{1}{t}\right) \right] ^n}{t}, &{}\quad t\in (0,t_0), \\ 1, &{}\quad t\ge t_0, \end{array} \right. \end{aligned}$$
(3.32)

then for all \(T>1\) we can find \(c_2=c_2(\delta ,T)>0\) fulfilling

$$\begin{aligned} |\nabla v_\varepsilon | \le c_2 f(t) \quad \text{ in } (\varOmega {\setminus } \overline{B}_\delta (0)) \times (0,T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(3.33)

For arbitrary \(t\in (0,T)\) and \(\psi \in C_0^\infty (\varOmega {\setminus } \overline{B}_\delta (0))\), by going back to (2.6) we can thus use (3.31) and (3.33) to estimate

$$\begin{aligned} \bigg | \int _\varOmega u_{\varepsilon t}(x,t) \psi (x) \mathrm{d}x \bigg |= & {} \bigg | \int _\varOmega u_\varepsilon \Delta \psi + \chi \int _\varOmega u_\varepsilon \nabla v_\varepsilon \cdot \nabla \psi + \kappa \int _\varOmega u_\varepsilon \psi - \mu \int _\varOmega u_\varepsilon ^2 \psi \bigg | \\\le & {} c_1\Vert \Delta \psi \Vert _{L^1(\varOmega )} + c_1 c_2 \chi f(t) \Vert \nabla \psi \Vert _{L^1(\varOmega )} + c_1 |\kappa | \cdot \Vert \psi \Vert _{L^1(\varOmega )} \\&+ c_1^2 \mu \Vert \psi \Vert _{L^1(\varOmega )} \end{aligned}$$

for all \(\varepsilon \in (0,1)\), whence by completion we obtain that with some \(c_3=c_3(\delta ,T)>0\),

$$\begin{aligned} \Vert u_{\varepsilon t}(\cdot ,t)\Vert _{(W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star } \le c_3 \cdot (f(t)+1) \qquad \text{ for } \text{ all }\, t\in (0,T)\, \hbox {and any}\, \varepsilon \in (0,1). \end{aligned}$$

In particular, this entails that whenever \(t\in [0,T]\) and \(s\in [0,t]\),

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)-u_\varepsilon (\cdot ,s)\Vert _{(W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star }= & {} \bigg \Vert \int _s^t u_{\varepsilon t}(\cdot ,\sigma ) \mathrm{d}\sigma \bigg \Vert _{(W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star } \nonumber \\\le & {} c_3 \int _s^t (f(\sigma )+1) \mathrm{d}\sigma \qquad \text{ for } \text{ all } \varepsilon \in (0,1), \end{aligned}$$
(3.34)

where we note that since \(f+1\) belongs to \(L^1((0,T))\) due to (3.32) and Lemma 3.4, the uniform continuity of \([0,T] \ni \widehat{t} \mapsto \int _0^{\widehat{t}} (f(\sigma )+1) \mathrm{d}\sigma \) thereby implied ensures that

$$\begin{aligned} \sup _{\begin{array}{c} 0 \le s \le t \le T \\ |s-t| < \eta \end{array}} \int _s^t (f(\sigma )+1)\mathrm{d}\sigma \rightarrow 0 \qquad \text{ as } \eta \searrow 0. \end{aligned}$$

Accordingly, (3.34) entails that \((u_\varepsilon )_{\varepsilon \in (0,1)}\) is equi-continuous on [0, T] as a family of \((W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \)-valued functions, so that since furthermore, for each fixed \(t\in [0,T]\), \((u_\varepsilon (\cdot ,t))_{\varepsilon \in (0,1)}\) is bounded in \(L^\infty (\varOmega {\setminus } \overline{B}_\delta (0)) \hookrightarrow \hookrightarrow (W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \) by (3.31), the claimed compactness property (3.30) becomes a consequence of the Arzelà–Ascoli theorem. \(\square \)

A natural consequence of the latter completes the following key step in our reasoning.

Lemma 3.7

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that (1.4) and (1.5) hold as well as (1.6)–(1.9) with some positive constants \(K, K_1\) and \(K_2\). Then the limit function u obtained in Lemma 2.7 has the property that for all \(\delta \in (0,R)\),

$$\begin{aligned} u(\cdot ,t) \rightarrow u_0 \quad \text{ in } \Big (W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0))\Big )^\star \qquad \text{ as } t\searrow 0. \end{aligned}$$
(3.35)

Proof

This is immediately implied by Lemma 3.6 when combined with Lemma 2.6 and the fact that \(u_\varepsilon (\cdot ,0)\rightarrow u_0\) in \(L^\infty (\varOmega {\setminus } \overline{B}_\delta (0)))\) as \(\varepsilon \searrow 0\) according to (2.5). \(\square \)

3.5 Additional consequences of uniform mass integrability on regularity near \(t=0\)

Let us now make sure that outside the spatial origin the upper bound for \(u_\varepsilon \) from Lemma 3.2, together with the mass control implied by Lemmas 3.3 and 3.4, actually implies further compactness properties of \((u_\varepsilon (\cdot ,t))_{t\in (0,1)}\). By means of a standard testing procedure involving appropriate localization, Lemma 3.5 together with Lemma 3.2 indeed entails the following.

Lemma 3.8

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that (1.4) and (1.5) are valid with \(K>0\) and \(\phi \) fulfilling (1.6) and (1.7) with some \(K_1>0\). Then for each \(\delta \in (0,R)\) and any \(T>0\) one can find \(C(\delta ,T)>0\) such that

$$\begin{aligned} \int _0^{\mathrm{T}} \int _{\varOmega {\setminus } B_\delta (0)} |\nabla u_\varepsilon (x,t)|^2 \mathrm{d}x \le C(\delta ,T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(3.36)

Proof

Given \(\delta \in (0,R)\) and \(T>0\), we once more invoke Lemma 3.2, (1.6), Lemmas 2.6, 3.4 and 3.5 to find \(c_1=c_1(\delta )>0\), \(c_2=c_2(\delta ,T)>0\) and \(f\in L^1((0,T))\) such that

$$\begin{aligned} u_\varepsilon \le c_1 \quad \text{ in } (\varOmega {\setminus } B_{\frac{\delta }{2}}(0)) \times (0,\infty ) \qquad \text{ for } \text{ all } \varepsilon \in (0,1) \end{aligned}$$
(3.37)

and

$$\begin{aligned} |\nabla v_\varepsilon | \le c_2 f(t) \quad \text{ in } (\varOmega {\setminus } B_{\frac{\delta }{2}}(0)) \times (0,T) \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(3.38)

We then pick a cutoff function \(\zeta \in C^\infty (\overline{\varOmega })\) such that \(0\le \zeta \le 1\) in \(\varOmega \), \(\zeta \equiv 1\) in \(\varOmega {\setminus } B_\delta (0)\) and \(\zeta \equiv 0\) in \(B_\frac{\delta }{2}(0)\), and test the first equation in (2.6) against \(\zeta u_\varepsilon \) in a standard manner to see that for each \(\varepsilon \in (0,1)\),

$$\begin{aligned} \frac{1}{2} \frac{d}{\mathrm{d}t} \int _\varOmega \zeta (x) u_\varepsilon ^2(x,t) \mathrm{d}x= & {} \int _\varOmega \zeta u_\varepsilon \cdot \Big \{ \Delta u_\varepsilon - \chi \nabla \cdot (u_\varepsilon \nabla v_\varepsilon ) + \kappa u_\varepsilon - \mu u_\varepsilon ^2\Big \} \nonumber \\= & {} - \int _\varOmega \zeta |\nabla u_\varepsilon |^2 - \int _\varOmega u_\varepsilon \nabla u_\varepsilon \cdot \nabla \zeta \nonumber \\&+\, \chi \int _\varOmega \zeta u_\varepsilon \nabla u_\varepsilon \cdot \nabla v_\varepsilon + \chi \int _\varOmega u_\varepsilon ^2 \nabla v_\varepsilon \cdot \nabla \zeta \nonumber \\&+\, \kappa \int _\varOmega \zeta u_\varepsilon ^2 - \mu \int _\varOmega \zeta u_\varepsilon ^3 \qquad \text{ for } \text{ all } t>0. \end{aligned}$$
(3.39)

Here upon another integration by parts, we can use (3.37) to estimate

$$\begin{aligned} - \int _\varOmega u_\varepsilon \nabla u_\varepsilon \cdot \nabla \zeta = \frac{1}{2} \int _\varOmega u_\varepsilon ^2\Delta \zeta \le \frac{c_1^2}{2} \Vert \Delta \zeta \Vert _{L^1(\varOmega )} \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
(3.40)

and similarly we may treat the third summand on the right of (3.39) to find that due to the second equation from (2.6) and our assumption that \(\mu>\chi >\frac{\chi }{2}\),

$$\begin{aligned}&\chi \int _\varOmega \zeta u_\varepsilon \nabla u_\varepsilon \cdot \nabla v_\varepsilon + \chi u_\varepsilon ^2 \nabla v_\varepsilon \cdot \nabla \zeta - \mu \int _\varOmega \zeta u_\varepsilon ^3 \nonumber \\&\quad = - \frac{\chi }{2} \int _\varOmega \zeta u_\varepsilon ^2 \Delta v_\varepsilon + \frac{\chi }{2} \int _\varOmega u_\varepsilon ^2 \nabla v_\varepsilon \cdot \nabla \zeta - \mu \int _\varOmega \zeta u_\varepsilon ^3 \nonumber \\&\quad = - \frac{\chi }{2} m_\varepsilon (t) \cdot \int _\varOmega \zeta u_\varepsilon ^2 - \Big (\mu -\frac{\chi }{2}\Big ) \int _\varOmega \zeta u_\varepsilon ^3 + \frac{\chi }{2} \int _\varOmega u_\varepsilon ^2 \nabla v_\varepsilon \cdot \nabla \zeta \nonumber \\&\quad \le \frac{\chi }{2} \int _\varOmega u_\varepsilon ^2 \nabla v_\varepsilon \cdot \nabla \zeta \nonumber \\&\quad \le \frac{\chi }{2} c_1^2 c_2\Vert \nabla \zeta \Vert _{L^1(\varOmega )} \cdot f(t) \qquad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$
(3.41)

As clearly, also by (3.37),

$$\begin{aligned} \int _\varOmega \zeta u_\varepsilon ^2 \le c_1^2|\varOmega | \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
(3.42)

from (3.39)–(3.41) we infer the existence of \(c_3=c_3(\delta ,T)>0\) such that for arbitrary \(\varepsilon \in (0,1)\),

$$\begin{aligned} \frac{d}{\mathrm{d}t} \int _\varOmega \zeta u_\varepsilon ^2 + 2 \int _\varOmega \zeta |\nabla u_\varepsilon |^2 \le c_3\cdot (f(t)+1) \qquad \text{ for } \text{ all } t\in (0,T), \end{aligned}$$

which on integration yields

$$\begin{aligned} \int _\varOmega \zeta u_\varepsilon ^2(\cdot ,T) + 2\int _0^{\mathrm{T}} \int _\varOmega \zeta |\nabla u_\varepsilon |^2 \le \int _\varOmega \zeta u_{0\varepsilon }^2 + c_3 \int _0^{\mathrm{T}} (f(t)+1) \mathrm{d}t \qquad \text{ for } \text{ all } \varepsilon \in (0,1). \end{aligned}$$
(3.43)

Since (2.4) and (1.5) along with the fact that \(\mathrm{supp} \, \zeta \subset \overline{\varOmega }{\setminus } B_\frac{\delta }{2}(0)\) enforce that \(\sup _{\varepsilon \in (0,1)} \int _\varOmega \zeta u_{0\varepsilon }^2\) is finite, and since \(f+1\) belongs to \(L^1((0,T))\), recalling that \(\zeta \) is nonnegative and satisfies \(\zeta \equiv 1\) in \(\varOmega {\setminus } B_\delta (0)\) we obtain (3.36) as a consequence of (3.43). \(\square \)

One final regularity argument, now using a localization slightly more subtle and thereby favorably cooperating with our knowledge on radial symmetry and monotonicity, yields even certain temporally uniform bounds for \(\nabla u_\varepsilon \) in annular regions.

Lemma 3.9

Let \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and let (1.4)–(1.7) be satisfied with some \(K>0\) and \(K_1>0\). Then for any \(\delta \in (0,R)\) and \(T>0\) there exists \(C(\delta ,T)>0\) such that whenever \(\varepsilon \in (0,1)\),

$$\begin{aligned} \int _{\varOmega {\setminus } B_\delta (0)} |\nabla u_\varepsilon (x,t)|^2 \mathrm{d}x \le C(\delta ,T) \qquad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$
(3.44)

Proof

For fixed \(\delta \in (0,R)\), we choose a cutoff function \(\zeta \in C^\infty (\overline{\varOmega })\) such that again \(\zeta \equiv 0\) in \(B_\frac{\delta }{2}(0)\) and \(\zeta \equiv 1\) in \(\varOmega {\setminus } B_\delta (0)\), but such that now, in contrast to the requirements from the proof of Lemma 3.8, \(\zeta \) additionally is radially symmetric and nondecreasing with respect to \(r=|x|\in [0,R]\). Then upon several integrations by parts, we obtain from (2.8) that for all \(\varepsilon \in (0,1)\),

$$\begin{aligned} \frac{1}{2} \frac{d}{\mathrm{d}t} \int _\varOmega \zeta ^2(x) |\nabla u_\varepsilon (x,t)|^2 \mathrm{d}x= & {} \int _\varOmega \zeta ^2 \nabla u_\varepsilon \cdot \nabla \Big \{ \Delta u_\varepsilon - \chi \nabla u_\varepsilon \cdot \nabla v_\varepsilon - \chi m_\varepsilon (t) u_\varepsilon \nonumber \\+ & {} \kappa u_\varepsilon - (\mu -\chi )u_\varepsilon ^2 \Big \} \nonumber \\= & {} - \int _\varOmega \zeta ^2 |\Delta u_\varepsilon |^2 - 2\int _\varOmega \zeta (\nabla u_\varepsilon \cdot \nabla \zeta )\Delta u_\varepsilon \nonumber \\&- \chi \int _\varOmega \zeta ^2 \nabla u_\varepsilon \cdot \nabla (\nabla u_\varepsilon \cdot \nabla v_\varepsilon ) - \chi m_\varepsilon (t) \int _\varOmega \zeta ^2 |\nabla u_\varepsilon |^2 \nonumber \\&+ \kappa \int _\varOmega \zeta ^2 |\nabla u_\varepsilon |^2 - 2(\mu -\chi ) \int _\varOmega \zeta ^2 u_\varepsilon |\nabla u_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0,\nonumber \\ \end{aligned}$$
(3.45)

where by Young’s inequality,

$$\begin{aligned} - 2\int _\varOmega \zeta (\nabla u_\varepsilon \cdot \nabla \zeta ) \Delta u_\varepsilon \le \int _\varOmega \zeta ^2 |\Delta u_\varepsilon |^2 + \int _\varOmega |\nabla \zeta |^2 |\nabla u_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0. \end{aligned}$$
(3.46)

In the crucial third summand on the right of (3.45), we resort to the radial notation again to see upon further integration by parts that thanks to the boundary condition \(u_{\varepsilon r}(R,\cdot )\equiv 0\) and the second equation in (2.6),

$$\begin{aligned}&-\int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r} \cdot (u_{\varepsilon r} v_{\varepsilon r})_r \mathrm{d}r \\&\quad = - \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r} u_{\varepsilon rr} v_{\varepsilon r} \mathrm{d}r - \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r}^2 v_{\varepsilon rr} \mathrm{d}r \\&\quad = \frac{1}{2} \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r}^2 \Big ( v_{\varepsilon rr} + \frac{n-1}{r} v_{\varepsilon r}\Big ) \mathrm{d}r + \int _0^R r^{n-1} \zeta (r) \zeta _r(r) u_{\varepsilon r}^2 v_{\varepsilon r} \mathrm{d}r \\&\quad \quad - \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r}^2 v_{\varepsilon rr} \mathrm{d}r \\&\quad = -\frac{1}{2} \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r}^2 \Big ( v_{\varepsilon rr} + \frac{n-1}{r} v_{\varepsilon r}\Big ) \mathrm{d}r + \int _0^R r^{n-1} \zeta (r)\zeta _r(r) u_{\varepsilon r}^2 v_{\varepsilon r} \mathrm{d}r \\&\quad \quad + (n-1) \int _0^R r^{n-2} \zeta ^2(r) u_{\varepsilon r}^2 v_{\varepsilon r} \mathrm{d}r \\&\quad = - \frac{1}{2} m_\varepsilon (t) \int _0^R r^{n-1} \zeta ^2(r) u_{\varepsilon r}^2 \mathrm{d}r + \frac{1}{2} \int _0^R r^{n-1} \zeta ^2(r) u_\varepsilon u_{\varepsilon r}^2 \mathrm{d}r \\&\quad \quad + \int _0^R r^{n-1} \zeta (r) \zeta _r(r) u_{\varepsilon r}^2 v_{\varepsilon r} \mathrm{d}r + (n-1) \int _0^R r^{n-2} \zeta ^2(r) u_{\varepsilon r}^2 v_{\varepsilon r} \mathrm{d}r \qquad \text{ for } \text{ all } t>0. \end{aligned}$$

Since \(u_{\varepsilon r} \le 0\) and \(v_{\varepsilon r} \le 0\) by Lemma 2.4 and Lemma 2.5, our requirement on upward radial monotonicity of \(\zeta \) thus ensures that, besides the fourth last and the last, also the second last summand herein is nonpositive for all \(\varepsilon \in (0,1)\) and \(t>0\). Therefore,

$$\begin{aligned} - \chi \int _\varOmega \zeta ^2 \nabla u_\varepsilon \cdot \nabla (\nabla u_\varepsilon \cdot \nabla v_\varepsilon ) \le \frac{\chi }{2} \int _\varOmega \zeta ^2 u_\varepsilon |\nabla u_\varepsilon |^2 \qquad \text{ for } \text{ all } t>0, \end{aligned}$$

whence (3.45) and (3.46) along with our assumption that \(\mu >\chi \) imply that

$$\begin{aligned}&\frac{1}{2} \frac{d}{\mathrm{d}t} \int _\varOmega \zeta ^2 |\nabla u_\varepsilon |^2 \le \int _\varOmega |\nabla \zeta |^2 |\nabla u_\varepsilon |^2 + \kappa _+ \int _\varOmega \zeta ^2 |\nabla u_\varepsilon |^2 + \frac{\chi }{2} \int _\varOmega \zeta ^2 u_\varepsilon |\nabla u_\varepsilon |^2 \\&\qquad \text{ for } \text{ all } t>0. \end{aligned}$$

Again relying on Lemma 3.2 and (1.6), we thus conclude that for any such \(\delta \) and each \(T>0\) we can find \(c_1=c_1(\delta ,T)>0\) fulfilling

$$\begin{aligned} \frac{d}{\mathrm{d}t} \int _\varOmega \zeta ^2 |\nabla u_\varepsilon |^2 \le c_1 \int _{\varOmega {\setminus } B_\frac{\delta }{2}(0)} |\nabla u_\varepsilon |^2 \qquad \text{ for } \text{ all } t\in (0,T) \text{ and } \text{ any } \varepsilon \in (0,1) \end{aligned}$$

and that accordingly, for each \(\varepsilon \in (0,1)\),

$$\begin{aligned}&\int _\varOmega \zeta ^2(x) |\nabla u_\varepsilon (x,t)|^2 \mathrm{d}x \le \int _\varOmega \zeta ^2(x) |\nabla u_{0\varepsilon }(x)|^2 \mathrm{d}x + c_1 \int _0^{\mathrm{T}} \int _{\varOmega {\setminus } B_\frac{\delta }{2}(0)} |\nabla u_\varepsilon |^2 \\&\qquad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$

In view of (2.5), the outcome of Lemma 3.8 therefore warrants validity of (3.44) with some suitably large \(C(\delta ,T)>0\). \(\square \)

3.6 Locally uniform initial trace attainment: proof of the main results

One last time making use of radial symmetry, from Lemma 3.9 and the compactness of the embedding \(W^{1,2}((\delta ,R)) \hookrightarrow C^0([\delta ,R])\) for \(\delta \in (0,R)\), we can finally infer the following.

Lemma 3.10

Assume that \(\kappa \in {\mathbb {R}}\) and \(0<\chi <\mu \), and suppose that (1.4) and (1.5) as well as (1.6)–(1.9) hold with some positive constants \(K, K_1\) and \(K_2\). Then the function u given by Lemma 2.7 belongs to \(C^0((\overline{\varOmega }{\setminus }\{0\}) \times [0,\infty ))\) and satisfies

$$\begin{aligned} u(\cdot ,t) \rightarrow u_0 \quad \text{ in } C^0_{\mathrm{loc}}(\overline{\varOmega }{\setminus } \{0\}) \qquad \text{ as } t\searrow 0. \end{aligned}$$
(3.47)

Proof

As \(u_0\) is continuous in \(\overline{\varOmega }{\setminus } \{0\}\), in view of the regularity properties asserted by Lemma 2.7, it is sufficient to verify (3.47). In fact, if this was false then there would exist \(\delta \in (0,R)\) and \((t_k)_{k\in {\mathbb {N}}}\subset (0,1)\) such that \(t_k\searrow 0\) as \(k\rightarrow \infty \) and

$$\begin{aligned} \inf _{k\in {\mathbb {N}}} \Vert u(\cdot ,t_k)-u_0\Vert _{C^0(\overline{\varOmega }{\setminus } B_\delta (0))} >0. \end{aligned}$$
(3.48)

On the other hand, combining Lemma 3.9 with the convergence statement from Lemma 2.7 provides \(c_1>0\) such that, again in radial coordinates, we have

$$\begin{aligned} \int _\delta ^R r^{n-1} u_r^2(r,t) \mathrm{d}r \le c_1 \qquad \text{ for } \text{ all } t\in (0,1) \end{aligned}$$

and hence, by the Cauchy–Schwarz inequality,

$$\begin{aligned} \Big | u(r_2,t)-u(r_1,t)\Big |= & {} \bigg | \int _{r_1}^{r_2} u_r(r,t) \mathrm{d}r \bigg | \\\le & {} \bigg \{ \int _{r_1}^{r_2} r^{n-1} u_r^2(r,t) \mathrm{d}r \bigg \}^\frac{1}{2} \cdot \bigg \{ \int _{r_1}^{r_2} r^{1-n} \mathrm{d}r \bigg \}^\frac{1}{2} \\\le & {} c_1^\frac{1}{2} \delta ^\frac{1-n}{2} (r_2-r_1)^\frac{1}{2} \qquad \text{ for } \text{ all }\,r_1\in [\delta ,R],\,r_2\in [r_1,R]\,\hbox {and}\, t\in (0,1). \end{aligned}$$

Together with Lemma 3.2 and (1.6), by the Arzelà–Ascoli theorem this equi-continuity property warrants that \((u(\cdot ,t_k))_{k\in {\mathbb {N}}}\) is relatively compact in \(C^0(\overline{\varOmega }{\setminus } B_\delta (0))\), so that for some subsequence \((t_{k_j})_{j\in {\mathbb {N}}}\) of \((t_k)_{k\in {\mathbb {N}}}\) we can find \(z\in C^0(\overline{\varOmega }{\setminus } B_\delta (0))\) such that \(u(\cdot ,t_{k_j}) \rightarrow z\) in \(C^0(\overline{\varOmega }{\setminus } B_\delta (0))\) as \(j\rightarrow \infty \). But since from Lemma 3.7 we already know that \(u(\cdot ,t)\rightarrow u_0\) in \((W_0^{2,1}(\varOmega {\setminus } \overline{B}_\delta (0)))^\star \) as \(t\searrow 0\), this necessarily implies that \(z=u_0\) and thereby contradicts (3.48). \(\square \)

Proving our main results thereby essentially reduces to collecting tesserae:

Proof of Theorem 1.1

We only need to combine the results from Lemma 2.7, 3.3 and 3.4 with the convergence statement asserted by Lemma 3.10. \(\square \)

Proof of Corollary 1.2

Setting \(\phi (r):=e^{\lambda r^{-\alpha }}\), \(r>0\), we immediately see that (1.5) and (1.6) hold, and computing

$$\begin{aligned}&\phi '(r) = -\alpha \lambda r^{-\alpha -1} e^{\lambda r^{-\alpha }} \quad \text{ and } \quad \phi ''(r) = \alpha ^2 \lambda ^2 r^{-2\alpha -2} e^{\lambda r^{-\alpha }} + \alpha (\alpha +1) \lambda r^{-\alpha -2} e^{\lambda r^{-\alpha }}, \\&\qquad r>0, \end{aligned}$$

we see that

$$\begin{aligned} \frac{\phi ''(r)}{\phi ^2(r)} = \alpha ^2 \lambda ^2 r^{-2\alpha -2} e^{-\lambda r^{-\alpha }} + \alpha (\alpha +1) \lambda r^{-\alpha -2} e^{-\lambda r^{-\alpha }} \rightarrow 0 \qquad \text{ as } r\searrow 0, \end{aligned}$$

which implies (1.7) for some suitably large \(K_1>0\). Likewise, (1.8) can be achieved upon observing that

$$\begin{aligned} \frac{\phi (2r)}{r^n \phi (r)} = r^{-n} e^{-(1-2^{-\alpha }) \lambda r^{-\alpha }} \rightarrow 0 \qquad \text{ as } r\searrow 0, \end{aligned}$$

whereas (1.9) is a direct consequence of the hypothesis (1.15). \(\square \)