1 Introduction

The dynamical features of the nonlinear parabolic equation

$$\begin{aligned} u_t = u^p \Delta u \end{aligned}$$
(1.1)

are known to depend quite crucially on the exponent \(p> 0\) that quantifies the strength of diffusion degeneracies in regions where the solution is small; indeed, a considerable literature has rigorously revealed various parabolictiy-diminishing effects going along with an increase of p. Among the most comprehensively understood aspects in this regard seem to be phenomena related to propagation of positivity: In striking difference to the borderline case \(p=0\) of the linear heat equation, throughout the range \(p\in (0,1)\) in which (1.1) is equivalent to the porous medium equation \(v_t=\Delta v^m\) with \(m=\frac{1}{1-p}>1\), compactly supported initial data evolve into continuous solutions [8] which at each point in the considered domain do eventually become positive, but the spatial positivity set of which propagates at finite speed ([5, 15]; see also [2, 7, 9, 13] for more detailed information, and [1] or [16] for an overview).

In this respect, a second sharp transition in behavior can be observed when further increasing p: Yet more drastically, namely, the support of solutions remains constant in time whenever \(p\ge 1\) ([21]; cf. also Proposition 3.1 below), and in the case \(p>2\) there even exist classical solutions to an associated homogeneous Dirichlet problem in domains \(\Omega \subset {\mathbb {R}}^n\) which satisfy \(u(\cdot ,t)\in C_0^\infty (\Omega )\) for all \(t>0\) [22]. Two examples addressing a relative of (1.1) with \(p\ge 3\) in such Dirichlet problems, augmented by the zero-oder source term \(u^{p+1}\), have unveiled that in such very strongly degenerate cases, the global behavior may be influenced quite substantially, up to an enforcement of repeated oscillations between vanishing and everywhere infinite profiles, by the particular manner in which the boundary value 0 is approached by the initial data [19, 20].

Beyond this, however, increasing the degeneracy in (1.1) may considerably affect the dynamics even of solutions which are strictly positive, and for which (1.1) hence actually is non-degenerate near each fixed point (x, t). In the context of the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=u^p \Delta u, &{}\quad x\in {\mathbb {R}}^n, \ t>0, \\ u(x,0)=u_0(x), &{} \quad x\in {\mathbb {R}}^n, \end{array} \right. \end{aligned}$$
(1.2)

for instance, the large time asymptotics of positive classical solutions emanating from positive and sufficiently fast decaying initial data \(u_0\) in general differs from that in the heat equation by some quantitative corrections already in the porous medium regime: When \(p\in (0,1)\), namely, any such solution with \(u_0^{1-p}\in L^1({\mathbb {R}}^n)\) satisfies \(\frac{1}{C} t^{-\frac{n}{2+(n-2)p}} \le \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \le C t^{-\frac{n}{2+(n-2)p}}\) for all \(t>1\) with some \(C>0\) ([16, Theorem I.2.5], [12]), meaning that temporal decay properties of widely arbitrary solutions with rapidly decreasing initial data rather closely parallel those of the particular explicit self-similar solutions that form the celebrated family of so-called Barenblatt solutions [1, 3].

As some more recent findings have been indicating, however, outside the range \(p\in (0,1)\) within which such Barenblatt solutions are available, some yet more subtle facets in the dependence of large time decay on spatial asymptotics need to be expected. When \(p\ge 1\), namely, decay rates of strictly positive solutions have some common lower bound which can be approached up to errors with arbitrarily small algebraic asymptotics, but which can never be attained exactly by any such solution. More precisely, the following has been shown in [10]:

Proposition A

Let \(n\ge 1, p\ge 1\) and \(u_0\in C^0({\mathbb {R}}^n) \cap L^\infty ({\mathbb {R}}^n)\) be such that \(u_0(x)>0\) for all \(x\in {\mathbb {R}}^n\). Then (1.2) possesses a classical solution \(u\in C^0({\mathbb {R}}^n \times [0,\infty )) \cap C^{2,1}({\mathbb {R}}^n\times (0,\infty ))\) which is such that \(u(x,t)>0\) for all \(x\in {\mathbb {R}}^n\) and \(t>0\), and which is minimal in the sense that whenever \(T\in (0,\infty ]\) and \(\widetilde{u} \in C^0({\mathbb {R}}^n\times [0,T)) \cap C^{2,1}({\mathbb {R}}^n\times (0,T))\) are such that \(\widetilde{u}\) is positive and solves (1.2) classically in \({\mathbb {R}}^n\times (0,T)\), we have \(u\le \widetilde{u}\) in \({\mathbb {R}}^n\times (0,T)\).

Moreover,

$$\begin{aligned} \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \}=\infty \end{aligned}$$

and if in addition \(u_0\in \bigcap _{q>0} L^q({\mathbb {R}}^n)\), then

$$\begin{aligned} \limsup _{t\rightarrow \infty } \Big \{ t^{\frac{1}{p}-\delta } \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty \quad \mathrm{for\, all\, } \delta >0. \end{aligned}$$

Apart from this, in [11] respective classes of suitably fast decreasing initial data have been identified within which actually any logarithmic, and even doubly logarithmic, corrections to the algebraic decay of \(t^{-\frac{1}{p}}\) is essentially attained by corresponding positive solutions to (1.2) (see [11, Corollaries 1.5, 1.6, 1.8 and 1.9]).

Main results I: Arbitrarily slow increase of \(t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )}\). The first objective of the present study now consists in examining whether beyond the latter particular examples, arbitrarily small deviations of the borderline decay rate indicated in Proposition A can be undercut by some positive solutions to (1.2). Our main results in this direction show that this indeed is possible in the following flavor that seems to be the least restrictive conceivable in this regard:

Theorem 1.1

Let \(n\ge 1\) and \(p\ge 1\), and suppose that \(f\in C^0([0,\infty ))\) is positive and such that \(f(t)\rightarrow + \infty \) as \(t\rightarrow \infty \). Then there exists a positive nonincreasing function \(\phi \in C^0([0,\infty ))\) with the property that whenever \(u_0\in C^0({\mathbb {R}}^n)\) is radially symmetric and such that

$$\begin{aligned} 0< u_0(x) < \phi (|x|) \qquad \mathrm{for\, all }\, x\in {\mathbb {R}}^n, \end{aligned}$$
(1.3)

the corresponding minimal solution u of (1.2) satisfies

$$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \qquad \mathrm{as }\, t\rightarrow \infty . \end{aligned}$$
(1.4)

Main results II: Attaining vs. remaining away from critical decay for solutions with \(\{u_0>0\} \ne {\mathbb {R}}^n\). We shall next address the question whether critical decay can be observed at least when the initial data are not strictly positive throughout \({\mathbb {R}}^n\). Here we note that already at the level of basic solution theories, the strong diffusion degeneracies present in the considered range \(p\ge 1\) give rise to significant challenges, for the caveat documented in [14] indicates that within straightforward and seemingly natural adaptations of weak solution concepts to the framework of (1.2), uniqueness of solutions can not even be expected for initial data from \(C_0^\infty ({\mathbb {R}}^n)\). We accordingly resort to a slightly modified notion of solvability, to be substantiated in Proposition 3.1 below, which inter alia requires continuity of the considered solution, and for which we thus, in accordance with known results on discontinuous solution behavior in the presence of initially isolated zeros [4, 6], impose some restrictions on the regularity of the positivity set \(\{u_0>0\}\) in order to assert the mere existence of such solutions. More precisely, in this part we shall assume that

$$\begin{aligned} \left\{ \begin{array}{l} 0\not \equiv u_0\in C^0({\mathbb {R}}^n) \hbox { is nonnegative such that}\\ \{u_0>0\} \hbox { coincides with the interior of } \mathrm{supp} \, u_0, \hbox { and that}\\ \hbox {each } \Omega \in {{{\mathcal {C}}}}(u_0) \hbox { is a bounded domain with Lipschitz boundary}, \end{array} \right. \end{aligned}$$
(1.5)

where for \(0\le \varphi \in C^0({\mathbb {R}}^n)\) we have set

$$\begin{aligned} {{{\mathcal {C}}}}(\varphi ):=\Big \{ \Omega \subset {\mathbb {R}}^n \ \Big | \ \Omega \hbox { is a connected component of } \{\varphi >0\} \Big \}, \end{aligned}$$
(1.6)

and note that under these hypotheses, a uniquely determined continous weak solution can indeed be found (Proposition 3.1).

Now our intention in this part is to relate the possibility of exhibiting critical decay to some properties exclusively referring to features of the connected components of \(\{u_0>0\}\), rather than to the size of \(u_0\) nor its overall decay in space. Specifically, our first result in this respect reads as follows.

Proposition 1.2

Let \(n\ge 1\) and \(p\ge 1\), and suppose that \(u_0\) satisfies (1.5) with

$$\begin{aligned} \inf _{\Omega \in {{{\mathcal {C}}}}(u_0)} \sup _{\begin{array}{c} 0 \le \varphi \in C^0({{\overline{\Omega }}})\cap C^2(\Omega ) \\ \Vert \varphi \Vert _{L^\infty (\Omega )}=1 \end{array}} \inf _{x\in \Omega } \Big \{ - \varphi ^{p-1}(x)\Delta \varphi (x)\Big \} >0. \end{aligned}$$
(1.7)

Then the continous weak solution u of (1.2) has the property that

$$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$
(1.8)

Indeed, the following consequence thereof establishes a link to the maximum size of all the members from \({{{\mathcal {C}}}}(u_0)\):

Corollary 1.3

Let \(n\ge 1\) and \(p\ge 1\), and suppose that \(u_0\) is such that (1.5) holds, and that there exists \(K>0\) with the property that each \(\Omega \in {{{\mathcal {C}}}}(u_0)\) lies between two parallel hyperplanes with distance K, that is, for any such \(\Omega \) one can find \(x_0\in {\mathbb {R}}^n\) and \(A\in SO(n)\) such that

$$\begin{aligned} \Omega \subset x_0+AS \quad \mathrm{with} \quad S:= \Big \{ x=(x_1,\ldots ,x_n) \in {\mathbb {R}}^n \ \Big | \ 0<x_1<K\Big \}. \end{aligned}$$

Then the continuous weak solution u of (1.2) satisfies (1.8). In particular, this conclusion holds if

$$\begin{aligned} \sup _{\Omega \in {{{\mathcal {C}}}}(u_0)} \mathrm{diam} \, \Omega < \infty . \end{aligned}$$

We shall next identify a criterion, partially complementary to that from Proposition 1.2, as sufficient to ensure absence of critical decay speeds also for some not strictly positive initial data. In formulating this, for notational convenience we abbreviate \(C_0^0({{\overline{\Omega }}}):=\{\varphi \in C^0({{\overline{\Omega }}}) \ | \ \varphi |_{\partial \Omega }=0\}\) for open sets \(\Omega \subset {\mathbb {R}}^n\).

Proposition 1.4

Let \(n\ge 1\) and \(p\ge 1\), and let \(u_0\) be such that (1.5) holds, and that

$$\begin{aligned} \inf _{\Omega \in {{{\mathcal {C}}}}(u_0)} \inf _{\begin{array}{c} 0 \le \varphi \in C_0^0({{\overline{\Omega }}})\cap {C^2(\{\varphi>0\})} \\ \Vert \varphi \Vert _{L^\infty (\Omega )}=1 \end{array}} \sup _{x\in {\{\varphi >0\}} } \Big \{ - \varphi ^{p-1}(x)\Delta \varphi (x)\Big \} =0. \end{aligned}$$
(1.9)

Then for the continous weak solution u of (1.2) we have

$$\begin{aligned} \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} =\infty . \end{aligned}$$
(1.10)

To finally indicate that here the requirement (1.9) again is in close relationship to the component sizes of \(\{u_0>0\}\), let us adopt the standard notation

$$\begin{aligned} \lambda _1(\Omega ) := \inf _{0\not \equiv \varphi \in W_0^{1,2}(\Omega )} \frac{\int _\Omega |\nabla \varphi |^2}{\int _\Omega \varphi ^2} \end{aligned}$$
(1.11)

for the principal Dirichlet eigenvalue of \(-\Delta \) in a bounded domain \(\Omega \subset {\mathbb {R}}^n\). In fact, we shall see that the conclusion of Proposition 1.4 holds whenever \(\{u_0>0\}\) contains components with arbitrarily small values of these eigenvalues, and hence whenever \(\{u_0>0\}\) has infinite inradius:

Corollary 1.5

Let \(n\ge 1\) and \(p\ge 1\), and let \(u_0\) be such that (1.5) holds, and that

$$\begin{aligned} \inf _{\Omega \in {{{\mathcal {C}}}}(u_0)} \lambda _1(\Omega ) =0. \end{aligned}$$
(1.12)

Then the continuous weak solution u of (1.2) satisfies (1.10). This especially follows if

$$\begin{aligned} \sup \bigg \{ R>0 \ \bigg | \, \mathrm{There\,exist } \,\Omega \in {{{\mathcal {C}}}}(u_0) \,\mathrm{and } \, x_0\in {\mathbb {R}}^n \,\mathrm{such \, that }\, B_R(x_0)\subset \Omega \bigg \} = \infty . \end{aligned}$$
(1.13)

2 Slow Increase of   \(t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )}\). Proof of Theorem 1.1

2.1 Specifying the Objective

Our approach toward the derivation of Theorem 1.1 will be based on the following fundamental observation made in [11, Theorem 1.3].

Theorem B

Assume that \(n\ge 1\) and \(p\ge 1\), that \(s_0>0\), and that \({{{\mathcal {L}}}}\in C^0([0,\infty )) \cap L^\infty ((0,\infty )) \cap C^2((0,s_0))\) is positive and nondecreasing on \((0,\infty )\) and such that \({{{\mathcal {L}}}}(0)=0\), that there exist \(a>0\) and \(\lambda _0>0\) fulfilling

$$\begin{aligned} {{{\mathcal {L}}}}(s) \le (1+a\lambda ) {{{\mathcal {L}}}}(s^{1+\lambda }) \quad \,\mathrm{for \, all}\, s\in (0,s_0)\,\mathrm{and }\, \lambda \in (0,\lambda _0), \end{aligned}$$
(2.1)

and that furthermore

$$\begin{aligned} s{{{\mathcal {L}}}}''(s) \ge - \frac{3p+q_0-2}{p+q_0} {{{\mathcal {L}}}}'(s) \quad \mathrm{for\, all }\, s\in (0,s_0) \end{aligned}$$
(2.2)

with a certain \(q_0>0\). Then whenever \(u_0\in C^0({\mathbb {R}}^n)\) is positive, radially symmetric and nonincreasing with respect to |x| and such that \(u_0 < \min \Big \{ s_0^\frac{2}{p}, s_0^\frac{2}{p+q_0}\Big \}\) in \({\mathbb {R}}^n\) as well as

$$\begin{aligned} \int _{{\mathbb {R}}^n} {{{\mathcal {L}}}}(u_0) < \infty , \end{aligned}$$

there exist \(t_0>0\) and \(C>0\) such that the minimal classical solution u of (1.2) satisfies

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \le C t^{-\frac{1}{p}} {{{\mathcal {L}}}}^{-\frac{2}{np}} \Big (\frac{1}{t}\Big ) \quad \,\mathrm{for\, all }\, t\ge t_0. \end{aligned}$$
(2.3)

Now in order to appropriately prepare a construction of a function \({{{\mathcal {L}}}}\) which on the one hand satisfies the requirements in Theorem B, and especially the inequalities (2.1) and (2.2), but for which, on the other hand, the correction factor \({{{\mathcal {L}}}}^{-\frac{2}{np}}(\frac{1}{t})\) in (2.3) remains small relative to a given divergent function f in the style of Theorem 1.1, let us firstly derive a handy criterion sufficient for (2.1).

Lemma 2.1

Let \(s_0\in (0,1]\) and \(a\in (0,1]\), and suppose that \({{{\mathcal {L}}}}\in C^1((0,s_0))\) is positive and such that

$$\begin{aligned} {{{\mathcal {L}}}}'(s) \le a\cdot \frac{{{{\mathcal {L}}}}(s)}{s\ln \frac{s_0}{s}} \quad \,\mathrm{for\, all }\, s\in (0,s_0). \end{aligned}$$
(2.4)

Then

$$\begin{aligned} {{{\mathcal {L}}}}(s) \le (1+a\lambda ) \cdot {{{\mathcal {L}}}}(s^{1+\lambda }) \quad \,\mathrm{for\, all }\, s\in (0,s_0) \,\mathrm{and }\, \lambda >0. \end{aligned}$$
(2.5)

Proof

Since \({{{\mathcal {L}}}}\) is positive, letting

$$\begin{aligned} H(s):=\ln \frac{1}{{{{\mathcal {L}}}}(s)}, \quad s\in (0,s_0), \end{aligned}$$

we obtain a well-defined element H of \(C^1((0,s_0))\) which according to (2.4) satisfies

$$\begin{aligned} H'(s) = - \frac{{{{\mathcal {L}}}}'(s)}{{{{\mathcal {L}}}}(s)} \ge - \frac{a}{s\ln \frac{s_0}{s}} \quad \,\mathrm{for\, all }\, s\in (0,s_0). \end{aligned}$$

Using that \(s_0\le 1\), and that thus \(s^{a+\lambda }\le s\) for all \(s\in (0,s_0)\) and \(\lambda >0\), we can therefore estimate

$$\begin{aligned} H(s) - H(s^{1+\lambda })= & {} \int _{s^{1+\lambda }}^s H'(\sigma )d\sigma \nonumber \\\ge & {} -a \int _{s^{1+\lambda }}^s \frac{d\sigma }{\sigma \ln \frac{s_0}{\sigma }} \nonumber \\= & {} - a \int _{(\frac{s}{s_0})^{1+\lambda }}^{\frac{s}{s_0}} \frac{d\xi }{\xi \ln \frac{1}{\xi }} \quad \,\mathrm{for\, all }\, s\in (0,s_0) \hbox { and } \lambda >0. \end{aligned}$$
(2.6)

Since

$$\begin{aligned} \int _{\Sigma ^{1+\lambda }}^\Sigma \frac{d\xi }{\xi \ln \frac{1}{\xi }}= & {} - \bigg [ \ln \ln \frac{1}{\xi } \bigg ]_{\xi =\Sigma ^{1+\lambda }}^{\xi =\Sigma } \\= & {} - \ln \left( \frac{\ln \frac{1}{\Sigma }}{(1+\lambda ) \ln \frac{1}{\Sigma }} \right) \\= & {} \ln (1+\lambda ) \quad \,\mathrm{for\, all }\, \Sigma \in (0,1) \hbox { and } \lambda >0, \end{aligned}$$

and since

$$\begin{aligned} a\ln (1+\lambda ) = \ln \Big \{ (1+\lambda )^a \Big \} \le \ln (1+a\lambda ) \quad \,\mathrm{for\, all }\, \lambda >0 \end{aligned}$$

due to the fact that \(a\le 1\) ensures that \((1+\lambda )^a \le 1+a\lambda \) for all \(\lambda >0\), from (2.6) we thus obtain that

$$\begin{aligned} H(s)-H(s^{1+\lambda }) \ge - \ln (1+a\lambda ) \quad \,\mathrm{for\, all }\, s\in (0,s_0) \hbox { and } \lambda >0. \end{aligned}$$

According to the definition of H, this implies that

$$\begin{aligned} \ln \left( \frac{{{{\mathcal {L}}}}(s)}{(1+a\lambda ) {{{\mathcal {L}}}}(s^{1+\lambda })} \right) = H(s^{1+\lambda })-H(s)-\ln (1+a\lambda ) \le 0 \quad \,\mathrm{for\, all }\, s\in (0,s_0) \hbox { and } \lambda >0 \end{aligned}$$

and hence establishes (2.5). \(\square \)

Fortunately, both this condition (2.4) and (2.2) can be reformulated in a rather convenient manner after a simple variable transformation:

Lemma 2.2

Let \(h\in C^2([0,\infty ))\). Then for any choice of \(s_0>0\), writing

$$\begin{aligned} z\equiv z(s):=\ln \frac{s_0}{s} \quad \,\mathrm{and}\, \quad {{{\mathcal {L}}}}(s):=e^{-h(z)}, \quad s\in (0,s_0], \end{aligned}$$
(2.7)

defines a positive function \({{{\mathcal {L}}}}\in C^2((0,s_0])\) which is such that whenever \(\kappa \in {\mathbb {R}}\), for all \(s\in (0,s_0)\) we have

$$\begin{aligned} s\ln \frac{s_0}{s} \cdot \frac{{{{\mathcal {L}}}}'(s)}{{{{\mathcal {L}}}}(s)} = zh'(z) \end{aligned}$$
(2.8)

and

$$\begin{aligned} s{{{\mathcal {L}}}}''(s) + \kappa {{{\mathcal {L}}}}'(s) = \frac{1}{s} \cdot e^{-h(z)} \cdot \Big \{ -h''(z) + (\kappa -1)h'(z) + h'^2(z)\Big \} \end{aligned}$$
(2.9)

with \(z=z(s)\).

Proof

On the basis of (2.7), for \(s\in (0,s_0)\) we compute \(z'(s)=-\frac{1}{s}\) and

$$\begin{aligned} {{{\mathcal {L}}}}'(s)=\frac{1}{s} \cdot e^{-h(z)} h'(z) \end{aligned}$$

and

$$\begin{aligned} {{{\mathcal {L}}}}''(s)=-\frac{1}{s^2} \cdot e^{-h(z)} h''(z) + \frac{1}{s^2} \cdot e^{-h(z)} h'^2(z) - \frac{1}{s^2} \cdot e^{-h(z)} h'(z) \end{aligned}$$

with \(z=z(s)\), so that both (2.8) and (2.9) readily follow. \(\square \)

In line with Theorem B, Lemmas 2.1 and 2.2, we will thus subsequently intend to make sure that given any function \(f=f(t)\) exhibiting arbitrarily slow unbounded growth, after transformation to a positive divergent function \(F=F(z)\) on \([0,\infty )\) in the style suggested by (2.3) and (2.7), we can find a yet unbounded minorant h for which the correspondingly translated version \({{{\mathcal {L}}}}\), as defined through (2.7), satisfies the requirement in (2.1) in the sharpened sense expressed in Lemma 2.1 and (2.8), and which simultaneously complies with (2.2) via (2.9). Here we observe that since fortunately the rightmost summand in (2.9) is nonnegative, and since the factor appearing on the right of (2.2) satisfies \(\frac{3p+q_0-2}{p+q_0}\ge 1\) for any choice of \(p\ge 1\) and \(q_0>0\), with regard to (2.2) it will be sufficient to construct h in such a way that \(h'\ge 0\) and \(h''\le 0\).

2.2 Construction of Slowly Increasing Minorants

To accomplish the first among two major steps in our design of a smooth minorant with the desired properties, let us construct a piecewise linear but already concave preliminary candidate.

Lemma 2.3

Let \(F\in C^0([0,\infty ))\) be positive and such that \(F(z)\rightarrow + \infty \) as \(z\rightarrow \infty \). Then there exist a strictly increasing sequence \((z_j)_{j\in {\mathbb {N}}} \subset [0,\infty )\) and a positive concave function \(h_0\in W^{1,\infty }_{loc}([0,\infty ))\) such that \(z_1=0\) and \(z_j\rightarrow \infty \) as \(j\rightarrow \infty \), that for all \(j\in {\mathbb {N}}\) we have

$$\begin{aligned} h_0\in C^2((z_j,z_{j+1})) \quad \mathrm{with} \quad h_0''(z)=0 \quad \,\mathrm{for\, all }\, z\in (z_j,z_{j+1}) \end{aligned}$$
(2.10)

and

$$\begin{aligned} 0<h_0'(z) \le \frac{1}{z+1} \quad \,\mathrm{for\, all }\, z\in (z_j,z_{j+1}), \end{aligned}$$
(2.11)

and that

$$\begin{aligned} h(z) \le F(z) \quad \,\mathrm{for\, all }\, z>0 \end{aligned}$$
(2.12)

and

$$\begin{aligned} h_0(z) \rightarrow + \infty \quad \mathrm{as } z\rightarrow \infty . \end{aligned}$$
(2.13)

Proof

We pick \(b\in (0,1)\) in such a way that \(F(z)\ge 2b\) for all \(z\ge 0\), and construct \((z_j)_{j\in {\mathbb {N}}}\) and \(h_0\) recursively as follows: Taking \(z_1:=0\), and for notational convenience also introducing \(z_0:=-b\), we assume that for some \(j\ge 1\) we already have found \((z_i)_{i\in \{0,\ldots ,j\}} \subset {\mathbb {R}}\) such that \(z_{i+1}>z_i\) for all \(i\in \{0,\ldots ,j-1\}\) and

$$\begin{aligned} F(z) \ge (i+1)b \quad \,\mathrm{for\, all }\, z\ge z_i \hbox { and } i\in \{1,\ldots ,j\}, \end{aligned}$$
(2.14)

and that letting

$$\begin{aligned} h_0(z):=m_i \cdot (z-z_i) + i\cdot b, \quad z\in (z_i,z_{i+1}], \quad i\in \{0,\ldots ,j-1\}, \end{aligned}$$
(2.15)

with

$$\begin{aligned} m_i:=\frac{b}{z_{i+1}-z_i}, \quad i\in \{0,\ldots ,j-1\}, \end{aligned}$$
(2.16)

defines a continuous and concave function \(h_0\) on \((z_0,z_j]\) which satisfies

$$\begin{aligned} 0 < h_0'(z) \le \frac{1}{z+1} \quad \,\mathrm{for\, all }\, z\in (z_0,z_j) {\setminus } \Big \{ z_i \ \Big | \ i\in \{1,\ldots ,j-1\} \Big \}. \end{aligned}$$
(2.17)

Now since \(F(z)\rightarrow +\infty \) as \(z\rightarrow \infty \), and since \(b<1\), we can fix \(z_{j+1}>z_j\) large enough such that

$$\begin{aligned} F(z) \ge (j+2)b \quad \,\mathrm{for\, all }\, z\ge z_{j+1}, \end{aligned}$$
(2.18)

and that furthermore

$$\begin{aligned} z_{j+1} > 2z_j - z_{j-1} \end{aligned}$$
(2.19)

as well as

$$\begin{aligned} {(1-b)} z_{j+1} \ge z_j+b. \end{aligned}$$
(2.20)

Then letting

$$\begin{aligned} m_j:=\frac{b}{{z_{j+1}}-z_j} \quad \,\mathrm{and}\, \quad h_0(z):=m_j\cdot (z-z_j) + j\cdot b, \quad z\in (z_j,z_{j+1}], \end{aligned}$$
(2.21)

evidently extends \(h_0\) to a function defined on all of \((z_0,z_{j+1}]\), in a manner consistent with (2.15) and (2.16), which is continuous on \((z_0,z_{j+1}]\) due to the fact that according to (2.21) and (2.15) we have \(h_0(z) \rightarrow j\cdot b=h_0(z_j)\) as \((z_j,z_{j+1}] \ni z \searrow z_j\). To see that \(h_0\) is concave on \((0,z_{j+1}]\), in view of (2.15) and (2.21) it is sufficient to observe that thanks to the definition of \((m_i)_{i\in \{0,\ldots ,j\}}\) in (2.16) and (2.21), the requirement in (2.19) guarantees that

$$\begin{aligned} \frac{1}{m_j} - \frac{1}{m_{j-1}} = \frac{z_{j+1}-2z_j + z_{j-1}}{b}>0, \end{aligned}$$

which namely asserts that

$$\begin{aligned} \lim _{z\searrow z_j} h_0'(z) = m_j < m_{j-1} = \lim _{z\nearrow z_j} h_0'(z). \end{aligned}$$

We next make use of (2.20) to confirm that

$$\begin{aligned} (z+1) \cdot h_0'(z)= & {} (z+1) m_j \\\le & {} (z_{j+1}+1) m_j \\= & {} \frac{b(z_{j+1}+1)}{z_{j+1}-z_j} \\= & {} b \cdot \left\{ 1 +\frac{z_j +1}{z_{j+1}-z_j} \right\} \\\le & {} b\cdot \left\{ 1 + \frac{z_j +1}{\frac{z_j+b}{1-b} - z_j} \right\} \\= & {} 1 { \quad \,\mathrm{for\, all }\, z\in (z_j,z_{j+1}),} \end{aligned}$$

which together with (2.17) implies that

$$\begin{aligned} 0 < h_0'(z) \le \frac{1}{z+1} \quad \,\mathrm{for\, all }\, z\in (z_0,z_{j+1}) {\setminus } \Big \{ z_i \ \Big | \ i\in \{1,\ldots ,j\} \Big \}. \end{aligned}$$

Since (2.18) along with (2.14) clearly ensures that

$$\begin{aligned} F(z) { \ge } (i+1) b \quad \,\mathrm{for\, all }\, z\ge z_i \hbox { and any} \, i\in \{1,\ldots ,j+1\}, \end{aligned}$$

this completes our inductive construction of a strictly increasing sequence \((z_j)_{j\ge 0} \subset {\mathbb {R}}\) which satisfies \(z_1=0\) and is such that (2.14) holds for all \(j\in {\mathbb {N}}\), in particular meaning that necessarily \(z_j\rightarrow \infty \) as \(j\rightarrow \infty \).

In view of the fact that (2.15)–(2.17) are valid for all \(j\in {\mathbb {N}}\), we therefore moreover obtain a function \(h_0: (z_0,\infty )\rightarrow {\mathbb {R}}\) which when restricted to \([0,\infty )\) belongs to \(W^{1,\infty }([0,\infty ))\) and satisfies (2.10) and (2.11) for all \(j\in {\mathbb {N}}\), which is concave by construction, and for which (2.13) holds thanks to the circumstance that \(h_0(z_j)=j\cdot b\rightarrow + \infty \) as \(j\rightarrow \infty \).

In order to finally verify (2.12), given \(z>0\) we fix \(j\in {\mathbb {N}}\) such that \(z\in (z_j,z_{j+1}]\), and use the definition of \(h_0\) in \((z_j,z_{j+1}]\) implied by (2.15) in estimating

$$\begin{aligned} h_0(z) = b\cdot \frac{z-z_j}{z_{j+1}-z_j} + j\cdot b \le b+j\cdot b, \end{aligned}$$

because \(z\le z_{j+1}\). Since, on the other hand, the inequality \(z\ge z_j\) enables us to conclude from (2.14) that

$$\begin{aligned} F(z) \ge (j+1) b, \end{aligned}$$

this completes the proof. \(\square \)

We next prepare an appropriate smoothing procedure, to be finally performed near each discontinuity point of the function \(h_0\) from Lemma 2.3, by means of an explicit construction concentrating on cases in which only one point of such nonsmooth behavior is present.

Lemma 2.4

Let \(z_\star \in {\mathbb {R}}\) and \(h_\star \in C^0({\mathbb {R}}) \cap C^2({\mathbb {R}}{\setminus } \{z_\star \})\) be concave and such that \(h_\star ''(z)=0\) for all \(z\in {\mathbb {R}}{\setminus } \{z_\star \}\). Then for any \(\varepsilon >0\) there exists \(h_\varepsilon \in C^2({\mathbb {R}})\) such that

$$\begin{aligned} h_\varepsilon (z)=h_\star (z) \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}{\setminus } (z_\star -\varepsilon ,z_\star +\varepsilon ) \end{aligned}$$
(2.22)

and

$$\begin{aligned} h_\varepsilon (z) \le h_\star (z) \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}\end{aligned}$$
(2.23)

as well as

$$\begin{aligned} h_\varepsilon ''(z) \le 0 \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}. \end{aligned}$$
(2.24)

Proof

Our hypotheses precisely mean that there exist \(c_1\in {\mathbb {R}}, {\underline{m}}\in {\mathbb {R}}\) and \({\overline{m}}\le {\underline{m}}\) such that

$$\begin{aligned} h_\star (z) = \left\{ \begin{array}{ll} {\underline{m}}\cdot (z-z_\star ) + c_1 &{}\quad \,\mathrm{for\, all }\, z\le z_\star , \\ {\overline{m}}\cdot (z-z_\star ) + c_1 &{}\quad \,\mathrm{for\, all }\, z>z_\star , \end{array} \right. \end{aligned}$$

and we may assume that actually \({\overline{m}}<{\underline{m}}\), for otherwise choosing \(h_\varepsilon \equiv h_\star \) clearly warrants validity of (2.22)–(2.24).

For fixed \(\varepsilon >0\), we then let

$$\begin{aligned} A_1:=\frac{\varepsilon }{2} ({\underline{m}}-{\overline{m}}), \quad A_2:=\frac{{\underline{m}}+{\overline{m}}}{2} \quad \,\mathrm{and}\, \quad A_3:=c_1-\frac{\varepsilon }{2} ({\underline{m}}-{\overline{m}}) - \frac{1}{2} ({\underline{m}}+{\overline{m}}) z_\star \end{aligned}$$

and observe that, as can easily be verified, these selections ensure that

$$\begin{aligned} h_\star (z)= A_1 \cdot {\widehat{h}}_\star \Big (\frac{z-z_\star }{\varepsilon }\Big ) + A_2 z + A_3 \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}, \end{aligned}$$
(2.25)

where

$$\begin{aligned} {\widehat{h}}_\star (\xi ):=1-|\xi |, \quad \xi \in {\mathbb {R}}. \end{aligned}$$

To see that the normalized situation thus obtained can be coped with by means of an explicit construction, we introduce

$$\begin{aligned} \widehat{h}_1(\xi ):=\left\{ \begin{array}{ll} 1+\xi , &{}\quad \xi \in (-\infty ,-1], \\ - \frac{1}{3} \xi ^3 - \xi ^2 + \frac{2}{3}, &{}\quad \xi \in (-1,0], \\ \frac{1}{3} \xi ^3 - \xi ^2 + \frac{2}{3}, &{}\quad \xi \in (0,1], \\ 1-\xi , &{} \quad \xi \in (1,\infty ). \end{array} \right. \end{aligned}$$

Then straightforward computation shows that \(\widehat{h}_1\) belongs to \(C^2({\mathbb {R}})\) and satisfies

$$\begin{aligned} \widehat{h}_1(\xi ) = {\widehat{h}}_\star (\xi ) \quad \,\mathrm{for\, all }\, \xi \in {\mathbb {R}}{\setminus } (-1,1) \end{aligned}$$
(2.26)

and

$$\begin{aligned} \widehat{h}_1(\xi )-{\widehat{h}}_\star (\xi )= & {} \Big \{ \frac{1}{3} |\xi |^3 - \xi ^2 + \frac{2}{3}\Big \} - \Big \{ 1-|\xi |\Big \} \nonumber \\\le & {} \frac{1}{3} \cdot (|\xi |-1)^3 \nonumber \\\le & {} 0 \quad \,\mathrm{for\, all }\, \xi \in (-1,1) \end{aligned}$$
(2.27)

as well as

$$\begin{aligned} \widehat{h}_1''(\xi ) = 2(|\xi |-1) \le 0 \quad \,\mathrm{for\, all }\, \xi \in (-1,1). \end{aligned}$$
(2.28)

Therefore, if in reminiscence of (2.25) we let

$$\begin{aligned} h_\varepsilon (z):=A_1 \cdot \widehat{h}_1 \Big (\frac{z-z_\star }{\varepsilon }\Big ) + A_2 z + A_3, \quad z\in {\mathbb {R}}, \end{aligned}$$
(2.29)

then (2.22) and (2.23) directly result from (2.26), (2.27) and (2.25), whereas (2.24) is a consequence of (2.28), because

$$\begin{aligned} h_\varepsilon ''(z) =\frac{A_1}{\varepsilon } \cdot \widehat{h}_1'' \Big (\frac{z-z_\star }{{ \varepsilon ^2}}\Big ) \quad \,\mathrm{for\, all }\, z\in (z_\star -\varepsilon ,z_\star +\varepsilon ) \end{aligned}$$

by (2.29). \(\square \)

Now suitable application of the latter to the function gained in Lemma 2.3 yields the following.

Lemma 2.5

Let \(F \in C^0([0,\infty ))\) be such that \(F(z)>0\) for all \(z\ge 0\) and \(F(z)\rightarrow + \infty \) as \(z\rightarrow \infty \). Then there exists \(h\in C^2([0,\infty ))\) such that

$$\begin{aligned} 0<h(z) \le F(z) \quad \,\mathrm{for\, all }\, z\ge 0 \end{aligned}$$
(2.30)

and

$$\begin{aligned} 0 < h'(z) \le \frac{1}{z} \quad \,\mathrm{for\, all }\, z> 0, \end{aligned}$$
(2.31)

and such that moreover

$$\begin{aligned} h''(z)\le 0 \quad \,\mathrm{for\, all }\, z> 0 \end{aligned}$$
(2.32)

and

$$\begin{aligned} h(z)\rightarrow +\infty \quad \mathrm{as } z\rightarrow \infty . \end{aligned}$$
(2.33)

Proof

We take \((z_j)_{j\in {\mathbb {N}}} \subset [0,\infty )\) and \(h_0\in W^{1,\infty }_{loc}([0,\infty ))\) as provided by Lemma 2.3, and for \(j\in {\mathbb {N}}\) with \(j\ge 2\) we then obtain from the linearity of \(h_0\) on \([z_{j-1},z_j]\) and on \([z_j,z_{j+1}]\) that

$$\begin{aligned} h_0(z)=h_\star ^{(j)}(z) \quad \,\mathrm{for\, all }\, z\in [z_{j-1},z_{j+1}], \end{aligned}$$
(2.34)

where

$$\begin{aligned} h_\star ^{(j)}(z) := \left\{ \begin{array}{ll} {\underline{m}}_j \cdot (z-z_j) + b_j, &{}\quad z \in (-\infty , z_j], \\ {\overline{m}}_j \cdot (z-z_j) + b_j, &{}\quad z\in (z_j,\infty ), \end{array} \right. \end{aligned}$$

with \(b_j:=h_0(z_j)\), and with \({\underline{m}}_j\in {\mathbb {R}}\) and \({\overline{m}}_j\in {\mathbb {R}}\) being the well-defined constants fulfilling \({ h_0'}\equiv {\underline{m}}_j\) in \((z_{j-1},z_j)\) and \({ h_0'}\equiv {\overline{m}}_j\) in \((z_j,z_{j+1})\). As the concavity of \(h_0\) requires that \({\underline{m}}_j \ge {\overline{m}}_j\) and that thus also \(h_\star ^{(j)}\) is concave for any such j, fixing any \(\varepsilon _j>0\) such that

$$\begin{aligned} \varepsilon _j < \min \bigg \{ \frac{z_j-z_{j-1}}{2} \, , \, \frac{z_{j+1}-z_j}{2} \, , \, \frac{1}{2} \bigg \}, \end{aligned}$$
(2.35)

we may employ Lemma 2.4 to find

$$\begin{aligned} h^{(j)} \equiv h^{(j)}_{\varepsilon _j} \in C^2({\mathbb {R}}) \end{aligned}$$
(2.36)

such that

$$\begin{aligned} h^{(j)}(z) = h_\star ^{(j)}(z) \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}{\setminus } (z_j-\varepsilon _j,z_j+\varepsilon _j), \end{aligned}$$
(2.37)

that

$$\begin{aligned} h^{(j)}(z) \le h_\star ^{(j)}(z) \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}, \end{aligned}$$
(2.38)

and that

$$\begin{aligned} (h^{(j)})''(z) \le 0 \quad \,\mathrm{for\, all }\, z\in {\mathbb {R}}. \end{aligned}$$
(2.39)

Then since (2.35) ensures that for all \(j\ge 2\) we have

$$\begin{aligned} z_j+\varepsilon _j< z_j + \frac{z_{j+1}-z_j}{2} = \frac{z_j+z_{j+1}}{2} = z_{j+1} - \frac{z_{j+1}-z_j}{2} < { z_{j+1}}-\varepsilon _{j+1}, \end{aligned}$$

it follows that

$$\begin{aligned}&[z_j-\varepsilon _j,z_j+\varepsilon _j] \cap [z_k-\varepsilon _k,z_k+\varepsilon _k] = \emptyset \quad \,\mathrm{for\, all }\, j\in {\mathbb {N}}\hbox { and } k\in {\mathbb {N}}\nonumber \\&\quad \hbox { such that } j\ge 2, k\ge 2 \hbox { and } j\ne k, \end{aligned}$$
(2.40)

and that thus

$$\begin{aligned} h(z):=\left\{ \begin{array}{ll} h^{(j)}(z) &{} \quad \hbox {if } z\in [z_j-\varepsilon _j,z_j+\varepsilon _j] \hbox { for some } j\ge 2, \\ h_0(z) &{} \quad \hbox {if } z\in [0,\infty ) {\setminus } \bigcup _{j=2}^\infty [z_j-\varepsilon _j,z_j+\varepsilon _j], \end{array} \right. \end{aligned}$$
(2.41)

introduces a well-defined function h on \([0,\infty )\) which due to (2.36), (2.37) and (2.40) belongs to \(C^2([0,\infty ))\), and for which from (2.39) and the piecewise linearity of \(h_0\) we know that (2.32) holds.

This concavity property also entails the left inequality in (2.31) as a particular consequence, because given any \(z>0\) we can rely on the unboundedness of \((z_j)_{j\in {\mathbb {N}}}\) to find \(j\ge 2\) such that \(z\le z_j+\varepsilon _j\), so that by (2.32), (2.41) and the left inequality in (2.11),

$$\begin{aligned} h'(z) \ge h'(z_j+\varepsilon _j) = h_0'(z_j+\varepsilon _j)>0. \end{aligned}$$

Likewise, combining (2.32) with the right inequality in (2.11) we see that if \(z>0\) is such that \(z\in [z_j-\varepsilon _j,z_j+\varepsilon _j]\) for some \(j\ge 2\), then due to the rightmost restriction expressed in (2.35),

$$\begin{aligned} zh'(z)-1\le & {} zh'(z_j-\varepsilon _j) -1 \\= & {} zh_0'(z_j-\varepsilon _j) -1 \\\le & {} \frac{z}{(z_j-\varepsilon _j)+1} -1 \\\le & {} \frac{z_j+\varepsilon _j}{z_j-\varepsilon _j+1} -1 \\= & {} \frac{2\varepsilon _j-1}{z_j-\varepsilon _j+1} \\\le & {} 0, \end{aligned}$$

whereas if \(z\in [0,\infty ) {\setminus } \bigcup _{j\ge 2} [z_j-\varepsilon _j,z_j+\varepsilon _j]\), then clearly \(zh'(z) \le \frac{z}{z+1} \le 1\) by (2.11).

Having thereby asserted both inequalities in (2.31) for all \(z\ge 0\), we proceed to observe that again thanks to (2.40), we may draw on (2.13) to infer that

$$\begin{aligned} \limsup _{z\rightarrow \infty } h(z) \ge \limsup _{j\rightarrow \infty } h(z_j+\varepsilon _j) = \limsup _{j\rightarrow \infty } h_0(z_j+\varepsilon _j) = + \infty , \end{aligned}$$

whence (2.33) becomes a consequence of the upward monotonicity of h guaranteed by (2.31).

Thus left with the verification of (2.30), we first note that for all \(z\in [0,\infty ) {\setminus } \bigcup _{j=2}^\infty [z_j-\varepsilon _j,z_j+\varepsilon _j]\) it directly follows from (2.12) that

$$\begin{aligned} h(z) = h_0(z)\le F(z), \end{aligned}$$

while if \(z\in [z_j-\varepsilon _j,z_j+\varepsilon _j]\) for some \(j\ge 2\), then (2.38) and (2.34) enable us to again invoke (2.12) when concluding that

$$\begin{aligned} h(z) = h^{(j)}(z) \le h_\star ^{(j)}(z) = h_0(z) \le F(z). \end{aligned}$$

Finally, the left inequality in (2.30) also results from the nonnegativity of \(h'\) when combined with the observation that since \(\varepsilon _2<\frac{z_2-z_1}{2}=\frac{z_2}{2}\) by (2.35), and since thus \(z_2-\varepsilon _2>\frac{z_2}{2}>0\), the definition (2.41) ensures that \(h(0)=h_0(0)\) and that hence \(h(0)>0\) due to the positivity of \(h_0\), as warranted by Lemma 2.3. \(\square \)

We can now return to Lemmas 2.1 and 2.2 to verify that indeed for essentially any given f diverging to \(+\infty \) we can find a function \({{{\mathcal {L}}}}\) that simultaneously possesses all the intended properties.

Lemma 2.6

Let \(n\ge 1\) and \(p\ge 1\), and suppose that \(f\in C^0([1,\infty ))\) is such that \(f(t)>1\) for all \(t\ge 1\), and that \(f(t)\rightarrow +\infty \) as \(t\rightarrow \infty \). Then one can find \({{{\mathcal {L}}}}\in C^0([0,\infty )) \cap C^2((0,1))\) with the properties that

$$\begin{aligned} {{{\mathcal {L}}}}(0)=0, \quad {{{\mathcal {L}}}}(s)>0 \quad \mathrm{for \, all}\, s\in (0,1] \quad \,\mathrm{and}\, \quad {{{\mathcal {L}}}}(s)={{{\mathcal {L}}}}(1) \quad \,\mathrm{for\, all }\, s>1, \end{aligned}$$
(2.42)

that

$$\begin{aligned} 0<{{{\mathcal {L}}}}'(s) \le \frac{{{{\mathcal {L}}}}(s)}{s\ln \frac{1}{s}} \quad \,\mathrm{for\, all }\, s\in (0,1), \end{aligned}$$
(2.43)

that

$$\begin{aligned} s{{{\mathcal {L}}}}''(s) \ge -{{{\mathcal {L}}}}'(s) \quad \,\mathrm{for\, all }\, s\in (0,1), \end{aligned}$$
(2.44)

and that

$$\begin{aligned} {{{\mathcal {L}}}}(s) \ge f^{-\frac{np}{4}} \Big (\frac{1}{s}\Big ) \quad \,\mathrm{for\, all }\, s\in (0,1). \end{aligned}$$
(2.45)

In particular,

$$\begin{aligned} \frac{{{{\mathcal {L}}}}^{-\frac{2}{np}} \Big (\frac{1}{t}\Big )}{f(t)} \rightarrow 0 \quad \mathrm{as } 1<t\rightarrow \infty . \end{aligned}$$
(2.46)

Proof

We let

$$\begin{aligned} F(z):=\frac{np}{4} \ln f(e^z), \quad z\ge 0, \end{aligned}$$
(2.47)

and observe that our assumptions on f ensure that F is continuous and positive on \([0,\infty )\) with \(F(z)\rightarrow + \infty \) as \(z\rightarrow \infty \). We may therefore employ Lemma 2.5 to obtain a function \(h\in C^2([0,\infty ))\) which satisfies (2.30)–(2.33), and thereupon define

$$\begin{aligned} {{{\mathcal {L}}}}(s):=\left\{ \begin{array}{ll} 0 &{}\quad \hbox {if } s=0, \\ e^{-h(z)}, \quad z\equiv z(s):=\ln \frac{1}{s}, &{} \quad \hbox {if } s\in (0,1], \\ e^{-h(0)} &{} \quad \hbox {if } s>1. \end{array} \right. \end{aligned}$$
(2.48)

Then since \(h(z)\rightarrow + \infty \) as \(z\rightarrow \infty \), it follows that \({{{\mathcal {L}}}}\) is continuous, whereas the inclusion \(h\in C^2([0,\infty ))\) clearly implies that \({{{\mathcal {L}}}}\) moreover belongs to \(C^2((0,1))\). All three properties in (2.42) and the left inequality in (2.43) are evident from (2.48) and the strict positivity of \(h'\), and the right inequality in (2.43) results from the identity in (2.8), applied to \(s_0:=1\), and the fact that \(zh'(z)\le 1\) for all \(z>0\) by (2.31). To verify (2.44), we only need to invoke (2.9) with \(\kappa :=1\) and use that \(h''(z)\le 0\) for all \(z>0\), and (2.45) can be seen by combining (2.48) with (2.47), which thanks to the right inequality in (2.30), namely, guarantees that

$$\begin{aligned} \ln {{{\mathcal {L}}}}(s)= & {} - h(z(s)) \\\ge & {} - F(z(s)) \\= & {} - \frac{np}{4} \ln f(e^{z(s)}) \\= & {} \ln f^{-\frac{np}{4}} \Big (\frac{1}{s}\Big ) \quad \,\mathrm{for\, all }\, s\in (0,1). \end{aligned}$$

Finally, since \(f(t)\rightarrow + \infty \) as \(t\rightarrow \infty \), this indeed entails (2.46) as a particular consequence, for by (2.45),

$$\begin{aligned} \frac{{{{\mathcal {L}}}}^{-\frac{2}{np}}\Big (\frac{1}{t}\Big )}{f(t)} \le \frac{f^\frac{1}{2}(t)}{f(t)} = { f^{-\frac{1}{2}}} (t) \rightarrow 0 \end{aligned}$$

as \(1<t\rightarrow \infty \). \(\square \)

The derivation of our main result on arbitrarily small deviations from the critical decay rate thereupon becomes quite straightforward:

Proof of Theorem 1.1

We take \({{{\mathcal {L}}}}\) as given by Lemma 2.6, and note that as a strictly increasing function, \({{{\mathcal {L}}}}|_{[0,1]}\) possesses a strictly increasing inverse \(\Lambda \) defined on \([0,{{{\mathcal {L}}}}(1)]\). Fixing any nonincreasing \(\psi \in C^0([0,\infty ))\) such that \(0<\psi (r)<{{{\mathcal {L}}}}(1)\) for all \(r\ge 0\) and

$$\begin{aligned} \int _0^\infty r^{n-1} \psi (r) dr < \infty , \end{aligned}$$
(2.49)

we then see that letting

$$\begin{aligned} \phi (r):=\Lambda (\psi (r)), \quad r\ge 0, \end{aligned}$$
(2.50)

introduces a well-defined \(\phi \in C^0([0,\infty ))\) which is positive and nonincreasing according to the monotonicity properties of \(\Lambda \).

Now (2.44) together with the nonnegativity of \({{{\mathcal {L}}}}'\) ensures that if we pick any \(q_0>0\), then

$$\begin{aligned} s{{{\mathcal {L}}}}''(s) + \frac{3p+q_0-2}{p+q_0} \cdot {{{\mathcal {L}}}}'(s)\ge & {} -{{{\mathcal {L}}}}'(s) + \frac{3p+q_0-2}{p+q_0} \cdot {{{\mathcal {L}}}}'(s) \\= & {} \frac{2(p-1)}{p+q_0} \cdot {{{\mathcal {L}}}}'(s) \quad \,\mathrm{for\, all }\, s\in (0,1), \end{aligned}$$

whereas (2.43) in conjunction with Lemma 2.1 warrants that

$$\begin{aligned} {{{\mathcal {L}}}}(s) \le (1+\lambda ) {{{\mathcal {L}}}}(s^{1+\lambda }) \quad \hbox {for all } s\in (0,1) \,\mathrm{and }\, \lambda >0. \end{aligned}$$

We may therefore employ Theorem B to conclude that whenever \(u_0\in C^0({\mathbb {R}}^n)\) is radially symmetric and such that (1.3) holds, then since especially also \(u_0(x) < \Lambda ({{{\mathcal {L}}}}(1)) =1=\max \big \{1^\frac{2}{p} , 1^\frac{2}{p+q_0}\big \}\) for all \(x\in {\mathbb {R}}^n\) by (2.50), and since

$$\begin{aligned} \int _{{\mathbb {R}}^n} {{{\mathcal {L}}}}(u_0)\le & {} \int _{{\mathbb {R}}^n} {{{\mathcal {L}}}}(\phi (|x|)) dx \\= & {} n|B_1(0)| \int _0^\infty r^{n-1} {{{\mathcal {L}}}}(\phi (r)) dr \\= & {} n|B_1(0)| \int _0^\infty r^{n-1} \psi (r) dr \\< & {} \infty \end{aligned}$$

due to (2.49), we can find \(t_0\ge 1\) and \(c_1>0\) such that

$$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \le c_1 {{{\mathcal {L}}}}^{-\frac{2}{np}} \Big (\frac{1}{t}\Big ) \quad \,\mathrm{for\, all }\, t\ge t_0. \end{aligned}$$

In view of (2.46), however, this already establishes (1.4). \(\square \)

3 Continuous Weak Solutions with Nontrivial Zero Sets

The basis for our investigation of solutions emanating from initial data \(u_0\) with \(\{u_0>0\} \ne {\mathbb {R}}^n\) will be formed by the following statement on existence and uniqueness of continuous weak solutions, as essentially contained already in the literature, together with a basic lower bound for their temporal decay.

Proposition 3.1

Given \(p\ge 1\), let \(\Xi (s):=\int _1^s \frac{d\sigma }{\sigma ^p}\), \(\sigma >0\), and assume that \(n\ge 1\) and that \(u_0\in C^0({\mathbb {R}}^n)\) is nonnegative and such that \(\{u_0>0\}\) coincides with the interior of \(\mathrm{supp} \, u_0\), and that each connected component of \(\{u_0>0\}\) is a bounded domain with Lipschitz boundary. Then there exists a nonnegative function \(u\in C^0({\mathbb {R}}^n\times [0,\infty )) \cap L^\infty ({\mathbb {R}}^n\times (0,\infty ))\), uniquely determined by the additional regularity requirements that

$$\begin{aligned} (u-\eta )_+\in W^{1,2}({\mathbb {R}}^n\times (t_1,t_2)) \quad \mathrm{for\, any }\, \eta>0, t_1>0 \,\mathrm{and}\, t_2>0, \end{aligned}$$

and that for all bounded domains \(\Omega \subset {\mathbb {R}}^n\) and any \(\varphi \in C^2({{\overline{\Omega }}})\) with \(\varphi >0\) in \(\Omega \) and \(\varphi |_{\partial \Omega }=0\),

$$\begin{aligned} 0\le t \mapsto \int _\Omega \Xi (u(\cdot ,t))\varphi \quad \mathrm{is\, continuous\, as\, a }\, [-\infty ,\infty )\text {-}\mathrm{valued \,mapping,} \end{aligned}$$

such that u forms a continuous weak solution of (1.2) in the sense that \(u|_{t=0}=u_0\) and that whenever \(\Omega \subset {\mathbb {R}}^n\) is a bounded domain with Lipschitz boundary and \(\varphi \in C^2({{\overline{\Omega }}})\) satisfies \(\varphi >0\) in \(\Omega \) with \(\varphi |_{\partial \Omega }=0\),

$$\begin{aligned} \int _\Omega \Xi (u(\cdot ,t_2))\varphi = \int _{t_1}^{t_2} \int _\Omega u\Delta \varphi - \int _{t_1}^{t_2} \int _{\partial \Omega } u\frac{\partial \varphi }{\partial \nu } + \int _\Omega \Xi (u(\cdot ,t_1))\varphi \end{aligned}$$

holds as an identity in \([-\infty ,\infty )\) for any \(t_1\ge 0\) and \(t_2>t_1\).

In addition, this solution satisfies

$$\begin{aligned} u(x,t)=0 \quad \,\mathrm{for\, all }\, x\in {\mathbb {R}}^n {\setminus } \{u_0>0\} \,\mathrm{and }\, t>0, \end{aligned}$$
(3.1)

and for each connected component \(\Omega _0\) of \(\{u_0>0\}\), u belongs to \(C^{2,1}(\Omega _0\times (0,\infty ))\) with \(u>0\) in \(\Omega _0\times (0,\infty )\). Furthermore, there exists \(C>0\) such that

$$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \ge C \quad \,\mathrm{for\, all }\, t>1. \end{aligned}$$
(3.2)

Proof

Except for (3.2), all statements can be obtained by means of an almost verbatim transfer of the arguments from [18, Theorem 1.2.4], as detailed there for homogeneous Dirichlet problems in bounded domains, to the present Cauchy problem situation (cf. also [21, Theorem 2.1] for a slightly simpler close relative involving marginally stronger regularity classes).

To derive (3.2), we fix any ball \(B\subset {\mathbb {R}}^n\) such that \(\overline{B}\subset \{u_0>0\}\), and let \(\Theta \in C^2(\overline{B})\) denote the principal Dirichlet eigenfunction of \(-\Delta \) in B with \(\max _{x\in \overline{B}} \Theta (x)=1\). Then defining

$$\begin{aligned} {\underline{u}}(x,t):=y(t)\Theta (x), \quad x\in \overline{B}, \ t\ge 0, \quad \hbox {where} \quad y(t):=\Big \{ y_0^{-p} + p\lambda _1(B)t \Big \}^{-\frac{1}{p}}, \quad t\ge 0,\nonumber \\ \end{aligned}$$
(3.3)

with \(y_0:=\frac{1}{2} \min _{x\in \overline{B}} u_0(x)\) being positive by continuity of \(u_0\), we immediately see that \({\underline{u}}(x,0)=y_0 { \Theta (x)} < u_0(x)\) for all \(x\in \overline{B}\) and \({\underline{u}}(x,t)=0<u(x,t)\) for all \(x\in \overline{B}\) and \(t\ge 0\) by positivity of u in \(\{u_0>0\} \times [0,\infty )\). As moreover the identities \(-\Delta \Theta =\lambda _1(B)\Theta \) and \(y'=-\lambda _1 y^{p+1}\) along with the inequalities \(0\le \Theta \le 1\) ensure that

$$\begin{aligned} {\underline{u}}_t - {\underline{u}}^p \Delta {\underline{u}}= \Theta \cdot \Big \{ y'(t) + \lambda _1(B) y^{p+1}(t)\Theta ^p\Big \} \le \Theta \cdot \Big \{ y'(t) + \lambda _1(B) y^{p+1}(t)\Big \} =0 \end{aligned}$$

in \(B\times (0,\infty )\), due to the fact that u classically solves \(u_t=u^p\Delta u\) in \(B\times (0,\infty )\) we may conclude by a comparison argument [17, Sect. 3.1] that \({\underline{u}}(x,t)\le u(x,t)\) for all \(x\in B\) and \(t>0\). Since \(\Vert {\underline{u}}(\cdot ,t)\Vert _{L^\infty (B)} = y(t)\) for all \(t\ge 0\), and since y is positive with \(t^\frac{1}{p} y(t) \rightarrow (p\lambda _1(B))^{-\frac{1}{p}}\) as \(t\rightarrow \infty \) by (3.3), this immediately yields (3.2) with suitably small \(C>0\). \(\square \)

3.1 Attaining Critical Decay. Proof of Proposition 1.2 and of Corollary 1.3

Now our general criterion ensuring attainment of critical speed is based on a comparison argument involving separated supersolutions:

Proof of Proposition 1.2

According to (1.7), there exists \(c_1>0\) with the property that for any \(\Omega \in {{{\mathcal {C}}}}(u_0)\) one can find \(\varphi \in C^0({{\overline{\Omega }}}) \cap C^2(\Omega )\) such that

$$\begin{aligned} 0 \le \varphi (x)\le 1 \quad \,\mathrm{for\, all }\, x\in \Omega \end{aligned}$$
(3.4)

and \(-\varphi ^{p-1}(x)\Delta \varphi (x) \ge c_1\) for all \(x\in \Omega \), where the latter clearly entails that actually \(\varphi (x)>0\) for all \(x\in \Omega \) and

$$\begin{aligned} - \frac{1}{c_1} \varphi ^{p-1}(x)\Delta \varphi (x) \ge 1 \quad \,\mathrm{for\, all }\, x\in \Omega . \end{aligned}$$
(3.5)

For any such \(\Omega \), we now define \({\overline{u}}\equiv {\overline{u}}_\Omega \) by letting

$$\begin{aligned} {\overline{u}}(x,t):=y(t)\cdot \big (\varphi (x)+1\big ), \quad x\in {{\overline{\Omega }}}, \ t\ge 0, \end{aligned}$$
(3.6)

where

$$\begin{aligned} y(t):=\Big \{ y_0^{-p} + pc_1 t\Big \}^{-\frac{1}{p}}, \quad t\ge 0, \end{aligned}$$
(3.7)

with

$$\begin{aligned} y_0:=\Vert u_0\Vert _{L^\infty ({\mathbb {R}}^n)}, \end{aligned}$$
(3.8)

and observe that then

$$\begin{aligned} {\overline{u}}(x,0) = y_0 \cdot \big (\varphi (x)+1\big ) > y_0 \ge u(x,0) \quad \,\mathrm{for\, all }\, x\in {{\overline{\Omega }}}\end{aligned}$$
(3.9)

by (3.6)–(3.8), and that

$$\begin{aligned} {\overline{u}}(x,t) > u(x,t) \quad \,\mathrm{for\, all }\, x\in \partial \Omega \hbox { and } t\ge 0 \end{aligned}$$
(3.10)

due to the fact that \(u|_{\partial \Omega \times [0,\infty )}=0\) thanks to Proposition 3.1. Apart from that, using that \(y'(t)=-c_1 y^{p+1}(t)\) for all \(t>0\) by (3.7), from (3.5) we obtain that

$$\begin{aligned} {\overline{u}}_t - {\overline{u}}^p \Delta {\overline{u}}= & {} y'(t) \cdot (\varphi +1) - y^{p+1}(t) \cdot (\varphi +1)^p\Delta \varphi \\= & {} (\varphi +1) \cdot \Big \{ y'(t) - y^{p+1}(t) \cdot (\varphi +1)^p \Delta \varphi \Big \} \\= & {} c_1 y^{p+1}(t) \cdot (\varphi +1) \cdot \Big \{ -1- \frac{1}{c_1} (\varphi +1)^p \Delta \varphi \Big \} \\\ge & {} c_1 y^{p+1}(t) \cdot (\varphi +1) \cdot \Big \{ -1 + \frac{(\varphi +1)^{p-1}}{\varphi ^{p-1}} \Big \} \\> & {} 0 \quad \hbox {in } \Omega \times (0,\infty ). \end{aligned}$$

Relying on the strictness of the inequalities both in (3.9) and (3.10), we may therefore employ the comparison principle from [17, Sect. 3.1] to conclude that whenever \(\Omega \in {{{\mathcal {C}}}}(u_0)\),

$$\begin{aligned} u(x,t) \le {\overline{u}}_\Omega (x,t) \quad \,\mathrm{for\, all }\, x\in \Omega \hbox { and } t>0, \end{aligned}$$

which again due to Proposition 3.1 implies that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}= & {} \sup _{\Omega \in {{{\mathcal {C}}}}(u_0)} \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \nonumber \\\le & {} \sup _{\Omega \in {{{\mathcal {C}}}}(u_0)} \Vert {\overline{u}}_\Omega (\cdot ,t)\Vert _{L^\infty (\Omega )} \nonumber \\\le & {} 2(pc_1)^{-\frac{1}{p}} t^{-\frac{1}{p}} \quad \,\mathrm{for\, all }\, t>0, \end{aligned}$$
(3.11)

because obviously \(y(t)\le (pc_1t)^{-\frac{1}{p}}\) for all \(t>0\) by (3.7), and because \(1\le \varphi +1\le 2\) in \(\Omega \) by (3.4). As \(c_1\) was positive, (1.8) thus results from (3.11) when combined with the lower estimate provided by (3.2). \(\square \)

Indeed, the requirement on boundedness in one direction made in Corollary 1.3 can readily be seen by means of an explicit construction to ensure a uniform elliptic inequality in the flavor of that required in (1.7):

Proof of Corollary 1.3

It is sufficient to verify that

$$\begin{aligned} \sup _{\begin{array}{c} 0 \le \varphi \in C^0({{\overline{\Omega }}})\cap C^2(\Omega ) \\ \Vert \varphi \Vert _{L^\infty (\Omega )}=1 \end{array}} \inf _{x\in \Omega } \Big \{ - \varphi ^{p-1}(x)\Delta \varphi (x)\Big \} \ge c_1:=\frac{\pi ^2}{2^\frac{p+4}{2} K^2} \quad \,\mathrm{for\, all }\, \Omega \in {{{\mathcal {C}}}}(u_0),\nonumber \\ \end{aligned}$$
(3.12)

and to achieve this, we let any \(\Omega \in {{{\mathcal {C}}}}(u_0)\) be given and first note that upon translating and rotating that \(\Omega \subset S\). Then

$$\begin{aligned} \varphi _0(x):=\cos \frac{\pi \cdot (2x_1-K)}{4K}, \quad x=(x_1,\ldots ,x_n)\in {{\overline{\Omega }}}, \end{aligned}$$

defines a function \(\varphi _0\in C^2({{\overline{\Omega }}})\) which satisfies

$$\begin{aligned} \varphi _0(x)\ge \cos \frac{\pi }{4}=2^{-\frac{1}{2}} \quad \,\mathrm{for\, all }\, x\in {{\overline{\Omega }}}, \end{aligned}$$

because

$$\begin{aligned} -\frac{\pi }{4} = \frac{\pi \cdot (-K)}{4K} \le \frac{\pi \cdot (2x_1-K)}{4K} \le \frac{\pi \cdot (2K-K)}{4K} =\frac{\pi }{4} \quad \,\mathrm{for\, all }\, x_1\in [0,K]. \end{aligned}$$

Since clearly \(\Delta \varphi _0(x) = -(\frac{\pi }{2K})^2 \varphi _0(x)\) for all \(x\in \Omega \), we therefore obtain that

$$\begin{aligned} - \varphi _0^{p-1}(x)\Delta \varphi _0(x) = \frac{\pi ^2}{4K^2} \varphi _0^p(x) \ge c_1 \quad \,\mathrm{for\, all }\, x\in \Omega , \end{aligned}$$
(3.13)

so that (3.12) results upon observing that

$$\begin{aligned} \varphi (x):=c_2 \varphi _0(x), \quad x\in {{\overline{\Omega }}}, \quad \mathrm{with }\, c_2:=\frac{1}{\Vert \varphi _0\Vert _{L^\infty (\Omega )}} \, \ge 1, \end{aligned}$$

thus defines a nonnegative function \(\varphi \in C^0({{\overline{\Omega }}})\cap C^2(\Omega )\) with \(\Vert \varphi \Vert _{L^\infty (\Omega )}=1\) and

$$\begin{aligned} -\varphi ^{p-1}(x)\Delta \varphi (x) = c_2^p \cdot \Big \{\varphi _0^{p-1}(x)\Delta \varphi _0(x)\Big \} \ge c_2^p c_1 \ge c_1 \quad \,\mathrm{for\, all }\, x\in \Omega \end{aligned}$$

by (3.13). Based on the inequality (3.12) thus derived, an application of Proposition 1.2 hence completes the proof. \(\square \)

3.2 Decay Slower than Critical. Proof of Proposition 1.4 and of Corollary 1.5

Conversely, the framework created in the formulation of Proposition 1.4 enables us to derive the claimed unboundedness feature through comparison from below with separated subsolutions, refining the corresponding procedure from the proof of Proposition 3.1 so as to yield suitably large lower bounds.

Proof of Proposition 1.4

Given \(M>0\), we let

$$\begin{aligned} \eta \equiv \eta _M:=\frac{1}{2^{p+1} p { M^p}}, \end{aligned}$$
(3.14)

and then may rely on (1.9) in choosing \(\Omega \in {{{\mathcal {C}}}}(u_0)\) and a nonnegative \(\varphi \in C_0^0({{\overline{\Omega }}})\cap { C^2(\{\varphi >0\})}\) such that

$$\begin{aligned} \max _{x\in {{\overline{\Omega }}}} \varphi (x)=1 \end{aligned}$$
(3.15)

and \(-\varphi ^{p-1}(x)\Delta \varphi (x) { < \eta }\) for all \(x\in { \{\varphi >0\}}\), that is,

$$\begin{aligned} -\frac{1}{\eta } \Delta \varphi (x) { < } \varphi ^{1-p}(x) \quad \,\mathrm{for\, all }\, x\in \{\varphi >0\}. \end{aligned}$$
(3.16)

Now since \(\varphi \) is continuous in \({{\overline{\Omega }}}\) with \(\varphi =0\) on \(\partial \Omega \), the open set \(\Omega _0:=\{\varphi >\frac{1}{2}\}\) satisfies \({{\overline{\Omega }}}_0 \subset \Omega \), and therefore the positivity of the continuous function \(u_0\) on \({{\overline{\Omega }}}_0\) ensures the existence of \(y_0>0\) such that

$$\begin{aligned} \frac{1}{2} y_0 < u_0(x) \quad \,\mathrm{for\, all }\, x\in {{\overline{\Omega }}}_0. \end{aligned}$$
(3.17)

Moreover, (3.15) guarantees that

$$\begin{aligned} {\underline{u}}(x,t):=y(t)\cdot \Big (\varphi (x)-\frac{1}{2}\Big ), \quad x\in {{\overline{\Omega }}}_0, \ t\ge 0, \end{aligned}$$
(3.18)

with

$$\begin{aligned} y(t):=\Big \{ y_0^{-p} + p\eta t\Big \}^{-\frac{1}{p}}, \quad t\ge 0, \end{aligned}$$
(3.19)

satisfies

$$\begin{aligned} {\underline{u}}(x,0) = y_0 \cdot \Big (\varphi (x)-\frac{1}{2}\Big ) \le \frac{1}{2} y_0 < u_0(x) \quad \,\mathrm{for\, all }\, x\in {{\overline{\Omega }}}_0 \end{aligned}$$
(3.20)

due to (3.17), and

(3.21)

according to the definition of \(\Omega _0\) and the positivity of u inside \(\Omega \times [0,\infty )\), as asserted by Proposition 3.1. Forthermore, since \(y'(t)=-\eta y^{p+1}(t)\) for all \(t>0\) by (3.19), using (3.16) we see that

$$\begin{aligned} {\underline{u}}_t - {\underline{u}}^p \Delta {\underline{u}}= & {} y'(t) \cdot \Big (\varphi -\frac{1}{2}\Big ) - \Big (\varphi -\frac{1}{2}\Big )^p \Delta \varphi \cdot y^{p+1}(t) \\= & {} \eta \cdot \Big (\varphi -\frac{1}{2}\Big ) \cdot y^{p+1}(t) \cdot \Big \{ - 1 - \frac{1}{\eta } \cdot \Big (\varphi -\frac{1}{2}\Big )^{p-1}\Delta \varphi \Big \} \\< & {} \eta \cdot \Big (\varphi -\frac{1}{2}\Big ) \cdot y^{p+1}(t) \cdot \Big \{ - 1 { + } \frac{\Big (\varphi -\frac{1}{2}\Big )^{p-1}}{\varphi ^{p-1}} \Big \} \\\le & {} 0 \quad \hbox {in } \Omega _0\times (0,\infty ), \end{aligned}$$

whence on the basis of (3.20) and (3.21) we may once more employ the comparison principle from [17, Sect. 3.1] to infer that

$$\begin{aligned} u(x,t) \ge {\underline{u}}(x,t) \quad \,\mathrm{for\, all }\, x\in \Omega _0 \hbox { and} \, t>0, \end{aligned}$$

and that thus

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}\ge & {} \Vert {\underline{u}}(\cdot ,t)\Vert _{L^\infty (\Omega _0)} \\= & {} y(t) \cdot \Big \Vert \varphi -\frac{1}{2}\Big \Vert _{L^\infty (\Omega _0)} \\= & {} \frac{1}{2} y(t) \quad \,\mathrm{for\, all }\, t>0 \end{aligned}$$

thanks to (3.18) and (3.15). Since (3.19) implies that

$$\begin{aligned} y(t) \ge (2p\eta _M t)^{-\frac{1}{p}} \quad \,\mathrm{for\, all }\, t\ge t_M:=\frac{1}{p\eta _M y_0^p}, \end{aligned}$$

and since (3.14) says that

$$\begin{aligned} \frac{1}{2} \cdot (2p\eta _M)^{-\frac{1}{p}} =M, \end{aligned}$$

this means that

$$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \ge M \quad \,\mathrm{for\, all }\, t\ge t_M \end{aligned}$$

and thereby establishes (1.10), for \(M>0\) was arbitrary. \(\square \)

Now in the presence of arbitrarily small principal eigenvalues within \({{{\mathcal {C}}}}(u_0)\), the validity of (1.9) can be verified by simply using appropriate eigenfunctions of \(-\Delta \):

Proof of Corollary 1.5

Given \(\varepsilon >0\), due to (1.12) we can find \(\Omega \in {{{\mathcal {C}}}}(u_0)\) such that \(\lambda _1(\Omega ) \le \frac{\varepsilon }{2}\). Then taking \(\Theta \in W_0^{1,2}(\Omega )\) such that \(0<{ \lambda _1(\Omega )} \int _\Omega \Theta ^2 = \int _\Omega |\nabla \Theta |^2\), by definition of \(W_0^{1,2}(\Omega )\) we can pick \((\varphi _j)_{j\in {\mathbb {N}}} \subset C_0^\infty (\Omega ){\setminus } \{0\}\) such that \(\varphi _j \rightarrow \Theta \) in \(W_0^{1,2}(\Omega )\) as \(j\rightarrow \infty \) and hence

$$\begin{aligned} \frac{\int _\Omega |\nabla \varphi _j|^2}{\int _\Omega \varphi _j^2} \rightarrow \frac{\int _\Omega |\nabla \Theta |^2}{\int _\Omega \Theta ^2} = \lambda _1(\Omega ) \end{aligned}$$

as \(j\rightarrow \infty \). We can therefore fix \(j_0\in {\mathbb {N}}\) such that

$$\begin{aligned} \frac{\int _\Omega |\nabla \varphi _{j_0}|^2}{\int _\Omega \varphi _{j_0}^2} \le \varepsilon , \end{aligned}$$
(3.22)

and use that then \(\overline{\{\varphi _{j_0}>0\}}\) is a compact subset of \(\Omega \) to construct a smoothly bounded subdomain \(\Omega _0\subset \Omega \) such that \(\overline{\{\varphi _{j_0}>0\}} \subset \Omega _0\). Since \(\varphi _{j_0}\) clearly belongs to \(W_0^{1,2}(\Omega _0)\), relying on the variational characterization of \(\lambda _1(\Omega _0)\) we thus infer from (3.22) that \(\lambda _1(\Omega _0) \le \frac{\int _{\Omega _0} |\nabla \varphi _{j_0}|^2}{\int _{\Omega _0} \varphi _{j_0}^2} = \frac{\int _\Omega |\nabla \varphi _{j_0}|^2}{\int _\Omega \varphi _{j_0}^2} \le \varepsilon \), and since \(\Omega _0\) has smooth boundary, standard elliptic regularity theory applies so as to ensure the existence of a function \(\Theta _0\in C^2({{\overline{\Omega }}}_0)\) fulfilling \(-\Delta \Theta _0(x)=\lambda _1(\Omega _0) \Theta _0(x)\) for all \(x\in \Omega _0\), \(\Theta _0(x)=0\) for all \(x\in \partial \Omega _0\) and \(0\le \Theta _0(x)\le 1=\max _{y\in {{\overline{\Omega }}}_0} \Theta _0(y)\) for all \(x\in \Omega _0\). Therefore, the nonnegative function \(\varphi \in C_0^0({{\overline{\Omega }}})\cap C^2(\{\varphi >0\})\) defined by

$$\begin{aligned} \varphi (x) := \left\{ \begin{array}{ll} \Theta (x), &{} \quad x\in \Omega _0, \\ 0 &{} \quad x\in {{\overline{\Omega }}}{\setminus } \Omega _0, \end{array} \right. \end{aligned}$$

satisfies

$$\begin{aligned} { -\varphi ^{p-1}(x) \Delta \varphi (x) = } -\Theta _0^{p-1}(x)\Delta { \Theta _0} (x) = \lambda _1(\Omega _0) \Theta _0^p(x) \le \lambda _1(\Omega _0) \le \varepsilon { \quad \,\mathrm{for\, all }\, x\in \{\varphi >0\},} \end{aligned}$$

so that since \(\varepsilon >0\) was arbitrary, we conclude that (1.9) holds, and that hence (1.12) implies (1.10) as a consequence of Proposition 1.4.

Finally, assuming (1.13) to be satisfied, for arbitrary \(\eta >0\) we can take \(R>0\) large enough such that \(\frac{\lambda _1(B_1(0))}{R^2}<\eta \), and then use (1.13) to choose \(\Omega \in {{{\mathcal {C}}}}(u_0)\) fulfilling \(\Omega \supset B_R(x_0)\) for some \(x_0\in {\mathbb {R}}^n\). Then, by evident monotonicity and scaling properties of \(\lambda _1(\cdot )\), it follows that

$$\begin{aligned} \lambda _1(\Omega ) \le \lambda _1(B_R(x_0)) =\frac{\lambda _1(B_1(0))}{R^2} < \eta , \end{aligned}$$

and that therefore (1.12) and hence the claimed conclusion holds.\(\square \)