Oscillatory decay in a degenerate parabolic equation

The Cauchy problem in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}, n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}, for the degenerate parabolic equation ut=upΔu(⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$\end{document}is considered for p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document}. It is shown that given any positive f∈C0([0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^0([0,\infty ))$$\end{document} and g∈C0([0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in C^0([0,\infty ))$$\end{document} satisfying f(t)→+∞andg(t)→0ast→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(t)\rightarrow + \infty \quad \text{ and } \quad g(t)\rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$\end{document}one can find positive and radially symmetric continuous initial data with the property that the initial value problem for (⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\star $$\end{document}) admits a positive classical solution such that t1p‖u(·,t)‖L∞(Rn)→∞and‖u(·,t)‖L∞(Rn)→0ast→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow \infty \qquad \text{ and } \qquad \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$\end{document}but that lim inft→∞t1p‖u(·,t)‖L∞(Rn)f(t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{f(t)} =0 \end{aligned}$$\end{document}and lim supt→∞‖u(·,t)‖L∞(Rn)g(t)=∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{\Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{g(t)} =\infty . \end{aligned}$$\end{document}


Introduction
The nonlinear diffusion equation v t = ∇ · (v m−1 ∇v), m ∈ R\{1}, (1.1) has been understood quite thoroughly in regimes of solutions emanating from L p initial data. Classical findings in this regard have asserted that the asymptotic behavior of finite-mass solutions to an associated Cauchy problem in R n essentially coincides with that of certain explicit self-similar solutions, the so-called Barenblatt solutions, both in the case m > 1 in which (1.1) becomes the porous medium equation, and in the case when m ∈ ( n−2 n , 1) [13,20,21]. Further examples have revealed a substantially more colorful picture in the more singular range m ≤ n−2 n , including more complex large time behavior of positive solutions [15,16], phenomena of nonuniqueness, mass loss and finite-time extinction [8,19,21], and even instantaneous extinction in the flavor of results on nonexistence when m ≤ 0 [6,7]; especially subtleties of finite-time extinction mechanisms in dependence of spatial decay features of the initial data have been the object of study in a considerable part of the recent literature [2-5, 9, 10, 14].
In comparison to this, the knowledge about possibly nontrivial facets of solution behavior in the presence of initial data which grow near spatial infinity seems noticeably more restricted, mainly concentrating on the analysis of explicit examples [1], and of essentially one-dimensional wave-like transport mechanisms ( [1]; see also [26,27] for examples on propagation at non-constant speeds). In order to describe two exceptions in this direction recently achieved for m < 0, let us reformulate (1.1) in a way that appears convenient in this framework by involving bounded quantities rather than unbounded functions. Specifically, we shall subsequently be concerned with the full initial value problem u t = u p u, x ∈ R n , t > 0, with p ≥ 1 and a prescribed positive function u 0 , recalling that when p > 1, via the In this setting, it is known that whenever u 0 ∈ C 0 (R n ) ∩ L ∞ (R n ) is positive, the problem (1.2) possesses a minimal classical solution u ( [6,11]; cf. also Lemma 2.1 below for a precise statement), and that if in addition [24]. On the other hand, in [11,Proposition 1.3] it has been found that any positive classical solution of (1.2) has the property that so that, roughly speaking, any initial data satisfying (1.3) will lead to solution behavior somewhere between mere decay, as expressed in (1.4), and decrease at a rate near that of p , subject to the limitation in (1.5). That this latter restriction indeed is essentially sharp has recently been confirmed in [12,28], where it has been seen that given any positive f ∈ C 0 ([0, ∞)) such that f (t) → +∞ as t → ∞, one can find initial data such that the corresponding minimal solution of (1.2) satisfies (see also Proposition 3.2 below).
Main results. The intention of the present note now is to provide an example which indicates that both the above two extremal types of decay behavior can actually be found approached by single trajectories. In fact, by means of a recursive design of initial data following the construction in the seminal work by Poláčik and Yanagida [18], we shall find a solution to (1.2), of quite simple basic structure by being radially symmetric and radially nonincreasing, which on the one hand exhibits arbitrarily slow decay along some unbounded sequence of times, and which on the other hand decreases at a close-to-maximum speed in the style of (1.6) along some further divergent time sequence. By fully acting within the realm of strictly positive smooth solutions, unlike previous discoveries of degeneracy-supported large-time oscillations in related problems [22,25] this result does not rely on the presence of prescribed singular behavior at any point in space. More precisely, our main result can be formulated as follows.

Minimal solutions: radial and radially nonincreasing data
To begin with, let us briefly recall from [11] a regularization-based construction of minimal solutions to (1.2) that will be appropriate for our purposes: Given an arbitrary family (u 0R ) R>0 of functions satisfying for R > 0 and ε ∈ (0, 1) we consider the non-degenerate initial-boundary value problems and the corresponding limit problems On the basis thereof, the following statement on global existence of minimal solutions to (1.2) enjoying some convenient approximation features can be found proved in [11, Lemma 2.1, Proposition 1.1] (cf. also [6]).
Moreover, there exists a positive function u ∈ C 0 (R n × [0, ∞)) ∩ C 2,1 (R n × (0, ∞)) which is such that u R u in R n × (0, ∞) as R ∞, and that u is a minimal solution of (1.2) in the sense that u solves (1.2) classically, and that whenever v Having thus singled out a uniquely identifiable object to be dealt with subsequently, let us attach to this a convenient label that will facilitate notation in some places below. Definition 2.1 Let n ≥ 1 and p ≥ 1. Then given any positive where u denotes the minimal solution of (1.2) according to Lemma 2.1.
These minimal solutions satisfy a favorable comparison principle: Lemma 2.2 Let n ≥ 1 and p ≥ 1, and suppose that u 0 and v 0 belong Proof Let (u 0R ) R>0 be such that (2.1) holds. Then for each R > 0, the assumed positivity of v in R n × [0, ∞) ensures that for the solutions of (2.3) from Lemma 2.1 we have the ordering for each T > 0, the comparison principle recorded in [23, Section 3.1] therefore applies and guarantees that u R ≤ v in B R (0) × (0, ∞) for each R > 0, so that the claim results from the approximation part in Lemma 2.1, according to which, namely, we know that u R (x, t) → u(x, t) as R → ∞ for all x ∈ R n and t > 0.
The following observation will later on, in conjunction with Lemma 2.2, be used to make sure that minimal solutions cannot become unexpectedly small when initially bounded from below by positive constants.
Proof We let u := Su 0 and then first obtain given such x 0 , t 0 and η we pick R > 0 large enough such that Then using that ϕ ≡ −2n R 2 , that ϕ p−1 ≤ 1 according to our assumption p ≥ 1, and that y ≡ − 2n R 2 · y p+1 , we see that while u lies below u on the corresponding parabolic boundary in that u( As D 2 u evidently is bounded, the comparison principle from [23, Section 3.1] therefore becomes applicable so as to ensure that u ≤ u inB R (x 0 ) × [0, ∞), and that thus (2.5) results upon an evaluation thereof at (x, t) = (x 0 , t 0 ), using (2.6) and recalling (2.7)-(2.9).
In what follows, for convenience in presentation we shall restrict most of our considerations to conveniently smooth radially symmetric and radially nonincreasing initial data by saying that a function ϕ : for all r > 0, and such that ϕ(x) ≤ ϕ(y) whenever x ∈ R n and y ∈ R n satisfy |x| ≥ |y|.
(H) By making use of the freedom to choose in (2.3) and (2.2) any family (u 0R ) R>0 fulfilling (2.1), we can readily verify that when present initially, this property (H) is conserved throughout evolution: Lemma 2.4 Let n ≥ 1 and p ≥ 1. If u 0 complies with (H), then the minimal solution u = Su 0 of (1.2) has the property that also u(·, t) satisfies (H) for all t > 0. In particular, u(·, t) L ∞ (R n ) = u(0, t) for all t > 0. (2.10)

Preparing the inductive step: a result on continuous dependence
Besides the basic comparison property from Lemma 2.2, a second ingredient of crucial importance in our recurvive construction will be the following statement on monotone approximation that can be viewed as documenting a certain type of continuous dependence of solutions on the initial data.

1)
and that either u 0 j u 0 in R n or u 0 j u 0 in R n as j → ∞ (3.2) with some positive u 0 ∈ C 0 (R n ). Then the corresponding minimal solutions u j := Su 0 j and u := Su 0 of (1.2) satisfy u j → u in C 0 loc (R n × (0, ∞)) as j → ∞.

(3.3)
Proof We detail the proof only for the case when u 0 j u 0 as j → ∞, as the situation in which u 0 j u 0 as j → ∞ can be covered by minor and obvious modification. We fix any α > n 2 and observe that then defines a positive function ϕ ∈ C ∞ (R n ) which is such that is finite, and which moreover satisfies for all x ∈ R n (3.5) and (3.6) for all x ∈ R n , because ξ ξ 2 +1 ≤ 1 2 and ξ (ξ +1) 2 ≤ 1 4 for all ξ ≥ 0. Apart from that, taking any nonincreasing ζ ∈ C ∞ ([0, ∞)) such that ζ ≡ 1 on [0, 1 2 ] and supp ζ ⊂ [0, 1) we let and note that then with c 2 := ζ L ∞ ((0,1)) and c 3 := ζ L ∞ ((0,1)) + 2(n − 1) ζ L ∞ ((0,1)) , because R for all x ∈ R n and R > 0, and because |x| ≥ R 2 whenever R > 0 and ζ ( |x| R ) = 0. We now introduce to rewrite the respective versions of (1.2) according to ∂ t H (u j ) = u j and ∂ t H (u) = u, and to thereby obtain that since supp (χ R ϕ) is bounded for all R > 0, for all t > 0, R > 0 and j ∈ N. Here we may use that from Lemma 2.2 and the monotone approximation feature in (3.2) we already know that for all x ∈ R n , t > 0 and j ∈ N, to see that due to (3.6) and the inequalities 0 ≤ χ R ≤ 1 for R > 0, for all t > 0, R > 0 and j ∈ N, (3.11) and that From (3.9)-(3.12) we thus infer that and that hence (3.14) We now make full use of (3.2) to conclude, again by Lemma 2.2, that u j (x, t) u(x, t) as j → ∞ for all x ∈ R n and t ≥ 0 (3. 15) with some limit function u on R n × [0, ∞) which satisfies for all x ∈ R n and t ≥ 0, (3.16) and hence particularly is strictly positive. We may therefore rely on Beppo Levi's theorem, as well as on (3.2) explicitly once again, in turning (3.14) into the inequality in which using the same token we may let R ∞ to obtain that But since (3.8) in conjunction with the mean value theorem, (3.15) and (3.13) guarantees that for all x ∈ R n and T > 0 we can find ξ( for all x ∈ R n and T > 0, this implies that As thus according to Grönwall's lemma, thanks to the strict positivity of ϕ this shows that, again in view of (3.16), we must have u = u a.e. in R n × (0, ∞).
To conclude as intended, we only need to finally note that (3.15) and (3.13) together with (1.2) and standard parabolic regularity theory [17, Theorem V.1.1] warrant that for each compact subset K of R n × (0, ∞), the sequence (u j ) j∈N is bounded in C θ, ϑ 2 (K ) with some θ ∈ (0, 1), and hence relatively compact in C 0 (K ) by the Arzelà-Ascoli theorem.
In order to prepare an appropriate application of the previous lemma to particular classes of initial data, let us first recall a known feature of (1.2) with respect to the large time behavior of solutions corresponding to initial data that decay sufficiently fast in space. The following statement in this regard extracts from [12, Theorem 1.3] and [28, Lemma 2.6] what will be needed here. Proposition 3.2 Let n ≥ 1 and p ≥ 1, and suppose that f ∈ C 0 ([0, ∞)) is positive and such that f (t) → ∞ as t → ∞. Then there exists L ∈ C 0 ([0, 1)) ∩ C 2 ((0, 1)) with the properties that L(0) = 0 as well as L(s) > 0 and L (s) > 0 for all s ∈ (0, 1), (3.17) and that whenever u 0 satisfies (H) and is such that u 0 < 1 in R n as well as the minimal classical solution u of (1.2) satisfies On the basis of this, we can design a template for infinitely many parts of the initial data to be finally used in the proof of Theorem 1.1.
Having these preliminaries at hand, we can proceed to design the two basic nuclei of our construction, throughout the sequel making repeated use of the two different truncation and extension processes described in the following: Lemma 3.4 Let n ≥ 1, and suppose that ϕ is such that (H) holds with ϕ(x) < 1 for all x ∈ R n . Then with (φ a ) a∈(0,1) taken from Lemma 3.3, for R > 0 letting

25)
defines functions T R ϕ and T R ϕ on R n which satisfy (H).
Proof This is an evident consequence of the fact that φ a is positive and nonincreasing for all a ∈ (0, 1) by Lemma 3.3.
Thanks to the continuous dependence feature documented in Lemma 3.1, deviations encountered when performing the first of these operations can conveniently be estimated: Lemma 3.5 Let n ≥ 1 and p ≥ 1, and let u 0 be such that (H) holds. Then given any R > 0 and T > 0, one can find R > R such that u 0 := T R u 0 has the property that u := S u 0 and u := Su 0 satisfy u(0, T ) ≤ 2 u(0, T ).
and fixing R 1 > 0 and R 2 > R 1 , we see that and that since 0 ≤ r → u 0 | ∂ B r (0) is nonincreasing by (H), as well as Therefore, 0 < R → h(r , R) is nonincreasing for all r ≥ 0, so that The claim hence results from Lemma 3.1 upon observing that u(0, T ) is positive.
The effects of the second manipulation type from Lemma 3.4 can be controlled in quite a similar fashion. Lemma 3.6 Let n ≥ 1 and p ≥ 1, and suppose that beyond satisfying (H), the function u 0 is such that u 0 ≡ a in R n \B R 0 (0) with some a ∈ (0, 1) and R 0 > 0. Then for each R > 0 and any T > 0 there exists R > R such that writing u 0 := T R u 0 , for the minimal solutions u := S u 0 and u := Su 0 of (1.2) we have and note that then according to (3.25) we have because u 0 | ∂ B r (0) = a for r ≥ R. Moreover, given R 1 > R 0 and R 2 > R 1 , we evidently have As furthermore, due to the inequality φ a ≤ 0, it follows that h(·, R 2 ) ≥ h(·, R 1 ) on [0, ∞) for any such R 1 and R 2 , meaning that T R u 0 u 0 in R n as R ∞. As a consequence of Lemma 3.1, the inequality in (3.27) can thus be achieved upon choosing R > R suitably large.