Lack of smoothing for bounded solutions of a semilinear parabolic equation

Marek Fila; Johannes Lankeit

doi:10.1515/anona-2020-0059

Article Open Access

Lack of smoothing for bounded solutions of a semilinear parabolic equation

Marek Fila and Johannes Lankeit

Published/Copyright: March 5, 2020

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all t > 0. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as t → ∞.

Keywords: singular gradient; semilinear parabolic equation

MSC 2010: 35K58; 35B44

For a bounded, smooth domain Ω ⊂ ℝⁿ, T > 0 and A ∈ ℝ, consider solutions of the problem

ut=Δu+g(u,∇u)in Ω×(0,T),u=Aon ∂Ω×(0,T),u(⋅,0)=u0in Ω¯. (0.1)

It is well known (see [1, Thm. VI.4.2]) that this problem has a unique classical solution for small T > 0 provided g ∈ C¹(ℝⁿ⁺¹), u₀ ∈ C¹(Ω) and u₀ = A on ∂Ω. In this paper we study a particular case of problem (0.1) in a radially symmetric setting in B_R: = {x ∈ ℝⁿ ∣ ∣x∣ < R}, R > 0, where g is a smooth function of u and u_r but u₀ is only Hölder continuous in B_R. In our example, the global bounded weak solution emanating from u₀ maintains the singularity of the gradient of u₀ for all t > 0. Thus, there is no smoothing effect which one usually expects from a semilinear uniformly parabolic equation.

The equation we will be interested in is the following:

ut=Δu+uur3in(BR∖0)×(0,∞). (0.2)

(Written in Cartesian coordinates, g hence corresponds to g(x,u,∇u)=u(∇ux|x|)3 for x ∈ B_R ∖ {0}.)

For n ∈ ℕ, n ≥ 2, the function

u∗(r)=−αr13forr>0,where α:=9n−153, (0.3)

forms a stationary solution of (0.2) (for any R > 0 both in B_R ∖ {0}, cf. Lemma 5, and – in the weak sense – in B_R, see Lemma 6).

We will impose several conditions on the initial data u₀ (and refer to (2.14) in Section 2 below for details) that, besides radial symmetry, essentially require that u₀ lies below the stationary solution, but is ’close’ to it in a suitable sense. Under these conditions we will be able to show the global existence of solutions that retain the singularity in their gradient throughout the evolution.

Theorem 1

Let n ≥ 2 and 0<R<38(3n−5)(2n−3)3. Assume that u₀ satisfies (2.14). Then there is a function

u∈C(B¯R×[0,∞))∩C2,1((BR∖0)×(0,∞)) (0.4)

which solves

ut=Δu+uur3in(BR∖0)×(0,∞),u(0,t)=0,u(R,t)=u∗(R)forallt>0,u(⋅,0)=u0inB¯R, (0.5)

in the classical sense. This solution is unique in the class of functions satisfying (0.4) and with u_r being bounded from above on (B_R ∖ {0}) × (0, ∞). Moreover, it holds that

limr↘0ur(r,t)=−∞foreveryt>0. (0.6)

For a more precise description of the singularity see Remark 7. Next we show that the function u from Theorem 1 solves the equation from (0.2) also in B_R × (0, ∞) in a suitable weak sense.

Theorem 2

In addition to the assumptions of Theorem 1 let n ≥ 3. Then the solution u from Theorem 1 is a weak solution of

ut=Δu+uur3inBR×(0,∞),u=u∗(R)on∂BR×(0,∞),u(⋅,0)=u0inB¯R. (0.7)

By this we mean that

uur3∈Lloc1(BR×[0,∞))and∇u∈Lloc1(BR×[0,∞)), (0.8)

and for every φ ∈ Cc∞ (B_R × (0, ∞)) we have

−∫0∞∫BRφtu=−∫0∞∫BR∇u⋅∇φ+∫0∞∫BRuur3φ. (0.9)

We note that Theorem 1 guarantees that the initial and boundary conditions are satisfied.

Concerning the large-time behavior we establish the following:

Theorem 3

Under the assumptions of Theorem 1,

u(⋅,t)→u∗ast→∞.

This convergence is uniform in B_R and occurs with an exponential rate.

An equation closely related to (0.2) has been studied before in [2, 3], see also [4]. It was shown in [2] that interior gradient blow-up may occur for solutions of the problem

ut=uxx+f(u)|ux|m−1ux,x∈(−1,1),u(±1,t)=A±,

where m > 2 and f(u) = u, for example. A global continuation after the interior gradient blow-up has been constructed recently in [3] for m = 3.

For various parabolic equations, solutions with a standing or moving singularity have been investigated by many authors. We shall give some references below. But in these references it is the solution itself that is unbounded while in the present work only the gradient stays unbounded.

For the equation

ut=∇⋅(um−1∇u), (0.10)

solutions with standing singularities were considered in [5, 6, 7, 8, 9, 10] for various ranges of m > 0, m ≠ 1, and some results on moving singularities for the same equation can be found in [11].

Results on moving singularities for the heat equation were established in [12, 13] and for semilinear equations of the form

ut=Δu±up,p>1, (0.11)

in [14, 15, 16, 17, 18, 19]. The behaviour of solutions with standing singularity for equation (0.11) with positive sign has been studied in [20, 21].

Counterexamples to the regularizing effect of (0.10) can be found in [22] There, again, it is the solution itself that is unbounded.

Next we describe the plan of the paper. Due to the gradient singularity that the solutions have at the spatial origin, the notion of classical solvability is restricted to (B_R ∖ {0}) × (0, ∞). In Section 1 we therefore begin by establishing a connection between classical solutions in (B_R ∖ {0}) × (0, ∞) and weak solutions in B_R × (0, ∞).

Section 2 will be concerned with the stationary solution u^* mentioned in (0.3) (and already use the result of Section 1). At the end of this section, we give a precise formulation of the conditions on u₀ that the theorems require (and that involve the stationary solution).

We will construct the solutions between a super- and a subsolution. As a supersolution we will use u^*, finding the subsolution will be the goal of Section 3. To this aim, we will find a solution v to a (formal) linearization of (0.5) (see Lemma 8) and then ensure that u^* − v is a subsolution (Lemma 10). (This is also the source of the restriction on R in the theorems.)

The actual construction of solutions takes place in Section 4. We first restrict the spatial domain to Ω_ε := B_R ∖ B_ε, for the choice of the boundary value on the new boundary ∂B_ε × (0, ∞) already relying on u^* − v from Section 3. In Section 4.1, we take care of the solvability of this problem. (Classical existence results become applicable after replacing the nonlinearity ur3 by f(u_r), see Lemma 18, and until Lemma 26, we will have derived sufficient estimates allowing for removal of f, though still ε-dependent.) Section 4.2 will then be concerned with ε-independent estimates in preparation of a compactness argument leading to the existence of solutions. The key to this part will lie in a comparison principle applied to high powers of u_r (see Lemma 27). This is a modification of a classical technique which involves ∣∇u∣² and originated in [23]. Section 4.3 will contain the passage to the limit ε ↘ 0 (Lemma 32) and deal with (0.5) and (0.7).

In Section 5, finally, we give the proofs of the theorems. By this time, they will only consist in collecting the right lemmata previously proven, and will be accordingly short.

1 Relation between classical and weak solutions

Of course, every classical solution of (0.5) is also a weak solution of (0.5) – in (B_R ∖ {0}) × (0, ∞), which means that the singularity appears on the boundary of the domain. In order to interpret classical solutions in (B_R ∖ {0}) × (0, ∞) as weak solutions in B_R × (0, ∞), we merely require suitable integrability properties of the derivative near 0:

Lemma 4

Let n ≥ 1 and R > 0. Assume that a radially symmetric function

u∈C(B¯R×[0,∞))∩C2,1((BR∖0)×(0,∞))

satisfies (0.8), (0.5), and for every T > 0 we have that

limε→01ε∫0T∫0εrn−1|ur(r,t)|drdt=0. (1.12)

Then (0.9) holds for every φ ∈ Cc∞ (B_R × (0, ∞)).

Proof

For every ψ ∈ Cc∞ ((B_R ∖ {0}) × (0, ∞)) we obtain

−∫0∞∫BRψtu=−∫0∞∫BR∇u⋅∇ψ+∫0∞∫BRuur3ψ,

as u solves the equation classically in (B_R ∖ {0}) × (0, ∞).

We introduce a non-decreasing cut-off function χ ∈ C^∞(ℝ) with 0 ≤ χ′ ≤ 2 and χ(0) = 0, χ ≡ 1 on [1, ∞) and let χε(x):=χ(|x|ε).

We let φ ∈ Cc∞ (B_R × (0, ∞)) and note that for every positive ε, ψ := χ_εφ belongs to Cc∞ ((B_R ∖ {0}) × (0, ∞)).

−∫0∞∫BRφtu=−∫0∞∫BRψtu−∫0∞∫BRφt(1−χε)u=−∫0∞∫BR∇u⋅∇ψ+∫0∞∫BRuur3ψ−∫0∞∫BRφt(1−χε)u=−∫0∞∫BRχε∇u⋅∇φ−∫0∞∫BRφ∇u⋅∇χε+∫0∞∫BRuur3φχε−∫0∞∫BRφt(1−χε)u

for every ε > 0. As χ_ε → 1 a.e. in supp φ and by (0.8) and boundedness of u each of the functions ∇u⋅ ∇ φ, uur3φ, φ_tu belongs to L¹(supp φ),

−∫0∞∫BRχε∇u⋅∇φ+∫0∞∫BRuur3φχε−∫0∞∫BRφt(1−χε)u→−∫0∞∫BR∇u⋅∇φ+∫0∞∫BRuur3φas ε→0

by Lebesgue’s dominated convergence theorem.

Moreover, |∇χε(x)|=|χεr(r)|=1εχ′(rε)≤2ε if r = ∣x∣ < ε and ∣∇ χ_ε(x)∣ = 0 if ∣x∣ > ε. With T > 0 such that supp φ ⊂ B_R × (0, T), we have

∫0∞∫BRφ∇u⋅∇χε≤∥φ∥∞∫0T∫BR|∇u||∇χε|≤2ε∥φ∥∞∫0T∫0εrn−1|ur(r,t)|drdt,

which vanishes as ε → 0 according to (1.12), and (0.9) follows.□

2 The stationary solution and conditions on the initial data

In (0.3), we have introduced a stationary solution u^* to (0.2). In this section we first prove that the function from (0.3) actually has this property (see Lemma 5 for the classical, Lemma 6 for the weak sense) and then formulate the conditions on the initial data, which involve relations with u^* and whose formulation we therefore had postponed.

Lemma 5

Let n ≥ 2. Then the function u^* from (0.3) solves

Δu∗+u∗(ur∗)3=0inRn∖0.

Proof

We use radial symmetry and the explicit form of u^* to write

Δu∗+u∗(ur∗)3=r1−nrn−1−α3r−23r+αr13α3r−233=α27r−5315−9n+α3=0.

□

Lemma 6

Let n ≥ 2. Then for any R > 0 the function u^* defined in (0.3) is a weak solution of (0.7).

Proof

In order to apply Lemma 4, we only have to check integrability of u∗(r)(ur∗)3(r)=α427r13−2 and ur∗(r)=α3r−23, which is satisfied, and

limε→01ε∫0εrn−1|ur∗(r)|dr=limε→0α3n−2εn−53=0.

□

Now and in the following, given any n ∈ ℕ we let

ν:=ν(n):=1636n2−96n+61. (2.13)

Having introduced u^* and ν, we are now in a position to give the conditions on initial data that Theorems 1, 2 and 3 have posed.

u0∈C2(B¯R∖0), (2.14a)

u0 is radially symmetric, (2.14b)

u∗≥u0, (2.14c)

lim supr↘0|r32−n−ν(u∗(r)−u0(r))|<∞, (2.14d)

u0(R)=u∗(R), (2.14e)

there is C>0 such that 0≥u0r(r)≥−Cr−23for every r∈(0,R). (2.14f)

Remark 7

The shape of the solution from Theorem 1 near the singularity of its gradient can be described more precisely than in (0.6) by saying that (2.14d) continues to hold for t > 0 in the sense that

lim supr↘0|r32−n−ν(u∗(r)−u(r,t))|<∞for all t>0.

We will include a proof in the proof of Theorem 1 in Section 5.

3 Finding a subsolution

In order to construct a subsolution of (0.5) near u^*, we first find a solution of the (formal) linearization of (0.5) around u^*.

Lemma 8

Let n ≥ 2, C > 0, ν as in (2.13), λ > 0. Then the function

v(r,t):=Ce−λ2trn−32Jν(λr),r>0,t>0, (3.15)

where J_ν denotes the Bessel function of the first kind of order ν, solves

vt=Δv+3u∗ur∗2vr+ur∗3vin(Rn∖0)×(0,∞)

with u^* taken from (0.3).

Proof

Let us recall that the function defined by χ(r) := J_ν(λr), r > 0, satisfies

r2χ″(r)+rχ′(r)+λ2r2−ν2χ=0for every r>0. (3.16)

We abbreviate A := 4 − 2n and B:=3n−59 and δ:=n−32 and note that

2δ+A=1 (3.17)

and

δ(δ−1)+Aδ−B=−n2+8n3−6136=−ν2, (3.18)

so that (3.16), (3.17) and (3.18) for ψ(r) := r^δχ(r), r > 0, entail

r2ψ″(r)+Arψ′(r)+λ2r2ψ(r)−Bψ(r)=r2(rδχ)″+Ar(rδχ)′−Brδχ=r2δ(δ−1)rδ−2χ+2r2δrδ−1χ′+r2rδχ″+Aδrδχ+Arδ+1χ′−Brδχ=rδr2χ″+(2δ+A)rχ′+(δ(δ−1)+Aδ−B)χ=rδr2χ″+rχ′−ν2χ=−rδr2λ2χ=−r2λ2ψfor r>0,

and

v(r,t)=Ce−λ2trn−32Jν(λr)=Ce−λ2tψ(r),r>0,t>0,

solves

vt=Ce−λ2t(−λ2ψ)=Ce−λ2tψ″+Arψ′−Br2ψ=vrr+4−2nrvr+5−3n9r2v=vrr+n−1rvr−α33rvr−α327r2v=Δv+3u∗ur∗2vr+ur∗3vin (Rn∖0)×(0,∞),

where we have used that α=9n−153 and u∗(r)=−αr13. □

Definition 9

With ν from (2.13), we let x₀ > 0 and x₁ ∈ (0, x₀) be the first positive roots of the Bessel function J_ν of the first kind and its derivative Jν′ , respectively. (As ν > 0, J_ν and Jν′ are positive on (0, x₀) and (0, x₁), respectively.)

Lemma 10

Let n ≥ 2, C > 0, λ > 0 and, with x₁ from Definition 9,

0<R<minx1λ,38(3n−5)(2n−3)3. (3.19)

With v from (3.15), the function

u:=u∗−v

then satisfies

ut≤Δu+uur3in(BR∖0)×(0,∞). (3.20)

Proof

For u = u^* − v, we have

−uur3=−(u∗−v)(u∗−v)r3=−u∗(u∗−v)r3+v(u∗−v)r3=−u∗(ur∗)3+3u∗(ur∗)2vr−3u∗ur∗vr2+u∗vr3+(ur∗)3v−3(ur∗)2vvr+3ur∗vvr2−vvr3in (BR∖0)×(0,∞).

As u^* is a stationary solution according to Lemma 5 and by Lemma 8 v solves the linearized equation, we conclude

ut∗−Δu∗−u∗(ur∗)3=0in (BR∖0)×(0,∞)

and

−vt+Δv+3u∗(ur∗)2vr+(ur∗)3v=0in (BR∖0)×(0,∞).

Accordingly, in (B_R ∖ {0}) × (0, ∞) we obtain

ut−Δu−uur3=ut∗−vt−Δu∗+Δv−u∗(ur∗)3+3u∗(ur∗)2vr−3u∗ur∗vr2+u∗vr3+(ur∗)3v−3(ur∗)2vvr+3ur∗vvr2−vvr3=−3u∗ur∗vr2−3(ur∗)2vvr+u∗vr3+3ur∗vvr2−vvr3=−α2C2r−13e−2λ2t(ψ′(r))2+αC2r−23e−2λ2tψ(r)ψ′(r)−αC3r13e−3λ2t(ψ′(r))3−αC3r−23e−3λ2tψ(r)(ψ′(r))2−C4e−4λ2tψ(r)(ψ′(r))3.

Due to rλ ≤ Rλ ≤ x₁ = min{x₀, x₁}, we have that

ψ′(r)=n−32rn−52Jν(λr)+λrn−32Jν′(λr)≥0for all r∈(0,R), (3.21)

hence

ut−Δu−uur3≤−α2C2r−13e−2λ2t(ψ′(r))2+αC2r−23e−2λ2tψ(r)ψ′(r)=αC2e−2λ2tψ′(r)r−23−αr13ψ′(r)+ψ(r) in (0,R). (3.22)

From (3.21) and λR < x₁ = min {x₀, x₁}, we can also infer

ψ′(r)ψ(r)=r−1(n−32)+rλJν′(rλ)Jν(rλ)≥n−32r−1for all r∈(0,R),

so that

−αr13ψ′(r)+ψ(r)≤−αr−23n−32+1ψ(r)≤−αR−23n−32+1ψ(r)≤0

for every r ∈ (0, R), because R−23≥(38(3n−5)(2n−3)3)−13=(α(n−32))−1, hence (3.22) turns into (3.20).□

4 Existence

4.1 An approximate problem

Construction of the solution to (0.2) will be based on an appropriately modified problem on (B_R ∖ B_ε) × (0, ∞). In preparation of suitable initial data, we first turn our attention to u₀.

Lemma 11

Let n ≥ 2, 0<R<38(3n−5)(2n−3)3. Assume that u₀ satisfies (2.14). Let λ > 0 be such that λR < x₁. There is C > 0 so that v from (3.15) satisfies

u0≥u∗−v(⋅,0)inBR. (4.23)

Proof

Since λR < x₀, known asymptotics of the Bessel function [24, p. 360, (9.1.7)] yields the existence of c₁ = c₁(λ) > 0 such that c₁r^ν ≤ J_ν(λr) for every r ∈ [0, R]. Therefore, (2.14d) implies that for some c₂ > 0 we obtain

|u0(r)−u∗(r)|rn−32Jν(λr)≤c2for every r∈(0,R).

If we let C ≥ c₂, this coincides with (4.23).□

Definition 12

Now and in all of the following, we let n, C, R, λ, v be as in Lemma 10 and Lemma 11.

Definition 13

Let ε > 0 and u₀ satisfy (2.14). We denote Ω_ε := B_R ∖ B_ε. Moreover, let u_0ε ∈ C²(Ω_ε) be radially symmetric and such that

u0ε(ε)=u∗(ε)−v(ε,0), (4.24a)

u0r≤u0εr≤0, (4.24b)

u∗≥u0ε≥u∗−v(⋅,0), (4.24c)

u0ε=u0on the set {r∈(ε,R]∣u0(r)<u∗(ε)−v(ε,0)−ε}. (4.24d)

Remark 14

For (4.24c), we rely on Lemma 11; that the other conditions can be fulfilled is more immediate from (2.14).

Remark 15

As u^*(ε) − v(ε, 0) − ε → 0 as ε → 0, (4.24d) ensures that for every δ > 0 there is ε₀ > 0 such that for all ε ∈ (0, ε₀) we have u_0ε = u₀ on B_R ∖ B_δ.

Definition 16

Let ε > 0. First let us note that

cv:=−eλ2tv(ε,t)

is positive and constant with respect to t according to (3.15).

We choose cε∗ > 1 large enough so as to satisfy

cε∗>sup[ε,R]|ur∗|, (4.25a)

cε∗>sup[ε,R]|(u∗−v(⋅,0))r|, (4.25b)

cε∗>sup[ε,R]|u0εr|, (4.25c)

cv+n−1εcε∗+u∗(ε)(cε∗)3≤0. (4.25d)

Definition 17

We let f_ε ∈ Cc∞ (ℝ) be such that f_ε(s) = s³ for every s∈[−cε∗,cε∗] (with cε∗ from Definition 16) and f_ε ≤ 0 on (−∞, 0).

With u_0ε and f_ε as in Definitions 13 and 17, we now consider

uεt=Δuε+uεfε(uεr)in Ωε×(0,∞),uε(⋅,t)|∂Bε=u∗−v(⋅,t)|∂Bεfor all t>0,uε(⋅,t)|∂BR=u0(R)=u∗(R)for all t>0,uε(⋅,0)=u0εin Ωε¯. (4.26)

By classical theory for parabolic PDEs, this problem has a solution.

Lemma 18

Let ε > 0. Then (4.26) has a unique solution

uε∈Cβ,β2(Ωε¯×[0,∞))∩C2+β,1+β2(Ωε×(0,∞))with∇uε∈Lloc∞(Ωε¯×[0,∞))

for some β ∈ (0, 1). This solution is radially symmetric.

Proof

Boundedness of f_ε and the regularity requirements on u_0ε ensure applicability of [1, Thm. V.6.2], which yields existence and uniqueness of the solution. Radial symmetry of u_0ε together with the uniqueness assertion implies radial symmetry of the solution.□

Later (in Lemma 25 and 27) we want to invoke comparison principles for the derivative. In order to make them applicable, we need slightly more regularity than provided by Lemma 18.

Lemma 19

Let ε > 0. Then there is β ∈ (0, 1) such that

uε∈C3+β,3+β2(Ωε×(0,∞))and∇uε∈Cβ,β2(Ωε¯×[0,∞)).

Proof

Letting η ∈ Cc∞ (Ω_ε × (0, ∞)) we observe that ηu solves (ηu)_t = Δ(ηu) + g, where g = −η_tu − 2∇η⋅∇u − uΔη + ηuf_ε(u_r) and that, thanks to u ∈ C2+β,1+β2 (supp η) by Lemma 18, g ∈ C1+β,1+β2 (Ω_ε × (0, ∞)). [1, Thm. IV.5.2] therefore implies ηu ∈ C3+β,3+β2 (Ω_ε × [0, ∞)). Hölder continuity of ∇u_ε up to t = 0 and to the spatial boundary follows from [25, Thm. 4.6].□

As a first estimate of u_ε, the following lemma not only affirms boundedness of u_ε, but also forms the foundation of estimate (4.30) for u.

Lemma 20

Let ε > 0. Then

u∗≥uε≥u∗−vinΩε×(0,∞). (4.27)

Proof

Due to (4.25a) and (4.25b), each of the functions w ∈ {u^*, u_ε, u^* − v} satisfies f_ε(w_r) = wr3 in Ω_ε × (0, ∞) and hence for w ∈ {u^*, u_ε} we have

wt=Δw+fε(wr)w,

whereas w_t ≤ Δw + f_ε(w_r)w for w = u^* − v (cf. Lemma 10). By construction, u^*(R) = u_ε(R, t) ≥ u^*(R) − v(R, t) and u^*(ε) ≥ u_ε(ε, t) = u^*(ε) − v(ε, t) for all t > 0, and u^* ≥ u_0ε ≥ u^* − v(⋅, 0), so that the comparison principle ([4, Prop. 52.6]) implies (4.27).□

We prepare for an estimate of u_εr by comparison, first providing some information on its value on the spatial boundary, beginning with the outer part ∂B_R × (0, ∞).

Lemma 21

For every ε > 0 and t > 0 we have

ur∗(R)≤uεr(R,t)≤0.

Proof

Since u^*(R) = u_ε(R, t) for all t > 0, (4.27) shows that ur∗ (R) ≤ u_εr(R, t) for all t > 0. Moreover, u(r, t) := u^*(R), (r, t) ∈ [ε, R] × [0, ∞), satisfies u_t ≤ Δu + f(u_r)u in (ε, R) × (0, ∞) and u(R, t) ≤ u_ε(R, t), u(ε, t) ≤ u_ε(ε, t) for all t > 0 and u(r, 0) ≤ u_ε(r, 0) for all r ∈ (ε, R). By the comparison principle [4, Prop. 52.6] therefore u_ε(r, t) ≥ u^*(R) = u_ε(R, t) for every (r, t) ∈ (0, R) × (0, ∞) so that u_εr(R, t) ≤ 0 for every t > 0.□

On the inner boundary, we first establish the sign of u_εr.

Lemma 22

For every ε > 0 and t > 0 it holds that

uεr(ε,t)≤0.

Proof

With 𝓜[ϕ] := ϕ_t − Δϕ − u_ε uεr2 ϕ_r and u(x, t) := u^*(ε) − v(ε, t) for (x, t) ∈ Ω_ε × [0, ∞), we have

M[uε]=0,M[u¯]=u¯t=−vt(ε,t)≥0in Ωε×(0,∞),

which together with u_ε(ε, t) = u(ε, t), u_ε(R, t) = u^*(R) ≤ u(R, t) for all t > 0 and the consequence u_0ε(r) ≤ u_0ε(ε) = u(r, 0) of (4.24b) and (4.24a) enables us to invoke [4, Prop. 52.6] once more to conclude u_ε(r, t) ≤ u(r, t) = u_ε(ε, t) for all r ∈ (ε, R) and t > 0, which implies u_εr(ε, t) ≤ 0 for all t > 0.□

The upper estimates in Lemma 21 and Lemma 22 determine the sign of u_εr throughout Ω_ε × [0, ∞).

Lemma 23

Let ε > 0. Then

uεr≤0inΩε×[0,∞).

Proof

As w := u_εr belongs to C(Ω_ε × (0, ∞)) ∩ C([0, ∞);L²(Ω_ε)) with w_t, ∇ w, D²w ∈ Lloc2 (Ω_ε × (0, ∞)) by Lemma 19, solves w_t = Δw + f_ε(u_εr)w + uεfε′ (u_εr) w_r in Ω_ε × (0, ∞), f(u_εr) is bounded in Ω_ε × (0, ∞) due to boundedness of f_ε, and so is uεfε′ (u_εr) because of Lemma 20, we can apply [4, Prop. 52.8] to conclude nonpositivity of w from nonpositivity of w on Ω_ε × {0} (see (4.24b)) and on ∂Ω_ε × (0, ∞) as guaranteed by Lemmata 21 and 22.□

We now turn our attention to the counterpart of Lemma 22.

Lemma 24

For every ε > 0 we obtain

uεr(ε,t)≥−cε∗

for every t ∈ (0, ∞), where cε∗ is as in Definition 16.

Proof

We define u(r, t) := (u^* − v)(ε, t) + cε∗ (ε − r). Then u(ε, t) = u_ε(ε, t) for all t > 0 due to the boundary condition in (4.26); by (4.24a) and (4.25c),

u_(r,0)=u0ε(ε)−cε∗(r−ε)≤u0ε(ε)−∫εrsup|u0εr|≤u0ε(r),

for every r ∈ (ε, R), and similarly by (4.25a),

u_(R,t)=u∗(ε)−v(ε,t)−cε∗(R−ε)≤u∗(ε)−cε∗(R−ε)≤u∗(R)=uε(R,t)

for every t > 0. Due to Definition 17, fε(cε∗)=(cε∗)3 and hence, by Lemma 20 and (4.25d),

u_t−Δu_−uεfε(u_r)=−vt(ε,t)+n−1rcε∗+(cε∗)3uε≤e−λ2tcv+n−1εcε∗+u∗(ε)(cε∗)3≤0.

Therefore, comparison ([4, Prop. 52.6]) implies

uε(r,t)≥u_(r,t)for all t>0,r∈(ε,R),

and as u_ε(ε, t) = u(ε, t) for every t > 0, this shows that u_εr(ε, t) ≥ u_r(ε, t) = − cε∗ for every t > 0.□

The previous lemmata and a first Bernstein-type comparison of uεr2 confirm that including f_ε in (4.26) – although necessary for application of the classical existence theorems – has not altered the equation.

Lemma 25

For every ε > 0 we have

supΩε×(0,∞)|∇uε|≤cε∗.

Proof

We let 𝓜[ϕ] := ϕ_t − Δϕ − fε′ (u_εr)u_ε ϕ_r. Then M[cε∗]=0 and

M[|∇uε|2]=2∇uε⋅∇Δuε+2|∇uε|2fε(uεr)+2uεfε′(uεr)∇uε⋅∇uεr−∇⋅(2D2uε∇uε)−2fε′(uεr)uε∇uε⋅∇uεr=2|∇uε|2fε(uεr)−2|D2uε|2in Ωε×(0,∞).

In view of Lemma 21, 𝓜[∣∇u_ε∣²] ≤ 0. Lemma 21 and (4.25a) together with Lemmata 22 and 24 show that (cε∗)2 ≥ ∣∇u_ε∣² on ∂Ω_ε × (0, ∞), and (4.25c) ensures the same on Ω_ε × {0}. Therefore, comparison (in the form of [4, Prop. 52.10], if one allows f to also depend on t there – the necessary adaptations in the corresponding proof are minor) proves sup_{Ω_ε×(0,∞)} ∣∇u_ε∣² ≤ (cε∗)2 and thus the lemma.□

Lemma 26

The function u_ε solves

uεt=Δuε+uεuεr3inΩε×(0,∞),uε|∂Bε(⋅,t)=u∗−v(⋅,t)|∂Bεforallt>0,uε|∂BR(⋅,t)=u0(R)=u∗(R)forallt>0,uε(⋅,0)=u0εinΩε¯. (4.28)

Proof

Lemma 25 guarantees that ∣u_εr∣ = ∣∇u_ε∣ ≤ cε∗ in Ω_ε × (0, ∞), therefore f(u_εr) = uεr3 by Definition 17, and Lemma 26 becomes a corollary of Lemma 18.□

4.2 A priori estimates

Inspired by the reasoning in [2, Sec. 2], which goes back to [23], we will now obtain an ε-independent bound for u_εr from a comparison principle applied to, essentially, a large, even power of u_εr. Lack of ε-independent control over u_εr on the inner boundary (for which we refer to Lemma 24 and which is natural if seen in light of the unbounded derivative of u^* near r = 0) makes inclusion of a cutoff function necessary.

Lemma 27

Let p ≥ 4 be an even integer. There is c > 0 such that

(r−δ)+p+3uεrp(r,t)≤c(1+supr>δ(r−δ)+p+3u0εrp+t) (4.29)

for every δ > 0, ε ∈ (0, δ) and t > 0, r ∈ (0, R).

Proof

We define c:=max1,Rp+3|ur∗(R)|p,(3(p+3))p+3|u∗(R)|p+3+(Rp(p+3)2p−1)p+33 and ε ∈ (0, δ). Letting w(r,t):=(r−δ)+p+3uεrp(r,t) for (r, t) ∈ (δ, R) × (0, ∞), in (δ, R) × (0, ∞) we compute

wr=(p+3)(r−δ)+p+2uεrp+p(r−δ)+p+3uεrp−1uεrr

and

wrr=(p+2)(p+3)(r−δ)+p+1uεrp+2p(p+3)(r−δ)+p+2uεrp−1uεrr+p(p−1)(r−δ)+p+3uεrp−2uεrr2+p(r−δ)+p+3uεrp−1uεrrr

as well as

uεrt=uεrrr+n−1ruεrr−n−1r2uεr+uεr4+3uεuεr2uεrr.

For 𝓜[ϕ] := ϕ_t − Δϕ − 3uεuεr2ϕr we thus obtain from (4.28)

M[w]=p(r−δ)+p+3uεrp−1uεrt−wrr−n−1rwr−3uεuεr2wr=p(r−δ)+p+3uεrp−1uεrrr+pn−1r(r−δ)+p+3uεrp−1uεrr−pn−1r2(r−δ)+p+3uεrp+p(r−δ)+p+3uεrp+3+3p(r−δ)+p+3uεuεrp+1uεrr−(p+2)(p+3)(r−δ)+p+1uεrp−2p(p+3)(r−δ)+p+2uεrp−1uεrr−p(p−1)(r−δ)+p+3uεrp−2uεrr2−p(r−δ)+p+3uεrp−1uεrrr−(p+3)n−1r(r−δ)+p+2uεrp−pn−1r(r−δ)+p+3uεrp−1uεrr−3(p+3)(r−δ)+p+2uεuεrp+2−3p(r−δ)+p+3uεuεrp+1uεrr=−pn−1r2(r−δ)+p+3uεrp+p(r−δ)+p+3uεrp+3−(p+3)(p+2)(r−δ)+p+1uεrp−2p(p+3)(r−δ)+p+2uεrp−1uεrr−p(p−1)(r−δ)+p+3uεrp−2uεrr2−(p+3)n−1r(r−δ)+p+2uεrp−3(p+3)(r−δ)+p+2uεuεrp+2≤p(r−δ)+p+3uεrp+3−2p(p+3)(r−δ)+p+2uεrp−1uεrr−p(p−1)(r−δ)+p+3uεrp−2uεrr2−3(p+3)(r−δ)+p+2uεuεrp+2 in (BR∖Bδ)×(0,∞).

Here, by Young’s inequality

−2p(p+3)(r−δ)+p+2uεrp−1uεrr≤p(p−1)(r−δ)+p+3uεrp−2uεrr2+p(p+3)2p−1(r−δ)p+1uεrp≤p(p−1)(r−δ)+p+3uεrp−2uεrr2+(r−δ)+p+3|uεr|p+3+(r−δ)+p(p+3)2p−1p+33

and

−3(p+3)(r−δ)+p+2uεuεrp+2≤(r−δ)+p+3|uεr|p+3+(3(p+3))p+3|uε|p+3

in (B_R ∖ B_δ) × (0, ∞). Recalling the sign of u_εr from Lemma 23 and setting c₁ := (3(p+3))^p+3 ∣u^*(R)∣^p+3 + Rp(p+3)2p−1p+33 we hence obtain

M[w]≤c1in (BR∖Bδ)×(0,∞).

Furthermore,

w(R,t)≤Rp+3uεrp(R,t)≤Rp+3(ur∗(R))p=:c2for all t>0

by Lemma 21. With c = max{c₁, c₂, 1} and w¯:=c(1+supr>δ(r−δ)+p+3u0εrp+t) we not only have 𝓜[w] = c ≥ 𝓜[w] in (B_R ∖ B_δ) × (0, ∞), but also w(R, t) ≥ c₂ ≥ w(R, t) for all t > 0 and w(r, 0) ≥ supr>δ(r−δ)+p+3u0εrp≥w(r,0) for all r ∈ (0, R) as well as w(δ, t) ≥ 0 = w(δ, t) for all t > 0. Comparison (again by means of an adaptation of [4, Prop. 52.10]) allows us to conclude (r−δ)+p+3uεrp=w≤w¯=c(1+supr>δ(r−δ)+p+3u0εrp+t) in (B_R ∖ B_δ) × (0, ∞). Additionally, for r ∈ (0, δ), the left-hand side of this inequality is zero, and (4.29) holds.□

Next we bring Lemma 27 in a more directly applicable form.

Lemma 28

Let p ≥ 4 be an even integer. For every T > 0 there is c > 0 such that

|uεr(r,t)|≤cr−p+3p

for every ε > 0, t ∈ [0, T], r ∈ (2ε, R).

Proof

Conditions (2.14f) and (4.24b) ensure the existence of c₁ > 0 such that

|u0εr|≤c1r−23on BR∖Bε

for every ε > 0, and hence

(r−δ)+p+3u0εrp≤c1(r−δ)+p+3r−2p3

for every r ∈ (δ, R) and ε < δ. Noting that r↦(r−δ)+p+3r−2p3 is increasing on (δ, R) due to p + 3 > 2p3, we conclude that

(r−δ)+p+3u0εrp≤c1Rp−2p3=c1Rp3

for every r ∈ (δ, R) and ε ∈ (0, δ). Lemma 27 hence implies that there is c₂ > 0 such that

(r−δ)+p+3uεrp(r,t)≤c2(1+t)

for every δ > 0, ε ∈ (0, δ) and t > 0, r ∈ (0, R). If we insert r = 2δ, we obtain

|uεr(2δ,t)|≤c3(2δ)−p+3p(1+t)

for every δ > 0, ε ∈ (0, δ), t > 0, where c3:=2p+3pc2. We conclude by letting c := c₃(1 + T).□

As preparation of the compactness argument that will finally establish existence of a solution of (0.5) in (B_R ∖ {0}) × (0, ∞), we use classical regularity theory for parabolic PDEs and rely on Lemma 28 as a starting point.

Lemma 29

Let β ∈ (0, 1). Let K be a compact subset of (B_R ∖ {0}) × (0, ∞). Then there are ε₀ > 0 and c > 0 such that for every ε ∈ (0, ε₀)

∥uε∥[C1+β,1+β2(K)]≤c.

Proof

Let us choose δ > 0 so small that (B_δ × (0, ∞)) ∩ K = ∅. Let η ∈ Cc∞ ((B_R ∖ B_δ) × (0, ∞)) be such that η ≡ 1 on K. Then for each ε ∈ (0, ε₀), ε₀ := δ2 , ηu_ε is well-defined on (B_R ∖ B_δ) × (0, ∞) and (ηu_ε)(δ, t) = 0, (ηu_ε)(R, t) = 0 for every t > 0, (ηu_ε)(r, 0) = 0 for every r ∈ (δ, R) and

(ηuε)t=Δ(ηuε)+gεin (BR∖Bδ)×(0,∞),

where g_ε := − u_ε Δη − 2∇u_ε ⋅ ∇η+ηu_ε uεr3 − η_tu_ε. Lemma 28 enables us to find c₁ > 0 satisfying

∥gε∥L∞((BR∖Bδ)×(0,∞))=∥gε∥L∞(supp η)≤c1

for every ε ∈ (0, ε₀). Consequently, [26, Thm. 7.4, p. 191] shows that with some c₂ > 0,

∥ηuε∥C1+β,1+β2((BR∖Bδ)×(0,∞))≤c2for every ε∈(0,ε0).

□

Leveraging Lemma 29, we can achieve higher regularity analogously.

Lemma 30

Let β ∈ (0, 1). Let K be a compact subset of (B_R ∖ {0}) × (0, ∞). Then there are ε₀ > 0 and c > 0 such that

∥uε∥C2+β,1+β2(K)≤cforeveryε∈(0,ε0).

Proof

Again, we choose δ > 0 so small that (B_δ × (0, ∞)) ∩ K = ∅, η ∈ Cc∞ ((B_R ∖ B_δ) × (0, ∞)) such that η ≡ 1 on K and ε₀ := δ and consider the Dirichlet problem of (ηu_ε)_t = Δ(ηu_ε) + g_ε in (B_R ∖ B_δ) × (0, ∞), with g_ε := − u_ε Δη − 2∇u_ε ⋅ ∇η+ηu_ε uεr3 − η_tu_ε. Thanks to Lemma 29, applied to the compact set supp η, there is c₁ > 0 fulfilling

∥gε∥Cβ,β2((BR∖Bδ)×(0,∞))=∥gε∥Cβ,β2(supp η)≤c1for every ε∈(0,ε0).

We can therefore rely on [26, Thm. 3.6, p. 65] so as to conclude the existence of c₂ > 0 such that

∥ηuε∥C2+β,1+β2((BR∖Bδ)×(0,∞))≤c2for every ε∈(0,ε0).

□

In the next step we aim for lower Hölder regularity, but strive to include the boundaries at r = R and t = 0.

Lemma 31

There is β ∈ (0, 1) such that for every compact subset K of (B_R ∖ {0}) × [0, ∞) there are ε₀ > 0 and c > 0 satisfying

∥uε∥Cβ,β2(K)≤cforeveryε∈(0,ε0).

Proof

We choose δ > 0 so small that (B_δ × (0, ∞)) ∩ K = ∅ and let ε₀ ∈ (0,δ2) be such that u_0ε = u₀ on B_R ∖ B_δ for every ε ∈ (0, ε₀) (cf. Remark 15). With η ∈ Cc∞ ((B_R ∖ B_δ) × (0, ∞)) such that η ≡ 1 on K and relying on Lemma 28, we can conclude from [1, Thm. III.10.1] that with some c > 0,

∥ηuε∥Cβ,β2(K)≤cfor every ε∈(0,ε0),

where β can be determined independently of δ, K and η.□

4.3 Solving the limit problem

With these estimates at hand, we are ready to carry out the existence proof.

Lemma 32

There is a function u ∈ C(B_R × [0, ∞)) ∩ C^2,1((B_R ∖ {0}) × (0, ∞)) solving (0.5). This function is radially symmetric, satisfies

u∗(r)≥u(r,t)≥u∗(r)−v(r,t)forall(r,t)∈[0,R]×[0,∞) (4.30)

and, in particular, with some c > 0 we have

0≥u≥−cr13inBR×[0,∞), (4.31)

as well as

ur≤0in(BR∖0)×(0,∞), (4.32)

and for every T > 0 there is some c = c(T) > 0 such that

ur>−cr−3128in(BR∖0)×(0,T). (4.33)

Proof

If we apply Lemmata 30 and 31 to sequences of compact sets exhausting (B_R ∖ {0}) × (0, ∞) and (B_R ∖ {0}) × [0, ∞), respectively, use the Arzelà-Ascoli theorem and a diagonalization procedure, we obtain a sequence (ε_j)_j∈ℕ ↘ 0 and a function u ∈ C((B_R ∖ {0}) × [0, ∞)) ∩ C^2,1((B_R ∖ {0}) × (0, ∞)) such that

uεj→u locally uniformly in (B¯R∖0)×[0,∞) (4.34)

and with respect to the topology of C2,1((BR∖0)×(0,∞)). (4.35)

The latter convergence statement (4.35) together with Lemma 23 already entails (4.32), whereas (4.33) similarly results from Lemma 28 upon the choice of p = 28.

Additionally, we define u(0, t) := 0. Then u is continuous in B_R × [0, ∞). In light of (4.34), only continuity at (0, t) for t ≥ 0 remains to be proven. Let η > 0. Choose δ > 0 such that u^*(δ) − v(δ, 0) > −η. Then for every ε ∈ (0, δ), every r ∈ (0, δ) and every t ≥ 0 we have 0 ≥ u_ε(r, t) ≥ u_ε(δ, t) ≥ u^*(δ) − v(δ, t) ≥ u^*(δ) − v(δ, 0) > −η and, by (4.34), hence 0 ≥ u(r, t) ≥ −η for every r ∈ (0, δ) and t ≥ 0.

Finally, (4.30) and hence (4.31) are obvious for r = 0 and easily obtained from Lemma 20 for r > 0.□

Theorem 1 also includes a uniqueness statement. The following lemma takes care of it.

Lemma 33

Let u, û be functions satisfying

u,u~∈C2,1((BR∖0)×(0,∞))∩C(B¯R×[0,∞)),supur<∞,supu~r<∞

that solve (0.5). (The suprema are supposed to be taken over (B_R ∖ {0}) × (0, ∞).) Then u = û.

Proof

The difference w := u − û solves w_t = Δw + bw_r + cw in (B_R ∖ {0}) × (0, ∞)), where b:=u~(ur2+uru~r+u~r2) and c:=ur3 has a finite supremum. Moreover, w = 0 on (B_R × {0}) ∪ (∂(B_R ∖ {0}) × (0, ∞)), and [4, Prop. 52.4] shows w ≤ 0.□

The final piece of the proof of Theorem 2 is the combination of Lemma 32 with Lemma 4.

Lemma 34

Let n ≥ 3. Then the function u obtained in Lemma 32 is a weak solution of (0.7).

Proof

We observe that according to (4.33) there is c₁ = c₁(T) such that

1ε∫0T∫0εrn−1|ur(r,t)|drdt≤c1Tε∫0εrn−1r−3128dr=c1Tn−3128εn−5928→0

as ε → 0. By (4.33) and (4.31)

|uur3|≤cr13r3⋅(−3128)=cr28−9⋅3184=cr−25184in BR×(0,∞),

and because −25184=−3+184≥−n, hence uur3∈Lloc1(BR×(0,∞)). Finally, |ur|≤cr−3128∈Lloc1 and Lemma 4 becomes applicable.□

5 Proofs of the theorems

Proof

Proof of Theorem 1 and Remark 7 Solvability is ensured by Lemma 32, which by means of (4.30) also ensures that for every t > 0 there are c₁ = c₁(t) > 0 and c₂ = c₂(t) > 0 such that

0≥u∗(r)−u(r,t)≥−v(r,t)≥−c1rn−32Jν(λr)≥−c2rn−32+νfor every r∈[0,R].

(The last estimate therein used λR < x₀ and [24, p. 360, (9.1.7)].) This proves Remark 7 and implies (0.6).

Uniqueness of solutions, on the other hand, has been asserted in Lemma 33.□

Proof

Proof of Theorem 2 This is the outcome of Lemma 34.□

Proof

Proof of Theorem 3 The construction of u during the proof of Theorem 1 had ensured that u^*(r) ≥ u(r, t) ≥ u^*(r) − v(r, t) for all (r, t) ∈ [0, R] × [0, ∞) (cf. (4.30)), and Theorem 3 can be seen from the explicit definition (3.15) of v.□

Acknowledgements

The first author was supported in part by the Slovak Research and Development Agency under the contract No. APVV-18-038 and by the VEGA grant 1/0347/18.

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Received: 2019-05-06

Accepted: 2019-12-01

Published Online: 2020-03-05

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https://doi.org/10.1515/anona-2020-0059

Keywords for this article

singular gradient; semilinear parabolic equation

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