Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay^{∗}

Filippo Gazzola Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano (Italy)

Hans-Christoph Grunau Fakultät für Mathematik Otto-von-Guericke-Universität Postfach 4120 39016 Magdeburg (Germany)

Abstract

We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u_{t} +Δ^{2}u = |u|^{p−1}u in ℝ^{n} ×[0, ∞), where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n − 4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t^{−1/(p−1)}.

1 Introduction

In the present paper we study existence and quantitative properties of global solutions of the following Cauchy problem for superlinear parabolic equations with the biharmonic operator as elliptic linear part

\{ \begin{matrix}u_{t} + Δ^{2}u = |u|^{p−1}u & in ℝ_{+}^{n+1} := ℝ^{n} × [0, ∞) \\ u(x, 0) = u_{0}(x) & in ℝ^{n} , \end{matrix}  (1)

where n ≥ 2, p > 1 + \frac{4}{n} and u_{0} is a bounded initial datum with suitable behaviour at ∞. The decay of these solutions with respect to space |x| → ∞ and time t → ∞ is investigated.

Before entering into the details of (1), we recall that the corresponding superlinear second order equation

\{ \begin{matrix}u_{t} − Δu = |u|^{p−1}u & in ℝ_{+}^{n+1} \\ u(x, 0) = u_{0}(x) & in ℝ^{n} \end{matrix}  (2)

was intensively studied in [7, 12, 14, 15, 19]. It was discovered by Fujita [7] that the exponent p = 1+ \frac{2}{n} plays a fundamental role for the stability of the trivial solution u ≡ 0 of problem (2). It turned out that 1 + \frac{4}{n} is the biharmonic analogue of this so-called “Fujita”-exponent. For the present paper, the following result relative to (2) is of particular relevance:

∗Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.

Proposition 1. [15, Theorem 3.8] Assume that n ≥ 2 and p > 1 + \frac{2}{n}. There exists α > 0 such that if

|u_{0}(x)| ≤ \frac{α}{1 + |x|^{2/(p−1)}}

then there exists a global strong solution u of (2). Moreover, there exists A > 0 such that

|u(x, t)| ≤ \frac{A}{1 + |x|^{2/(p−1)} + t^{1/(p−1)}} for all (x, t) ∈ ℝ_{+}^{n+1} .

In fact, a slightly different result was stated in [15] but with the very same arguments employed there, one also readily obtains Proposition 1. Subsequently, Wang [19] performed a more detailed study of (2) and obtained refined stability/instability results with a more precise description of the region of attraction of the trivial solution u ≡ 0. In this connection several “critical exponents” turned out to be of importance, being also related to the corresponding elliptic problem. For further results see e.g. [5, 6] and references therein.

Most of the methods employed for the proof of Proposition 1 and related subsequent results are special for second order equations and are in particular basing upon auxiliary functions satisfying suitable differential inequalities and the maximum principle. Such methods do not apply to (1), since not even a comparison principle is available here. The major difficulty is the change of sign of the kernel of the linear operator v ↦ v_{t} + Δ^{2}v, namely

b(x, t) = \frac{f(η)}{t^{n/4}} ,  η = \frac{x}{t^{1/4}} ,  f(η) = ω_{0}|η|^{1−n} ∫_{0}^{∞} e^{−s^{4}}(|η|s)^{n/2}J_{(n−2)/2}(|η|s) ds ,  (3)

where J_{m} denotes the m-th Bessel function and ω_{0} a suitable constant such that

∫_{ℝ^{n}} f(η) dη = 1.

In order to overcome this difficulty, Galaktionov–Pohožaev [9] introduced the following self-similar majorizing kernel associated to (1):

b̃(x, t) := ω_{1}t^{−n/4} exp (−μ (\frac{|x|^{4}}{t} )^{1/3})  (4)

with suitable constants μ, D > 0 and

ω_{1} = \frac{1}{∫_{ℝ^{n}} exp (−μ|y|^{4/3}) dy}

such that

|b(x, t)| ≤ Db̃(x, t) .  (5)

Let us now mention some results already known for the parabolic biharmonic problem (1) and the related Cauchy problem

\{ \begin{matrix}u_{t} + Δ^{2}u = |u|^{p} & in ℝ_{+}^{n+1} \\ u(x, 0) = u_{0}(x) & in ℝ^{n} . \end{matrix}  (6)

In the “sub-Fujita” case 1 < p ≤ 1 + \frac{4}{n}, Egorov et al. [3] show finite time blow up for solutions to (6) provided the initial datum satisfies

∫_{ℝ^{n}} u_{0}(x) dx ≥ 0.

Assuming that u_{0} ≥ 0 and that (1) has a positive solution – which is not obvious due to the oscillatory behaviour of the biharmonic “heat kernel” (3) – this would imply finite time blow up also in this situation. For p > 1 + \frac{4}{n}, Caristi and Mitidieri [1] obtain global solutions of (6), provided the initial datum u_{0} belongs to L^{1}(ℝ^{n}) and obeys the following growth condition at infinity with some constant c_{0} and the following smallness condition:

0 ≤ u_{0}(x) ≤ min \{α, \frac{c_{0}}{1 + |x|^{β}} \} , β > \frac{4}{p − 1}, α small enough in dependence of c_{0} and β. (7)

This result generalises and extends previous results in e.g. [2] and [3, 9, 10]. In the first work, for initial data being small enough in an L^{1}-sense, global existence of solutions decaying to 0 at rate t^{−n/4} was proved in a more general setting basing upon linear semigroup theory. In the latter works, for small exponentially fast decaying initial data, the same decay rate t^{−n/4} and, moreover, the asymptotic profile of the solution were calculated by means of perturbation theory. In particular, the time decay is governed by the linear principal part. By (18) below, these results also imply the existence of global solutions to (1) being majorised by those to (6).

Caristi and Mitidieri [1] also focussed on blow-up results for the modified equation (6) in the case p > 1 + \frac{4}{n}. They proved e.g. finite time blow up for initial data u_{0}(x) = \frac{α}{1+|x|^{4/(p−1)}} with large enough α. It does not seem to be obvious, however, to conclude blow-up also for (1) from these results.

Several questions were left open by the just mentioned results. In particular, we find most interesting and challenging to find out if β in (7) can be allowed to be β = \frac{4}{p−1}. In Theorem 1 we show that this is the case, namely if u_{0} satisfies (7) for β = \frac{4}{p−1} and a sufficiently small α, then the solution to (1) is global. Our proof relies on two crucial estimates (see Propositions 2 and 3 below) which seem to be a new unifying tool for the study of parabolic problems such as (1) whose kernel is sign-changing. In Remark 2 we also explain how our procedure may be (easily) extended to general higher order semilinear parabolic equations such as u_{t} + (−Δ)^{k}u = |u|^{p−1}u with p > 1 + \frac{2k}{n}. In some sense, we provide a unifying proof independent of the positivity of the kernel which applies for all k, included the “easy” case k = 1. Theorem 1 also states that the global solution converges uniformly to the (stable) stationary solution u ≡ 0 at a rate of t^{−1/(p−1)}, thereby giving the complete extension of Proposition 1 to higher order problems. Let us recall that in [4] stability of the trivial solution was obtained only for fast decay (exponential) initial data: here we extend it to the case of slowly decaying u_{0}.

In order to show that Theorem 1 is optimal with respect to the asymptotic decay of the initial datum, one should also prove finite time blow up if this assumption is violated. Theorem 2 below does not give the complete answer but it gives a strong hint that blow up should occur in finite time for initial data decaying slower than |x|^{−4/(p−1)}.

The asymptotic behaviour |x|^{−4/(p−1)} is also important in the corresponding elliptic problem

Δ^{2}u = |u|^{p−1}u in ℝ^{n}.  (8)

One might observe that a suitable multiple of x ↦ |x|^{−4/(p−1)} is a singular solution to (8) if p > n/(n − 4) and also an entire weak solution (in H_{loc}^{2}(ℝ^{n})) if p > (n + 4)/(n − 4), i.e. for supercritical exponents. In the latter case, the authors constructed entire regular radial solutions u to (8) and proved that |x|^{4/(p−1)} · u(x) converges to a suitable constant as |x| → ∞, so that the asymptotic behaviour of solutions to (8) is governed by its singular weak entire solution. See [11] and also Corollary 1 below.

2 Results

In Definitions 1 and 2 we distinguish between two kinds of solutions.

Definition 1. We say that u is a strong solution of (1) over [0, T) if u ∈ C^{4,1}(ℝ^{n} × (0, T)), if u is bounded on ℝ^{n}×[0, t] for every t ∈ (0, T), if u solves (in the classical sense) (1) and ∥u(t)−b(t)∗u_{0}∥_{∞} → 0 as t → 0 (in particular, this implies that u(x_{m}, t) → u_{0}(x) whenever t → 0 and x_{m} → x for a.e. x ∈ ℝ^{n}). The supremum T^{∗} of all T ’s for which u is a strong solution of (1) over [0, T) is called life span of the strong solution u. We say that u is a global strong solution of (1) if T^{∗} = +∞.

We first show the existence of global strong solutions:

Theorem 1. Assume that n ≥ 2 and p > 1 + \frac{4}{n}. There exists α > 0 such that if

|u_{0}(x)| ≤ \frac{α}{1 + |x|^{4/(p−1)}}  (9)

then there exists a global strong solution u of (1). Moreover, there exists A > 0 such that

|u(x, t)| ≤ \frac{A}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}}  for all (x, t) ∈ ℝ_{+}^{n+1} .  (10)

Theorem 1 is somehow optimal. Indeed it was shown in [1] that if α is large and equality holds in (9), then any positive solution of (1) blows up in finite time. Moreover, the asymptotic behaviour at infinity of the estimate in (9) is critical since in the case p > (n + 4)/(n − 4), it corresponds to the asymptotic behaviour of positive stationary radially symmetric regular and singular solutions of (1), see [11]. Therefore, Theorem 1 has the straightforward consequence:

Corollary 1. Assume that n ≥ 5 and p > (n + 4)/(n − 4). Let \overline{u} be a stationary positive radially symmetric solution of (1). There exists β > 0 such that if

|u_{0}(x)| ≤ β\overline{u}(x)

then the solution u of (1) is global. Moreover, there exists A > 0 such that (10) holds.

The strong solution found in Theorem 1 is globally bounded. We now deal with a weaker notion of solution:

Definition 2. We say that u is a weak solution of (1) over (0, T) if u ∈ L_{loc}^{p}(ℝ^{n} × [0, T)) and

∫_{0}^{T} ∫_{ℝ^{n}} |u|^{p−1}uϕ = ∫_{0}^{T} ∫_{ℝ^{n}} u(−ϕ_{t} + Δ^{2}ϕ) − ∫_{ℝ^{n}} u_{0}ϕ(0)  (11)

for all ϕ ∈ C_{c}^{∞}(ℝ^{n} × [0, T)). The supremum T^{∗} of all T ’s for which u is a weak solution over (0, T) is called the life span of the weak solution. We say that u is a global weak solution of (1) if T^{∗} = +∞.

We now introduce a suitable family of test functions.

Definition 3. We say that v ∈ H if the following facts occur:

(i) there exist ψ_{1} ∈ C_{c}^{∞}(ℝ^{n}) and ψ_{2} ∈ C_{c}^{∞}(ℝ_{+}) such that v(x, t) = ψ_{1}(x)ψ_{2}(t).

(ii) ψ_{1}(0) = ψ_{2}(0) = 1 and v(x, t) > 0 in the interior of spt(v), the support of v.

(iii) ∫_{spt(v)} |v_{t}|^{p′}|v|^{1−p′} < ∞ and ∫_{spt(v)} |Δ^{2}v|^{p′}|v|^{1−p′} < ∞ where p′ = \frac{p}{p−1}.

As pointed out in [17] (see also [1]), one has H ≠ ∅.

Then, we have

Theorem 2. Assume that u_{0} ∈ L^{∞}(ℝ^{n}) and that

λ := lim inf_{|x|→∞} |x|^{4/(p−1)}u_{0}(x) > 0 .  (12)

There exists Λ > 0 such that if λ > Λ, then any weak solution u of (1) with initial datum u_{0} satisfies one of the two following alternatives:

(i) u blows up in finite time, that is, T^{∗} < ∞;

(ii) for all v ∈ H and for all γ < 1 we have

lim inf_{R→∞} R^{4p/(p−1)} ∫_{spt(v)} (|u^{−}(Ry, R^{4}τ)|^{p} − γ|u^{+}(Ry, R^{4}τ)|^{p})v(y, τ) dydτ > 0 .  (13)

If (12) is replaced by lim sup_{|x|→∞} |x|^{4/(p−1)}u_{0}(x) < 0, then the same statement holds provided one switches u^{+} and u^{−} in (13).

Let us also briefly comment on Theorem 2. Under the further assumption that u_{0} ≥ 0 in ℝ^{n}, a byproduct of the results in [1] shows that any nonnegative solution of (1) blows up in finite time. Unfortunately, although we believe that for nonnegative initial data any global solution of (1) is eventually positive, as far as we are aware no condition is known which ensures this property of the solution.

Theorem 2 states that if (12) holds and the solution u is global, then the negative part of u is not neglectable with respect to the positive part. In some sense, the solution may be global only if the negative and positive parts are “perfectly balanced”. We believe that under further suitable assumptions on u_{0} (such as positivity) case (ii) never occurs and we have blow up in finite time.

Remark 1. By direct calculation, one can see that if there exists an entire smooth radial solution v, decaying at infinity, of the equation

Δ^{2}v = |v|^{p−1}v + \frac{1}{4}rv_{r} + \frac{1}{p − 1}v in ℝ^{n}

then

u(x, t) := (1 + t)^{−1/(p−1)}v (|x|(1 + t)^{−1/4}) , (x, t) ∈ ℝ_{+}^{n+1}

is a (self-similar) solution of (1). These solutions were studied in some detail in [8] in order to prove non-uniqueness results for (1) with unbounded initial data u_{0}.

3 Proof of Theorem 1

During the proof of Theorem 1 we will need the following crucial statements whose proofs are postponed, respectively, to Sections 6 and 7.

Proposition 2. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant C_{1} = C_{1}(n, p, μ) > 0 such that for all (x, t) ∈ ℝ_{+}^{n+1}, one has:

ω_{1} ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{t} )^{1/3}] \frac{dy}{t^{n/4}(1 + |x − y|^{4/(p−1)})} ≤ \frac{C_{1}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} .

Proposition 3. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant C_{2} = C_{2}(n, p, μ) > 0 such that for all (x, t) ∈ ℝ_{+}^{n+1}, one has:

ω_{1} ∫_{0}^{t} ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{s} )^{1/3}] \frac{dy ds}{s^{n/4} (1 + (t − s)^{1/(p−1)} + |x − y|^{4/(p−1)})^{p}} ≤ \frac{C_{2}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} .

The proof of Proposition 3 is quite lengthy and delicate. For this reason, in Section 8 we give a much simpler proof of it under the additional constraints that n ≥ 5 and p > \frac{n+4}{n−4}.

Let D be as in (5). Then, we will prove Theorem 1 by taking

α := \frac{1}{2^{p/(p−1)}D^{p/(p−1)}C_{1}C_{2}^{1/(p−1)}} .  (14)

Let b(t) = b(x, t) be the kernel defined in (3) and note that under the assumption (9) we have u_{0} ∈ L^{∞}(ℝ^{n}). It is then clear that any strong solution u of (1) also satisfies the integral equation

u(t) = b(t) ∗ u_{0} + ∫_{0}^{t} b(t − s) ∗ |u(s)|^{p−1}u(s) ds .  (15)

Conversely, it is well-known (see e.g. [7, Proposition A4]) that a bounded solution of the integral equation (15) is a strong solution of (1). Uniqueness of strong solutions to (15) follows by the standard contraction mapping principle, see e.g. [13, Section 3.3].

We study (15) by following the approach in [1, 20, 21] combined with the extremely powerful Propositions 2 and 3. Let D be as in (5) and let

v_{0}(x) := D|u_{0}(x)| .  (16)

Then, consider the equation

v(t) = b̃(t) ∗ v_{0} + D ∫_{0}^{t} b̃(t − s) ∗ v^{p}(s) ds .  (17)

In view of (5) and the existence proof below, it is clear that as long as v(t) exists, we have a solution u to (15)

|u(x, t)| ≤ v(x, t) .  (18)

In particular, if we can show that v is globally defined, this will also prove that u is globally defined. To this end, for all v ∈ L^{∞}(ℝ_{+}^{n+1}) we define

B̃v(x, t) := b̃(t) ∗ v_{0} + D ∫_{0}^{t} b̃(t − s) ∗ v^{p}(s) ds .  (19)

For T > 0 and for

M := \frac{1}{(2DC_{2})^{1/(p−1)}}  (20)

we introduce the set

S_{T} := \{v ∈ C(ℝ^{n} × [0, T]); 0 ≤ v(x, t) ≤ \frac{M}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} \} .

It is clear that

S_{T} is a nonempty, closed and bounded convex subset of C(ℝ^{n} × [0, T]) .  (21)

We claim that

B̃(S_{T}) ⊂ S_{T} .  (22)

In order to prove (22), take v ∈ S_{T} and consider B̃v as defined in (19). Clearly, B̃v(x, t) ≥ 0. Moreover, by (9), (16) and Proposition 2 we have

[b̃(t) ∗ v_{0}](x) ≤ \frac{DC_{1}α}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}}  for all (x, t) ∈ ℝ_{+}^{n+1} .  (23)

Furthermore, for v ∈ S_{T} we get

∫_{0}^{t} b̃(t − s) ∗ v^{p}(s) ds = ω_{1} ∫_{0}^{t} ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{s} )^{1/3}] \frac{v^{p}(x − y, t − s)}{s^{n/4}} dy ds ≤

≤ ω_{1}M^{p} ∫_{0}^{t} ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{s} )^{1/3}] \frac{dy ds}{s^{n/4} (1 + (t − s)^{1/(p−1)} + |x − y|^{4/(p−1)})^{p}} .

Using Proposition 3 we then obtain

∫_{0}^{t} b̃(t − s) ∗ v^{p}(s) ds ≤ \frac{C_{2}M^{p}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}}  for all (x, t) ∈ ℝ_{+}^{n+1} .  (24)

Inserting (23)-(24) into (19) and recalling (14)-(20), we finally obtain

B̃v(x, t) ≤ \frac{M}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} for all (x, t) ∈ ℝ_{+}^{n+1} ,

which proves (22).

Arguing as in [21, Lemma 3.1] (see also [1, Theorem 2.1]), we can also prove the two following facts:

B̃S_{T} is a compact subset (with respect to the L^{∞}-norm) of S_{T} ,  (25)
B̃ is continuous .  (26)

In view of (21)-(22)-(25)-(26), we may apply Schauder’s fixed point Theorem and infer that B̃ has a fixed point v_{T} ∈ S_{T}. For any T > 0, we now define

V_{T} (x, t) = \{ \begin{matrix}v_{T} (x, t) & if t ≤ T \\ v_{T} (x, T) & if t > T . \end{matrix}

By applying a local version of Ascoli-Arzelà’s Theorem as in [1, p.718] (see also [21, p.61]) we infer that, up to a subsequence, V_{T} converges uniformly on compact subsets of ℝ_{+}^{n+1} to a (global) solution of (17). This completes the proof of Theorem 1. □

Remark 2. The technique developed here may also serve to treat Cauchy problems with polyharmonic elliptic principal part

\{ \begin{matrix}u_{t} + (−Δ)^{k} u = |u|^{p−1}u & in ℝ_{+}^{n+1} \\ u(x, 0) = u_{0}(x) & in ℝ^{n} . \end{matrix}  (27)

As in the biharmonic case one may consider a majorising kernel associated to (27)

b̃_{k}(x, t) := ω_{1,k}t^{−n/2k} exp (−μ_{k} (\frac{|x|^{2k}}{t} )^{1/(2k−1)}) .

Propositions 2 and 3 continue to hold true provided one assumes p > 1 + 2k/n and provided (\frac{|y|^{4}}{t} )^{1/3}

is replaced by (\frac{|y|^{2k}}{t} )^{1/(2k−1)}, the exponent n/4 by n/2k and the exponent 4/(p − 1) by 2k/(p − 1). The proofs of these generalised propositions require almost only straightforward changes except for the end of the proof of Lemma 5 in the case k = 1, n = 2. In this special case one splits the last integral at 3T^{4/3} instead.

So, one may prove for (27) the same result as in Theorem 1 by changing 1 + 4/n into 1 + 2k/n and 4/(p − 1) into 2k/(p − 1). This general result contains also Proposition 1 as a special case thereby providing a unified proof independent of the maximum principle.

4 Proof of Theorem 2

Our proof is obtained by adapting the arguments in [1] to the case of sign-changing solutions. Let v(x, t) = ψ_{1}(x)ψ_{2}(t) ∈ H and let K = spt(ψ_{1}) and [0, T] = spt(ψ_{2}). For all R > 0 take

ϕ_{R}(x, t) := v (\frac{x}{R}, \frac{t}{R^{4}} ) = ψ_{1} (\frac{x}{R}) ψ_{2} (\frac{t}{R^{4}} )

as test function in (11). Then, we obtain

∫_{0}^{R^{4}T} ∫_{RK} |u(x, t)|^{p−1}u(x, t)v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt + ∫_{RK} u_{0}(x)ψ_{1} (\frac{x}{R}) dx =

= R^{−4} ∫_{0}^{R^{4}T} ∫_{RK} u(x, t) [−v_{t} (\frac{x}{R}, \frac{t}{R^{4}} ) + Δ^{2}v (\frac{x}{R}, \frac{t}{R^{4}} )] dxdt =: I .  (28)

In the sequel the c’s denote positive constants which may have different values also when they appear in the same formula. We estimate the right hand side I of (28) as follows. Fix γ < 1 and take δ := \frac{1−γ}{1+γ}; by Young’s inequality we know that there exists C_{δ} > 0 such that

I = R^{−4} ∫_{0}^{R^{4}T} ∫_{RK} u(x, t) v^{1/p} (\frac{x}{R}, \frac{t}{R^{4}} ) \frac{−v_{t} (\frac{x}{R}, \frac{t}{R^{4}} ) + Δ^{2}v (\frac{x}{R}, \frac{t}{R^{4}} )}{v^{1/p} (\frac{x}{R}, \frac{t}{R^{4}} )} dxdt ≤

≤ δ ∫_{0}^{R^{4}T} ∫_{RK} |u(x, t)|^{p} v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt + \frac{C_{δ}}{R^{4p′}} ∫_{0}^{R^{4}T} ∫_{RK} \frac{|v_{t} (\frac{x}{R}, \frac{t}{R^{4}} )|^{p′} + |Δ^{2}v (\frac{x}{R}, \frac{t}{R^{4}} )|^{p′}}{v^{p′−1} (\frac{x}{R}, \frac{t}{R^{4}} )} dxdt.

With the change of variables

x = Ry , t = R^{4}τ ,  (29)

we finally obtain

I ≤ δ ∫_{0}^{R^{4}T} ∫_{RK} |u(x, t)|^{p} v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt + C_{δ}R^{n−4/(p−1)} ∫_{0}^{T} ∫_{K} \frac{|v_{t}(y, τ)|^{p′} + |Δ^{2}v(y, τ)|^{p′}}{v^{p′−1}(y, τ)} dydτ ≤

≤ δ ∫_{0}^{R^{4}T} ∫_{RK} |u(x, t)|^{p} v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt + cR^{n−4/(p−1)} .  (30)

Next, by (12) we know that there exists C, ρ > 0 such that

|x|^{4/(p−1)}u_{0}(x) ≥ \frac{λ}{2} if |x| ≥ ρ .

Let R be sufficiently large so that B_{ρ} ⊂ RK. For such R we have

∫_{RK} u_{0}(x)ψ_{1} (\frac{x}{R}) dx ≥ −∥ψ_{1}∥_{∞} ∫_{B_{ρ}} |u_{0}(x)| dx + \frac{λ}{2} ∫_{RK\B_{ρ}} ψ_{1} (\frac{x}{R}) \frac{dx}{|x|^{4/(p−1)}} .

Clearly, there exist 0 < α < β such that if R is sufficiently large then X_{R} := {x ∈ ℝ^{n}; αR < |x| < βR} ⊂ RK \ B_{ρ}. Therefore, the last inequality becomes

∫_{RK} u_{0}(x)ψ_{1} (\frac{x}{R}) dx ≥ −c + \frac{λ}{2} min_{α≤|y|≤β} ψ_{1}(y) ∫_{X_{R}} \frac{dx}{|x|^{4/(p−1)}} ≥ −c + cλR^{n−4/(p−1)} .  (31)

Summarizing, by using (30) and (31) into (28), we arrive at

(1 − δ) ∫_{0}^{R^{4}T} ∫_{RK} |u^{+}(x, t)|^{p}v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt − (1 + δ) ∫_{0}^{R^{4}T} ∫_{RK} |u^{−}(x, t)|^{p}v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt ≤

≤ cR^{n−4/(p−1)} − cλR^{n−4/(p−1)} + c .

If we divide by 1 + δ and we take λ sufficiently large (say λ ≥ Λ), the previous inequality becomes

∫_{0}^{R^{4}T} ∫_{RK} |u^{−}(x, t)|^{p}v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt − γ ∫_{0}^{R^{4}T} ∫_{RK} |u^{+}(x, t)|^{p}v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt ≥ cR^{n−4/(p−1)}

for sufficiently large R. Finally, performing the change of variables (29) and letting R → ∞, proves (13).

Assume now that (12) is replaced by lim sup_{|x|→∞} |x|^{4/(p−1)}u_{0}(x) < 0 and consider again (28). In this case, instead of (31) we obtain

∫_{RK} u_{0}(x)ψ_{1} (\frac{x}{R}) dx ≤ c − cλR^{n−4/(p−1)} .

Moreover, we may replace (30) with

I ≥ −δ ∫_{0}^{R^{4}T} ∫_{RK} |u(x, t)|^{p} v (\frac{x}{R}, \frac{t}{R^{4}} ) dxdt − cR^{n−4/(p−1)} .

The proof may now be completed as in the case where (12) holds.

5 Some technical results

Here and in all the remaining of the paper, with c we denote positive constants which may have different values also within the same line. Moreover, in order to avoid heavy notations, when we write 1/αβ, we mean \frac{1}{αβ}.

For both the proofs of Propositions 2 and 3 we will make use of the following trivial facts:

min \{\frac{1}{a}, \frac{1}{b}\} ≤ \frac{2}{a + b}  for all a, b > 0 ,  (32)

and for all (m, q) ∈ [ℕ \ {0}] × (0, +∞) there exist γ_{1}, γ_{2} > 0 such that

γ_{1} (∑_{i=1}^{m} α_{i})^{q} ≤ ∑_{i=1}^{m} α_{i}^{q} ≤ γ_{2} (∑_{i=1}^{m} α_{i})^{q}  for all α_{i} ≥ 0 .  (33)

We now prove five technical statements (uniform bounds for 3-dimensional integrals depending on a parameter T > 0) which are needed for the proof of Proposition 3. Most delicate are Lemmas 3 and 4, where the essential idea consists in a suitable splitting of the domain of integration. The integrals Γ_{i} (i = 1, ..., 5) below depend on T but, for simplicity, we omit to emphasize this dependence.

Lemma 1. Assume that n ≥ 2 and p > 1 + 4/n. Then, there exists c > 0 such that

Γ_{1} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{0}^{1/2T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3} dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} ≤ c

for all T > 0.

Proof. First, we note that for the integral considered we have σ ≤ T ρ ≤ \frac{1}{2}. Hence, |σ − 1| ≥ \frac{1}{2} and

Γ_{1} ≤ ∫_{0}^{∞} \frac{dw}{(1 + w^{2})^{n/2}} · ∫_{0}^{1/2T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{T^{3}ρ^{3} dσ dρ}{(\frac{T^{3}}{ρ} (T ρ − σ) + \frac{1}{16})^{p/(p−1)}} ≤

≤ c ∫_{0}^{1/2T} ρ^{n−1}e^{−μρ^{4/3}} [\frac{1}{(\frac{T^{3}}{ρ} (T ρ − σ) + \frac{1}{16})^{1/(p−1)}} ]_{0}^{T ρ} dρ ≤ c ∫_{0}^{∞} ρ^{n−1}e^{−μρ^{4/3}} dρ = c

and the uniform upper bound follows. □

Lemma 2. Assume that n ≥ 2 and p > 1 + 4/n. Then, there exists c > 0 such that

Γ_{2} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{1/2T}^{3/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ/2} \frac{σ^{3} dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} ≤ c

for all T > 0.

Proof. As long as Γ_{2} is involved, we have σ^{3} ≤ \frac{T^{3}ρ^{3}}{8} . Then,

Γ_{2} ≤ c T^{3} ∫_{0}^{∞} \frac{dw}{(1 + w^{2})^{n/2}} · ∫_{1/2T}^{3/T} ρ^{n−2}e^{−μρ^{4/3}} ∫_{0}^{T ρ/2} \frac{dσ dρ}{[\frac{T^{3}}{ρ} (T ρ − σ)]^{p/(p−1)}} ≤

≤ c ∫_{1/2T}^{3/T} ρ^{n−1}e^{−μρ^{4/3}} [\frac{1}{(\frac{T^{3}}{ρ} (T ρ − σ))^{1/(p−1)}} ]_{0}^{T ρ/2} dρ ≤ \frac{c}{T^{4/(p−1)}} ∫_{1/2T}^{3/T} ρ^{n−1}e^{−μρ^{4/3}} dρ =

[τ = T ρ] = \frac{c}{T^{n+4/(p−1)}} ∫_{1/2}^{3} τ^{n−1} exp [−μ(\frac{τ}{T} )^{4/3}] dτ =: f(T).

It is clear that f(T) is well-defined (finite) for all T ∈ (0, ∞). Moreover, f(T) → 0 for both T → 0 and T → ∞ so that the uniform upper bound for Γ_{2} follows. □

Lemma 3. Assume that n ≥ 2 and p > 1 + 4/n. Then, there exists c > 0 such that

Γ_{3} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{σ^{3} dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} ≤ c

for all T > 0.

Proof. Since σ ≥ \frac{T ρ}{2} ≥ \frac{1}{4} and \frac{T^{3}}{ρ} ≥ \frac{1}{8ρ^{4}} , Γ_{3} converges uniformly in T whenever the integral

∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{σ^{3} dσ dρ dw}{[\frac{T ρ−σ}{8ρ^{4}} + (σ − 1)^{4} + \frac{1}{16(1+w^{2})^{2}} ]^{p/(p−1)}}

does so. Moreover, we remark that

∫_{0}^{1} \frac{1}{(1 + w^{2})^{n/2}} ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{σ^{3} dσ dρ dw}{[\frac{T ρ−σ}{8ρ^{4}} + (σ − 1)^{4} + \frac{1}{16(1+w^{2})^{2}} ]^{p/(p−1)}} ≤ c ∫_{0}^{∞} ρ^{n−5}e^{−μρ^{4/3}} dρ

so that the statement follows if there exists c > 0 (independent of T) such that

Γ′_{3} := ∫_{1}^{∞} \frac{1}{w^{n}} ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{dσ dρ dw}{[\frac{T ρ−σ}{8ρ^{4}} + (σ − 1)^{4} + \frac{1}{64w^{4}} ]^{p/(p−1)}} ≤ c  for all T > 0. (34)

Consider the map

h(ρ) := min{1, ρ^{4}} · (1 − T ρ)^{4} .  (35)

If ρ ∈ [\frac{1}{2T} , \frac{1}{T}], then T ρ − h(ρ) ≥ T ρ − (1 − T ρ)^{4} > \frac{T ρ}{2}. Hence, we can split the inner integral in (34) as follows

∫_{T ρ/2}^{T ρ} = ∫_{T ρ/2}^{T ρ−h(ρ)} + ∫_{T ρ−h(ρ)}^{T ρ} =: L_{1} + L_{2}  (36)

and we estimate L_{1} and L_{2}. This splitting is the essential idea in this proof in order to cover the full super-Fujita range p > 1 + 4/n.

First, we note that

L_{1} ≤ ∫_{T ρ/2}^{T ρ−h(ρ)} \frac{dσ}{[\frac{T ρ−σ}{8ρ^{4}} + \frac{1}{64w^{4}} ]^{p/(p−1)}} = [\frac{8(p − 1)ρ^{4}}{[\frac{T ρ−σ}{8ρ^{4}} + \frac{1}{64w^{4}} ]^{1/(p−1)}} ]_{T ρ/2}^{T ρ−h(ρ)} ≤

≤ \frac{8(p − 1)ρ^{4}}{[\frac{h(ρ)}{8ρ^{4}} + \frac{1}{64w^{4}} ]^{1/(p−1)}} ≤ \frac{c ρ^{4}}{[\frac{h(ρ)}{ρ^{4}} + \frac{1}{w^{4}} ]^{1/(p−1)}} .  (37)

Next, since σ ≤ T ρ ≤ 1, by using (33) we note that

L_{2} ≤ c ∫_{T ρ−h(ρ)}^{T ρ} \frac{dσ}{[1 − σ + \frac{1}{2\sqrt{2}w}]^{4p/(p−1)}} ≤ \frac{c h(ρ)}{[1 − T ρ + \frac{1}{w}]^{4p/(p−1)}} .  (38)

Inserting (37) and (38) into (36), and recalling the definition of Γ′_{3} in (34), entails

Γ′_{3} ≤ c ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{1}^{∞} \frac{1}{w^{n}} [\frac{ρ^{4}}{[\frac{h(ρ)}{ρ^{4}} + \frac{1}{w^{4}} ]^{1/(p−1)}} + \frac{h(ρ)}{[1 − T ρ + \frac{1}{w}]^{4p/(p−1)}} ] dwdρ (39)

and we now estimate the two inner integrals in (39). Since (1 − T ρ)^{−1} ≥ 2, for the first integral we have

∫_{1}^{∞} \frac{ρ^{4}}{[\frac{h(ρ)}{ρ^{4}} + \frac{1}{w^{4}} ]^{1/(p−1)}} \frac{dw}{w^{n}} ≤

≤ ρ^{4} (∫_{1}^{1/(1−T ρ)} w^{4/(p−1)−n} dw + \frac{ρ^{4/(p−1)}}{h(ρ)^{1/(p−1)}} ∫_{1/(1−T ρ)}^{∞} \frac{dw}{w^{n}} ) ≤

≤ ρ^{4} [Φ_{1}(1 − T ρ) + \frac{c ρ^{4/(p−1)}}{min\{1, ρ^{4/(p−1)}\} · (1 − T ρ)^{4/(p−1)−n+1}} ] ,  (40)

where (recall that \frac{4}{p−1} < n)

Φ_{1}(s) = c \{ \begin{matrix}s^{n−1−4/(p−1)} & if 4/(p − 1) − n > −1 \\ | log s| & if 4/(p − 1) − n = −1 \\ 1 & if 4/(p − 1) − n < −1 . \end{matrix}

For the second inner integral in (39) we have

∫_{1}^{∞} \frac{h(ρ)}{[1 − T ρ + \frac{1}{w}]^{4p/(p−1)}} \frac{dw}{w^{n}} ≤

≤ h(ρ) (∫_{1}^{1/(1−T ρ)} w^{4p/(p−1)−n} dw + \frac{1}{(1 − T ρ)^{4p/(p−1)}} ∫_{1/(1−T ρ)}^{∞} \frac{dw}{w^{n}} ) ≤

≤ min\{1, ρ^{4}\} · (Φ_{2}(1 − T ρ) + \frac{c}{(1 − T ρ)^{4/(p−1)−n+1}} ) ,  (41)

where

Φ_{2}(s) = c \{ \begin{matrix}s^{n−1−4/(p−1)} & if 4p/(p − 1) − n > −1 \\ s^{4}| log s| & if 4p/(p − 1) − n = −1 \\ s^{4} & if 4p/(p − 1) − n < −1 . \end{matrix}

Inserting (40)-(41) into (39) gives

Γ′_{3} ≤ c ∫_{1/2T}^{1/T} ρ^{n−1}e^{−μρ^{4/3}} (Φ_{1}(1 − T ρ) + \frac{c ρ^{4/(p−1)}}{min\{1, ρ^{4/(p−1)}\} · (1 − T ρ)^{4/(p−1)−n+1}} ) dρ+

+c ∫_{1/2T}^{1/T} ρ^{n−5}e^{−μρ^{4/3}} · min\{1, ρ^{4}\} · (Φ_{2}(1 − T ρ) + \frac{c}{(1 − T ρ)^{4/(p−1)−n+1}} ) dρ =: f(T).

Since 4/(p − 1) − n + 1 < 1 by the assumption on the exponent being “super-Fujita”, the function f is well-defined (finite) for all T ∈ (0, ∞). In order to study its behaviour as T → 0 and T → ∞, we perform the change of variables τ = T ρ. Then, we get

f(T) = \frac{c}{T^{n}} ∫_{1/2}^{1} τ^{n−1} exp [−μ(\frac{τ}{T} )^{4/3}] (Φ_{1}(1 − τ) + \frac{c τ^{4/(p−1)}}{min\{τ, T \}^{4/(p−1)} · (1 − τ)^{4/(p−1)−n+1}} ) dτ+

+\frac{c}{T^{n}} ∫_{1/2}^{1} τ^{n−5} exp [−μ(\frac{τ}{T} )^{4/3}] · min\{T, τ\}^{4} · (Φ_{2}(1 − τ) + \frac{c}{(1 − τ)^{4/(p−1)−n+1}} ) dτ.

For all T ≥ 1 we have min{τ, T } = τ so that

f(T) ≤ \frac{c}{T^{n}} ∫_{1/2}^{1} τ^{n−1} (Φ_{1}(1 − τ) + Φ_{2}(1 − τ) + \frac{c}{(1 − τ)^{4/(p−1)−n+1}} ) dτ

and it is clear that f(T) → 0 as T → ∞. For all T ≤ 1/2 we have min{τ, T } = T so that

f(T) ≤ \frac{c e^{−cT^{−4/3}}}{T^{n}} ∫_{1/2}^{1} τ^{n−1} (Φ_{1}(1 − τ) + \frac{c τ^{4/(p−1)}}{T^{4/(p−1)} · (1 − τ)^{4/(p−1)−n+1}} ) dτ+

+\frac{c e^{−cT^{−4/3}}}{T^{n−4}} ∫_{1/2}^{1} τ^{n−5} (Φ_{2}(1 − τ) + \frac{c}{(1 − τ)^{4/(p−1)−n+1}} ) dτ

and f(T) → 0 as T → 0. This proves (34) and shows that Γ_{3} is uniformly bounded. □ Lemma 4. Assume that n ≥ 2 and p > 1 + 4/n. Then, there exists c > 0 such that

Γ_{4} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{1/T}^{3/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{σ^{3} dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} ≤ c

for all T > 0. Proof. This proof follows the same lines as that of Lemma 3. Let us just briefly sketch it. As for (34), we see that the statement is equivalent to the existence of some c > 0 such that

Γ′_{4} := ∫_{1}^{∞} \frac{1}{w^{n}} ∫_{1/T}^{3/T} ρ^{n−5}e^{−μρ^{4/3}} ∫_{T ρ/2}^{T ρ} \frac{dσ dρ dw}{[\frac{T ρ−σ}{8ρ^{4}} + (σ − 1)^{4} + \frac{1}{64w^{4}} ]^{p/(p−1)}} ≤ c  for all T > 0. (42)

Consider again the map h defined in (35). Since 1 ≤ T ρ ≤ 3, the following three facts hold

T ρ − \frac{h(ρ)}{32} ≥ \frac{T ρ}{2} ,  T ρ − \frac{h(ρ)}{32} ≥ 1 ,  T ρ − 1 − \frac{h(ρ)}{32} ≥ \frac{T ρ − 1}{2} .  (43)

Instead of (36), the first inequality in (43) suggests to split here the inner integral in (42) as

∫_{T ρ/2}^{T ρ} = ∫_{T ρ/2}^{T ρ−h(ρ)/32} + ∫_{T ρ−h(ρ)/32}^{T ρ} =: M_{1} + M_{2}.

Arguing as for (37) and (38), we obtain

M_{1} ≤ \frac{c ρ^{4}}{[\frac{h(ρ)}{ρ^{4}} + \frac{1}{w^{4}} ]^{1/(p−1)}} , M_{2} ≤ \frac{c h(ρ)}{[T ρ − 1 + \frac{1}{w}]^{4p/(p−1)}} ,

where we used both the second and third inequality in (43).

Finally, repeating the arguments relative to (40)-(41), we conclude that

Γ′_{4} ≤ c ∫_{1/T}^{3/T} ρ^{n−1}e^{−μρ^{4/3}} (Φ_{1}(T ρ − 1) + \frac{c ρ^{4/(p−1)}}{min\{1, ρ^{4/(p−1)}\} · (T ρ − 1)^{4/(p−1)−n+1}} ) dρ+

+c ∫_{1/T}^{3/T} ρ^{n−5}e^{−μρ^{4/3}} · min\{1, ρ^{4}\} · (Φ_{2}(T ρ − 1) + \frac{c}{(T ρ − 1)^{4/(p−1)−n+1}} ) dρ.

With the change of variables τ = T ρ one can then show (42), namely uniform (with respect to T) boundedness of Γ′_{4}. □

Lemma 5. Assume that n ≥ 2 and p > 1 + 4/n. Then, there exists c > 0 such that

Γ_{5} := ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{3/T}^{∞} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3} dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} ≤ c

for all T > 0.

Proof. As long as Γ_{5} is involved, we have T ρ ≥ 3. Hence, we may estimate:

Γ_{5} ≤ ∫_{0}^{∞} \frac{dw}{(1 + w^{2})^{n/2}} · ∫_{3/T}^{∞} ρ^{n−5}e^{−μρ^{4/3}} (\frac{8ρ^{p/(p−1)}}{T^{3p/(p−1)}} ∫_{0}^{2} \frac{dσ}{(3 − σ)^{p/(p−1)}} + ∫_{2}^{T ρ} \frac{σ^{3} dσ}{(σ − 1)^{4p/(p−1)}} ) dρ ≤

≤ c ∫_{3/T}^{∞} ρ^{n−5}e^{−μρ^{4/3}} (\frac{c ρ^{p/(p−1)}}{T^{3p/(p−1)}} + ∫_{2}^{∞} \frac{σ^{3} dσ}{(σ − 1)^{4p/(p−1)}} ) dρ ≤

[τ = T ρ] ≤ \frac{c}{T^{n−4+4p/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} exp [−μ(\frac{τ}{T} )^{4/3}] dτ + c ∫_{0}^{∞} ρ^{n−5}e^{−μρ^{4/3}} dρ ≤

≤ c + \frac{c}{T^{n+4/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} exp [−μ(\frac{τ}{T} )^{4/3}] dτ .

Therefore, the statement will follow if we show that the function

g(T) := \frac{1}{T^{n+4/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} exp [−μ(\frac{τ}{T} )^{4/3}] dτ

remains bounded when T varies over (0, ∞). Since g(T) is finite for every T ∈ (0, ∞), it suffices to study the limits of g(T) when T → 0 and T → ∞. We first remark that for all T < 1 we may write

g(T) = \frac{1}{T^{n+4/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} exp [−\frac{μ}{2}(\frac{τ}{T} )^{4/3}] · exp [−\frac{μ}{2}(\frac{τ}{T} )^{4/3}] dτ ≤

≤ \frac{exp [−\frac{μ}{2}(\frac{3}{T} )^{4/3}]}{T^{n+4/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} · exp [ − \frac{μ}{2}τ^{4/3}] dτ = \frac{c}{T^{n+4/(p−1)}} · exp [−\frac{μ}{2}(\frac{3}{T} )^{4/3}]

and the last term tends to 0 as T → 0. Next, for all T > 1 we write

g(T) = \frac{1}{T^{n+4/(p−1)}} (∫_{3}^{3T^{n/(n−1)}} + ∫_{3T^{n/(n−1)}}^{∞}) .

Then, by using the two following facts

τ ≤ 3T^{n/(n−1)} ⟹ \frac{1}{T} ≤ \frac{c}{τ^{(n−1)/n}}  ,  τ ≥ 3T^{n/(n−1)} ⟹ \frac{τ}{T} ≥ cτ^{1/n} ,

we may estimate g(T) also when T > 1:

g(T) ≤ c ∫_{3}^{3T^{n/(n−1)}} \frac{τ^{n−5+p/(p−1)}}{τ^{n−1+4(n−1)/n(p−1)}} dτ + \frac{c}{T^{n+4/(p−1)}} ∫_{3T^{n/(n−1)}}^{∞} τ^{n−5+p/(p−1)} exp[−cτ^{4/3n}] dτ ≤

≤ c ∫_{3}^{∞} \frac{dτ}{τ^{(3np−4)/n(p−1)}} + \frac{c}{T^{n+4/(p−1)}} ∫_{3}^{∞} τ^{n−5+p/(p−1)} exp[−cτ^{4/3n}] dτ.

Since 3np − 4 > n(p − 1), the above term remains bounded as T → ∞. This completes the proof of the lemma. □

6 Proof of Proposition 2

In order to prove Proposition 2 we introduce the function

G_{1}(x, t) := ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{t} )^{1/3}] \frac{dy}{t^{n/4}(1 + |x − y|^{4/(p−1)})} , (x, t) ∈ ℝ_{+}^{n+1} .

With the change of variables y = s^{1/4}z, we may also rewrite G_{1} as

G_{1}(x, t) = ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] \frac{dz}{(1 + |x − t^{1/4}z|^{4/(p−1)})} , (x, t) ∈ ℝ_{+}^{n+1} .

We first estimate G_{1} for small values of |x|:

Lemma 6. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant K_{1} = K_{1}(n, p, μ) > 0 such that

G_{1}(x, t) ≤ \frac{K_{1}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} for all |x| ≤ 1 , t ≥ 0 .

Proof. For |x| ≤ 1, we have

G_{1}(x, t) ≤ ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] dz = c ≤ \frac{A_{1}}{1 + |x|^{4/(p−1)}}  for all |x| ≤ 1 , t ≥ 0 ,  (44)

for some A_{1} > 0. Moreover, with the change of variables w = z − t^{−1/4}x we obtain

G_{1}(x, t) ≤ \frac{1}{t^{1/(p−1)}} [∫_{|z−t^{−1/4}x|<1} \frac{dz}{|z − t^{−1/4}x|^{4/(p−1)}} + ∫_{|z−t^{−1/4}x|>1} exp [−μ|z|^{4/3}] dz] ≤

≤ \frac{1}{t^{1/(p−1)}} [∫_{|w|<1} \frac{dw}{|w|^{4/(p−1)}} + ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] dz] = \frac{A_{2}}{t^{1/(p−1)}}  for all (x, t) ∈ ℝ_{+}^{n+1} , (45)

with A_{2} being finite because \frac{4}{p−1} < n. Clearly, A_{2} is independent of x and t.

By combining (44) and (45), for all |x| ≤ 1 and t ≥ 0 we obtain

G_{1}(x, t) ≤ min \{\frac{A_{1}}{1 + |x|^{4/(p−1)}} , \frac{A_{2}}{t^{1/(p−1)}} \} ≤ max\{A_{1}, A_{2}\} · min \{\frac{1}{1 + |x|^{4/(p−1)}} , \frac{1}{t^{1/(p−1)}} \} .

By (32), this yields

G_{1}(x, t) ≤ \frac{2 · max\{A_{1}, A_{2}\}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} for all |x| ≤ 1 , t ≥ 0 ,

which proves the statement with K_{1} = 2 · max{A_{1}, A_{2}}. □

Assume now that |x| > 1. Then, we prove Lemma 7. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant K_{2} = K_{2}(n, p, μ) > 0 such that

G_{1}(x, t) ≤ \frac{K_{2}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} for all |x| > 1 , t ≥ 0 .

Proof. Put F (x, t) := (1 + |x|^{4/(p−1)} + t^{1/(p−1)}) · G_{1}(x, t). Then, the statement follows if we show that F (x, t) ≤ K_{2} for all |x| > 1 and t ≥ 0. When |x| > 1, we have

F (x, t) ≤ c ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] \frac{|x|^{4/(p−1)} + t^{1/(p−1)}}{|x − t^{1/4}z|^{4/(p−1)}} dz .

Therefore, simplifying by |x|^{4/(p−1)} and putting T := t^{1/4}|x|^{−1}, we obtain

F (x, t) ≤ c(1 + T^{4/(p−1)}) · ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] \frac{dz}{|\frac{x}{|x|} − T z|^{4/(p−1)}} .

At this stage, only the direction \frac{x}{|x|} of x is involved. Hence, with no loss of generality, we may assume that \frac{x}{|x|} = e_{1} = (1, 0, ..., 0), the first unit vector of the canonical basis of ℝ^{n}. In this case, the above estimate reads

F (x, t) ≤ c(1 + T^{4/(p−1)}) · ∫_{ℝ^{n}} exp [−μ|z|^{4/3}] \frac{dz}{|e_{1} − T z|^{4/(p−1)}} =: cΦ(T) .

Since \frac{4}{p−1} < n, the function Φ(T) is well-defined (finite) for all T ∈ (0, ∞). In order to prove the uniform boundedness of F, we have to show that Φ(T) remains bounded in both the cases T → 0 and T → ∞. When T → ∞, we have

Φ(T) = \frac{1 + T^{4/(p−1)}}{T^{4/(p−1)}} · [∫_{|z− \frac{e_{1}}{T} |<1} \frac{exp [−μ|z|^{4/3}] dz}{|z − \frac{e_{1}}{T} |^{4/(p−1)}} + ∫_{|z− \frac{e_{1}}{T} |>1} \frac{exp [−μ|z|^{4/3}] dz}{|z − \frac{e_{1}}{T} |^{4/(p−1)}} ] ≤

[w = z − \frac{e_{1}}{T} ] ≤ \frac{1 + T^{4/(p−1)}}{T^{4/(p−1)}} · [∫_{|w|<1} \frac{dw}{|w|^{4/(p−1)}} + ∫_{ℝ^{n}} e^{−μ|z|^{4/3}} dz] = O(1),

because (once more!) \frac{4}{p−1} < n. When T → 0, we have

Φ(T) ≤ O(1) [∫_{|e_{1}−T z|<1/2} \frac{exp [−μ|z|^{4/3}] dz}{|e_{1} − T z|^{4/(p−1)}} + ∫_{|e_{1}−T z|>1/2} \frac{exp [−μ|z|^{4/3}] dz}{|e_{1} − T z|^{4/(p−1)}} ] ≤

[w = e_{1} − T z] ≤ O(1) [\frac{exp[−cT^{−4/3}]}{T^{n}} ∫_{|w|<1/2} \frac{dw}{|w|^{4/(p−1)}} + c ∫_{ℝ^{n}} e^{−μ|z|^{4/3}} dz] = O(1),

where we also used the fact that |z| > \frac{1}{2T} whenever |e_{1} − T z| < 1/2. Therefore, Φ(T) is uniformly bounded on (0, ∞) and the proof of the lemma follows. □

Recalling the definition of G_{1}, Proposition 2 follows at once from Lemmas 6-7 by taking C_{1} = ω_{1} · max{K_{1}, K_{2}}.

7 Proof of Proposition 3

Our proof of Proposition 3 requires several lemmas. We define the function

G_{2}(x, t) := ∫_{0}^{t} ∫_{ℝ^{n}} exp [−μ (\frac{|y|^{4}}{s} )^{1/3}] \frac{dy ds}{s^{n/4} (1 + (t − s)^{1/(p−1)} + |x − y|^{4/(p−1)})^{p}} , (x, t) ∈ ℝ_{+}^{n+1}.

With the change of variables y = t^{1/4}z, we may also rewrite G_{2} as

G_{2}(x, t) = ∫_{0}^{t} ∫_{ℝ^{n}} \begin{matrix}exp & (−μ|z|^{4/3}) \\ [1 + (t − s)^{1/(p−1)} + |x − s^{1/4}z|^{4/(p−1)}] & p \end{matrix} dz ds , (x, t) ∈ ℝ_{+}^{n+1}.

We first obtain an upper bound for G_{2} in terms of the time variable: Lemma 8. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant B_{1} = B_{1}(n, p, μ) > 0 such that

G_{2}(x, t) ≤ \frac{B_{1}}{t^{1/(p−1)}} for all (x, t) ∈ ℝ_{+}^{n+1} .

Proof. We split the integral as follows:

G_{2}(x, t) = ∫_{0}^{t} ∫_{|z−s^{−1/4}x|<1/2} + ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} =: I_{1} + I_{2}  (46)

and we estimate separately I_{1} and I_{2}. First, note that

I_{1} ≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|<1/2} \begin{matrix}exp & (−μ|z|^{4/3}) \\ [(t − s)^{1/(p−1)} + |x − s^{1/4}z|^{4/(p−1)}] & p \end{matrix} dz ds ≤

[w = z − s^{−1/4}x] ≤ ∫_{0}^{t} ∫_{|w|<1/2} \frac{dw ds}{[(t − s)^{1/(p−1)} + s^{1/(p−1)}|w|^{4/(p−1)}]^{p}} ≤  (47)

[by (33)] ≤ c ∫_{|w|<1/2} ∫_{0}^{t} \frac{ds dw}{[t − s + s|w|^{4}]^{p/(p−1)}} ≤ \frac{c}{t^{1/(p−1)}} ∫_{|w|<1/2} \frac{dw}{(1 − |w|^{4})|w|^{4/(p−1)}} ≤ \frac{c}{t^{1/(p−1)}}

since \frac{4}{p−1} < n in view of the assumption p > 1 + \frac{4}{n}. Next, we have

I_{2} ≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} \begin{matrix}exp & (−μ|z|^{4/3}) \\ [(t − s)^{1/(p−1)} + s^{1/(p−1)}|xs^{−1/4} − z|^{4/(p−1)}] & p \end{matrix} dz ds ≤

≤ ∫_{0}^{t} ∫_{|z−s^{−1/4}x|>1/2} \begin{matrix}exp & (−μ|z|^{4/3}) \\ [(t − s)^{1/(p−1)} + (\frac{s}{16})^{1/(p−1)}] & p \end{matrix} dz ds ≤  (48)

[by (33)] ≤ c ∫_{ℝ^{n}} exp (−μ|z|^{4/3}) dz · ∫_{0}^{t} \frac{ds}{[16t − 15s]^{p/(p−1)}} ≤ \frac{c}{t^{1/(p−1)}} .

Combining (46)-(47)-(48) proves the statement. □

Next, we prove an upper bound for G_{2} for small values of |x|: Lemma 9. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant B_{2} = B_{2}(n, p, μ) > 0 such that

G_{2}(x, t) ≤ \frac{B_{2}}{1 + |x|^{4/(p−1)}} for all |x| ≤ 1 , t ≥ 0 .

Proof. Since |x| ≤ 1, it suffices to show that G_{2}(x, t) ≤ B_{2}/2 for some B_{2} > 0. This follows by performing the change of variables σ = t − s and the following trivial estimate

G_{2}(x, t) ≤ ∫_{ℝ^{n}} e^{−μ|z|^{4/3}} dz · ∫_{0}^{∞} \frac{ds}{[1 + σ^{1/(p−1)}]^{p}} := \frac{B_{2}}{2} ,

the constant B_{2} being independent of x and t. □

We now come to the most delicate estimate, an upper bound for G_{2} for large values of |x|:

Lemma 10. Assume that n ≥ 2 and p > 1 + 4/n. There exists a constant B_{3} = B_{3}(n, p, μ) > 0 such that

G_{2}(x, t) ≤ \frac{B_{3}}{1 + |x|^{4/(p−1)}} for all |x| > 1 , t ≥ 0 .

Proof. We will show that F (x, t) := (1 + |x|^{4/(p−1)})G_{2}(x, t) ≤ B_{3} for all (x, t) in their ranges. With the change of variables σ = \frac{|z|}{|x|} s^{1/4} and by simplifying by |x|^{4/(p−1)} we obtain

F (x, t) = 4 ∫_{ℝ^{n}} \frac{exp[−μ|z|^{4/3}]}{|z|^{4}} ∫_{0}^{\frac{|z|}{|x|} t^{1/4}} \frac{(1 + |x|^{−4/(p−1)}) σ^{3}}{[|x|^{−4/(p−1)} + (\frac{t}{|x|^{4}} − \frac{σ^{4}}{|z|^{4}} )^{1/(p−1)} + |\frac{x}{|x|} − \frac{z}{|z|}σ|^{4/(p−1)}]^{p}} dσ dz.

Put y := x/|x| so that |y| = 1 and y defines the direction of x; with no loss of generality we may take y = e_{1} = (1, 0, ..., 0), the first unit vector of the canonical basis. Let also T := \frac{t^{1/4}}{|x|}. Then, recalling |x| > 1, we obtain

F (x, t) ≤ c ∫_{ℝ^{n}} \frac{exp[−μ|z|^{4/3}]}{|z|^{4}} ∫_{0}^{|z|T} \frac{σ^{3} dσ dz}{(T^{4} − \frac{σ^{4}}{|z|^{4}} )^{p/(p−1)} + (σ − 1)^{4p/(p−1)} + [(1 − \frac{z_{1}}{|z|})2σ]^{2p/(p−1)}}

where we used (33) and the following fact:

|e_{1} − \frac{z}{|z|}σ|^{4/(p−1)} = (1 − 2\frac{z_{1}}{|z|}σ + σ^{2})^{2/(p−1)} = [(σ − 1)^{2} + (1 − \frac{z_{1}}{|z|})2σ]^{2/(p−1)} , z = (z_{1}, ..., z_{n}).

Now put z′ = (z_{2}, ..., z_{n}) ∈ ℝ^{n−1} so that z = (z_{1}, z′) and |z|^{2} = z_{1}^{2} +|z′|^{2}. Passing to radial coordinates r := |z′| in ℝ^{n−1} we obtain

F (x, t) ≤ c ∫_{0}^{∞} r^{n−2} ∫_{−∞}^{∞} \frac{exp[−μ(z_{1}^{2} + r^{2})^{2/3}]}{(z_{1}^{2} + r^{2})^{2}}

∫_{0}^{T \sqrt{z_{1}^{2}+r^{2}}} \frac{σ^{3} dσ dz_{1} dr}{(T^{4} − \frac{σ^{4}}{(z_{1}^{2}+r^{2})^{2}} )^{p/(p−1)} + (σ − 1)^{4p/(p−1)} + [(1 − _{\frac{z_{1}}{\sqrt{z_{1}^{2}+r^{2}}}} )2σ]^{2p/(p−1)}} .

With the change of variables z_{1} = rw the previous estimate becomes

F (x, t) ≤ c ∫_{0}^{∞} r^{n−5} ∫_{−∞}^{∞} \frac{exp[−μr^{4/3}(1 + w^{2})^{2/3}]}{(1 + w^{2})^{2}}

∫_{0}^{T r\sqrt{1+w^{2}}} \frac{σ^{3} dσ dw dr}{(T^{4} − \frac{σ^{4}}{r^{4}(1+w^{2})^{2}} )^{p/(p−1)} + (σ − 1)^{4p/(p−1)} + [(1 − \frac{w}{\sqrt{1+w^{2}}} )2σ]^{2p/(p−1)}} .

Making use of the two inequalities

1 − \frac{w}{\sqrt{1 + w^{2}}} = \begin{matrix}(\sqrt{1 + w^{2}} + w) & (\sqrt{1 + w^{2}} − w) \\ (\sqrt{1 + w^{2}} + w) & \sqrt{1 + w^{2}} \end{matrix} ≥ \frac{1}{2(1 + w^{2})} for all w ∈ ℝ

and

T^{4} − \frac{σ^{4}}{r^{4}(1 + w^{2})^{2}} = \frac{1}{r^{4}(1 + w^{2})^{2}} (T r\sqrt{1 + w^{2}} − σ)(T r\sqrt{1 + w^{2}} + σ)(T^{2}r^{2}[1 + w^{2}] + σ^{2}) ≥

≥ \frac{T^{3}}{r\sqrt{1 + w^{2}}} (T r\sqrt{1 + w^{2}} − σ) for all σ ∈ [0, T r\sqrt{1 + w^{2}}],

we may estimate further the above expression by

F (x, t) ≤ c ∫_{0}^{∞} r^{n−5} ∫_{0}^{∞} \frac{exp[−μr^{4/3}(1 + w^{2})^{2/3}]}{(1 + w^{2})^{2}}

∫_{0}^{T r\sqrt{1+w^{2}}} \frac{σ^{3} dσ dw dr}{(\frac{T^{3}}{r\sqrt{1+w^{2}}} )^{p/(p−1)}(T r\sqrt{1 + w^{2}} − σ)^{p/(p−1)} + (σ − 1)^{4p/(p−1)} + (\frac{σ}{1+w^{2}} )^{2p/(p−1)}} ,

where the integration with respect to w is now just over (0, ∞) since the integrand is even with respect to w. We now use (33) and we make the further change of variables r = ρ/\sqrt{1 + w^{2}} to obtain

F (x, t) ≤ c ∫_{0}^{∞} \frac{1}{(1 + w^{2})^{n/2}} ∫_{0}^{∞} ρ^{n−5}e^{−μρ^{4/3}} ∫_{0}^{T ρ} \frac{σ^{3}  dσ dρ dw}{(\frac{T^{3}}{ρ} (T ρ − σ) + (σ − 1)^{4} + \frac{σ^{2}}{(1+w^{2})^{2}} )^{p/(p−1)}} . (49)

We split the integral on the right hand side of (49) as follows:

∫_{0}^{∞} ∫_{0}^{1/2T} ∫_{0}^{T ρ} + ∫_{0}^{∞} ∫_{1/2T}^{3/T} ∫_{0}^{T ρ/2} + ∫_{0}^{∞} ∫_{1/2T}^{1/T} ∫_{T ρ/2}^{T ρ} + ∫_{0}^{∞} ∫_{1/T}^{3/T} ∫_{T ρ/2}^{T ρ} + ∫_{0}^{∞} ∫_{3/T}^{∞} ∫_{0}^{T ρ} .

Then, thanks to Lemmas 1-2-3-4-5 we know that there exists B_{3} > 0 such that F (x, t) ≤ B_{3} for all (x, t) in their ranges. This completes the proof of the lemma. □

We may now give the proof of Proposition 3. By Lemmas 8-9-10, for all (x, t) ∈ ℝ_{+}^{n+1} we obtain

G_{2}(x, t) ≤ min \{\frac{B_{1}}{t^{1/(p−1)}} , \frac{max\{B_{2}, B_{3}\}}{1 + |x|^{4/(p−1)}}\} ≤ max\{B_{1}, B_{2}, B_{3}\} · min \{\frac{1}{t^{1/(p−1)}} , \frac{1}{1 + |x|^{4/(p−1)}} \} .

Using (32), the last inequality yields

G_{2}(x, t) ≤ \frac{2 · max\{B_{1}, B_{2}, B_{3}\}}{1 + |x|^{4/(p−1)} + t^{1/(p−1)}} .

Recalling the definition of G_{2}, this proves Proposition 3 with C_{2} = 2ω_{1} · max{B_{1}, B_{2}, B_{3}}.

8 A simple proof of Proposition 3 when p > \frac{n+4}{n−4}

In this section we assume that n ≥ 5, that p > \frac{n+4}{n−4}, and we give a proof of Proposition 3 which is simpler (and shorter) than the one in the previous section. This proof bases upon the idea that in a certain sense – for time independent right hand sides – the limiting operator for the majorising kernel b̃ for t → ∞ is just a multiple of the Green operator for the bi-Laplacian in ℝ^{n}.

We denote by U_{0} an entire smooth positive radial (and radially decreasing) solution of

Δ^{2}U_{0} = U_{0}^{p}  in ℝ^{n},  (50)

which have been constructed in [11, Theorem 1]. According to [11, Theorem 3], there exists C_{0} > 0 such that

\frac{C_{0}^{−1}}{1 + |x|^{4/(p−1)}} ≤ U_{0}(x) ≤ \frac{C_{0}}{1 + |x|^{4/(p−1)}}  for all x ∈ ℝ^{n} .  (51)

In particular, the convolution of the fundamental solution of Δ^{2} with U_{0}^{p} is well-defined (the integral exists). Therefore, we may rewrite (50) as an integral equation

U_{0}(x) = \frac{1}{2n(n − 2)(n − 4)e_{n}} ∫_{ℝ^{n}} |x − y|^{4−n}U_{0}^{p}(y) dy  (52)

where e_{n} = |B_{1}|, the measure of the unit ball. Let G_{2} = G_{2}(x, t) denote the function defined in Section

7. Then, by making use of the coordinate transformation σ := (\frac{|x−y|^{4}}{s} )^{1/3}, we may estimate

G_{2}(x, t) ≤ ∫_{0}^{∞} ∫_{ℝ^{n}} s^{−n/4} exp [−μ (\frac{|x − y|^{4}}{s} )^{1/3}] \frac{dy ds}{(1 + |y|^{4/(p−1)})^{p}}

≤ C_{0}^{p} ∫_{ℝ^{n}} ∫_{0}^{∞} s^{−n/4} exp [−μ (\frac{|x − y|^{4}}{s} )^{1/3}] U_{0}^{p}(y) ds dy
= 3C_{0}^{p} ∫_{0}^{∞} exp(−μσ)σ^{\frac{3}{4} n−4} dσ · ∫_{ℝ^{n}} |x − y|^{4−n}U_{0}^{p}(y) dy.

We denote

Ĉ = Ĉ(n, μ) = ∫_{0}^{∞} exp(−μσ)σ^{\frac{3}{4} n−4} dσ < ∞,

which is finite since n > 4. We now proceed by means of (52):

G_{2}(x, t) ≤ (6n(n − 2)(n − 4)e_{n}ĈC_{0}^{p}) \frac{1}{2n(n − 2)(n − 4)e_{n}} ∫_{ℝ^{n}} |x − y|^{4−n}U_{0}^{p}(y) dy
= (6n(n − 2)(n − 4)e_{n}ĈC_{0}^{p}) U_{0}(x)
≤ \frac{K_{0}}{1 + |x|^{4/(p−1)}}  for all (x, t) ∈ ℝ_{+}^{n+1}

with K_{0} := 6n(n − 2)(n − 4)e_{n}ĈC_{0}^{p+1}. The just proved inequality, combined with Lemma 8 and (32) concludes the proof of Proposition 3 when p > \frac{n+4}{n−4}. □

Remark 3. 1)When p < \frac{n+4}{n−4} the arguments of the present section do not apply since (50) admits no positive solutions, see [16, Theorem 1.4]. On the other hand, if p = \frac{n+4}{n−4} positive entire radial solutions U_{0} = U_{0}(x) of (50) do exist: however, they behave at infinity as (1 + |x|^{n−4})^{−1} (see [18]) so that they do not satisfy the decay estimate (51). For this reason, also in the critical Sobolev case p = \frac{n+4}{n−4}, one cannot argue as in this section.

2) Under the assumption p > (n + 4)/(n − 4), the above proof replaces Lemmas 9 and 10. The proof simplifies because the term (t − s)^{1/(p−1)} in the denominator of G_{2} is dropped. This does not seem to be possible when trying to cover the full range p > 1 + 4/n.

References

[1] G. Caristi, E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl. 279, 710-722 (2003)

[2] Shangbin Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., Theory Methods Appl. 43 (A), 293-323 (2001)

[3] Yu.V. Egorov, V.A. Galaktionov, V.A. Kondratiev, S.I. Pohožaev, On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range, C. R. Acad. Sci. Paris 335, 805-810 (2002)

[4] Yu.V. Egorov, V.A. Galaktionov, V.A. Kondratiev, S.I. Pohožaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Diff. Eq. 9, 1009-1038 (2004)

[5] M. Fila, M. Winkler, E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity, J. Dyn. Differ. Equations 17, 249-269 (2005)

[6] M. Fila, M. Winkler, E. Yanagida, Grow up rate of solutions for a supercritical semilinear diffusion equation, J. Differ. Equations 205, 365-389 (2004)

[7] H. Fujita, On the blowing up of solutions of the Cauchy problem u_{t} = Δu + u^{1+α}, J. Fac. Sci. Univ. Tokyo, Sect. 1 A Math. 16, 105-113 (1966)

[8] V.A. Galaktionov, P.J. Harwin, Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation, preprint

[9] V.A. Galaktionov, S.I. Pohožaev, Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators, Indiana Univ. Math. J. 51, 1321-1338 (2002)

[10] V.A. Galaktionov, J.L. Vázquez, A stability technique for evolution partial differential equations. A dynamical systems approach, Progress in Nonlinear Differential Equations and their Applications 56, Boston (MA) etc.: Birkhäuser (2004)

[11] F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, to appear in Math. Annalen

[12] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49, 503-505 (1973)

[13] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes Math. 840, Springer (1981)

[14] K. Kobayashi, T. Sirao, H. Tanaka, On the growing problem for semilinear heat equations, J. Math. Soc. Japan 29, 407-424 (1977)

[15] T.-Y. Lee, W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Am. Math. Soc. 333, 365-378 (1992)

[16] C.S. Lin, A classification of solutions of a conformally invariant fourth order equation in ℝ^{n}, Comment. Math. Helv. 73, 206-231 (1998)

[17] E. Mitidieri, S. Pohožaev, Apriori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234, 1-362 (2001)

[18] C.A. Swanson, The best Sobolev constant, Appl. Anal. 47, 227-239 (1992)

[19] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337, 549-590 (1993)

[20] Qi S. Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. Part. Diff. Eq. 22, 1529-1557 (1997)

[21] Qi S. Zhang, The critical exponent of a reaction diffusion equation on some Lie groups, Math. Z. 228, 51-72 (1998)