May Pascal Da Costa
email: pascal.da-costa@ecp.fr

Semi-Endogenous Growth and Pollution: No Double Dividend in the Long Term

Keywords: O41, Q20 Innovation, Pollution, Semi-Endogenous Growth, Double Dividend

Literature on endogenous growth shows that a polluting economy can grow sustainably and that a double-dividend or win-win effect comprising growth and the environment is possible. Even with a semiendogenous growth approach, which occur when the knowledge stock yield falls below the unit in the production of innovations, what happens to sustainability and the double dividend? This paper presents the first semi-endogenous growth model with pollution which answers this very question. We first illustrate that the dynamics of this economy can be sustainable even if its long-term growth rate is exogenous. To ensure the latter, a knowledge stock yield that is greater than a certain strictly positive threshold is required. We then demonstrate that the double dividend is impossible since the level of support for innovation no longer has a positive impact on the long-term growth rate. And the environmental policy always has a negative effect on growth.

Introduction

Significant progress made in understanding growth mechanisms has been possible thanks to endogenous growth models. It is worth citing the founding models by [START_REF] Romer | Endogenous Technical Change[END_REF], [START_REF] Aghion | A Model of Growth Through Creative Destruction[END_REF] and [START_REF] Grossman | Quality Leaders in the Theory of Growth[END_REF]. The Jones growth model (1995 a, b) marks a turning point. In Jones' criticism, endogenous growth is based on the strong assumption that the yield of the knowledge stock in the production of innovations is unitary (unit yield on the accumulable factor). An equivalent theory is maintained by [START_REF] Kortum | Research, Patenting and Technological Change[END_REF] and [START_REF] Segerstrom | Endogenous Growth Without Scale Effects[END_REF]. Long-term growth becomes exogenous once the yield is considered below the unit, with the labour force increase and this yield acting as the determinants. Intuitively, a yield below the unit indicates the level of difficulty in research.

To date no study has explicitly investigated the growth and environment double-dividend or win-win effect of a polluting economy faced with the semi-endogenous growth approach. This double dividend appears when the economic policy makes it possible to both increase the growth rate of production and improve the quality of the environment in the long term. It is worth citing the existing semi-endogenous growth models by [START_REF] Eriksson | Population Density, Pollution and Growth[END_REF] 1 , [START_REF] Groth | A New-Growth Perspective on Non-Renewable Resources[END_REF] 2 and [START_REF] Jones | The Costs of Economic Growth[END_REF]. By studying the impact of various types of pollution and damage generated by economic growth on the mortality rate, the latter shows that policy changes do not affect growth in the long term. Only the levels are impacted. The economic policy issues related to semi-endogenous growth are therefore important and are worth studying in-depth, all the more so since numerous questions still remain unanswered: is the long-term efficiency of policies designed to reduce polluting emissions called into question by semi-endogeneity? Does the difficulty in research hinder the dissemination of technology and therefore represent an obstacle to sustainable growth? What are the new conditions (if they do indeed exist) for sustainable growth in the strongest sense of the term, i.e. leading to reduced emissions compatible with sustained growth? Is the 'growth and environment' double-dividend which is so dear to endogenous growth economists (since it justifies pollution reduction efforts thanks to the future high growth rate) still possible? 3This paper develops and analyses the model of an economy experiencing both semi-endogenous growth and polluting emissions in an attempt to answer these questions. Our model is based on Stokey's polluting economy (1998) which is applied to endogenous growth by [START_REF] Aghion | Endogenous Growth Theory[END_REF] on the one hand, and on R&D with Segerstrom's semi-endogenous growth (1998) on the other hand. Within this scope, the environment is degraded during the production process and the total factor productivity (TFP) increases thanks to R&D. The following new principles are enabled by our model.

It will be shown that with semi-endogenous growth, the new conditions to a path of sustainable growth in the strongest sense mainly depend on the difficulty in research (yield value on the accumulable factor in the innovation production function). A suitable substitution between the natural capital (source of polluting emissions) and intangible capital (TFP) can lead the economy onto a growth path that is advantageous to the environmental quality (reduced pollution stock) when the research difficulties are not excessive. This appropriate substitution is made possible by the implementation of an incentive policy internalising the externalities: tax on polluting emissions (for negative externalities) and R&D subsidies or even taxation (for positive externalities). Like in other endogenous growth models with pollution (Aghion andHowitt 1998, Grimaud and[START_REF] Grimaud | The Growth-Environment Trade-off : Horizontal versus Vertical Innovations[END_REF], the optimal taxation is increasing in our model. It has a negative effect on the production growth rate, while increasing the intensity of R&D. This paper shows that the last point nonetheless does not validate the Porter hypothesis [START_REF] Porter | America's green strategy[END_REF][START_REF] Porter | Toward a new conception of the environment -competitiveness relationship[END_REF] according to which environmental policy may improve the environmental quality while aiding the growth rate: the intensity of R&D under semi-endogenous growth has no effect on the long-term growth rate of innovation, nor does it affect that of production. R&D is no longer an endogenous drive for the growth of production in the long term. R&D subsidies only have an impact on the production level. The impact of economic policy is therefore significantly modified with semi-endogenous growth: the 'growth and environment' double dividend proves to be impossible.

We have to recognise that the complementarity of economic policy instruments in the double dividend is postulated in a great many models applied to the environment. The conclusions of numerical simulations by [START_REF] Acemoglu | The Environment and Directed Technical Change[END_REF] can be cited to illustrate this statement. 4They make it possible to gauge the effects economic instruments when several endogenous R&D sectors co-exist within the scope of a climate model (with two sectors aiming to improve the productivity of non-renewable and renewable resources respectively). These authors show that the R&D support policy (second-best) alone, i.e. without the combined implementation of the carbon tax, does have a direct effect on the allocation of labour in favour of research sectors, but very little (or no) indirect effect on the polluting emissions. It should be pointed out that the polar case is also true. The combination of two tools appears to be the only efficient policy capable of producing a double-dividend effect in the long term according to [START_REF] Acemoglu | The Environment and Directed Technical Change[END_REF]. Though it is based on a single R&D sector, the theoretical approach applied in this article is sufficient to start a debate on such results. This article is structured in the following manner: firstly, it determines the equilibrium of the optimal economy and the conditions required for an optimal sustainable growth path to exist (section 2), before defining the equilibrium of a decentralised economy (section 3) and then defining the optimal policy which matches with the two previous equilibriums (section 4). The conclusion is given in section 5.

Social Optimum

This section will first lay down the model's assumptions.

Final Good

The production function of the final good is given by the following relationship [START_REF] Stokey | Are There Limits To Growth?[END_REF]Aghion and[START_REF] Aghion | Endogenous Growth Theory[END_REF]:

Y t = z t [A t (1 -n t ) L t ] 1-α K α t (1)
with α ∈ [0, 1], z γ t ∈ [0, 1] denoting the polluting intensity of the economy which is a controlled variable of the final goods sector, (1 -n t ) ∈ [0, 1] denoting the fraction of the labour input devoted to producing the final good, L t denoting the size of the labour force, A t denoting the average quality or TFP, and K t denoting the capital. When z t = 1, the polluting intensity is maximal and production of the final good reaches its potential level. On the contrary, however, when z t < 1, the level of production is reduced.

The variation in the capital stock is given by: Kt = Y t -L t c t , with c t representing the per capita consumption.

Households and Pollution

The pollution stock accumulation equation is written as follows: Ṡt = P t -ΘS t , with Θ ≥ 0 denoting the natural assimilation rate which is kept constant for the purpose of simplification, and P t represents the pollution flow.

The representative consumer's utility function is given by:

U (c t , S t ) = c 1-ε t 1-ε - S 1+̟ t 1+̟
. Its arguments are: per capita consumption c t , the pollution stock level S t , and parameters for which 1 ε is the intertemporal elasticity of substitution so that:

∂U (ct,St) ∂ct > 0, ∂U (ct,St) ∂St < 0, ∂ 2 U (ct,St) ∂ct∂St = 0.
The pollution flow is expressed as follows [START_REF] Stokey | Are There Limits To Growth?[END_REF]Aghion andHewitt 1998)

: P t = z γ t Y t , with γ > 0.

R&D

The R&D sector comprises a continuum of specialised laboratories. Each laboratory is looking to take into account the best available quality -i.e. the maximum productivity parameter or the frontier productivity -at date t, Āt = max j {A jt }, into its intermediate good j (j ∈ [0, 1] for the purpose of simplification). Under the semi-endogenous growth approach [START_REF] Segerstrom | Endogenous Growth Without Scale Effects[END_REF], the variation in the frontier quality Āt depends on both the labour flow allocated to the R&D sector and on the past frontier quality whose yield is below the unit:

Ȧt = κψ t ≡ δ (n t L t ) Āφ t (2)
with κ > 0 and δ ≡ κβ > 0 (β is relative to Poisson's process which we will discuss later on), ψ t denotes the instantaneous flow of improvement in the frontier quality, while n t L t denotes the R&D effort with n t ∈ [0, 1] denoting the fraction of work allocated to R&D (intensity of R&D) and φ < 1 the yield over the accumulable factor (unitary under endogenous growth hypothesis). This yield indicates the level of difficulty in R&D: the higher φ is, the easier research proves to be. Therefore, when 1 > φ > 0, the productivity of R&D increases with the innovations and an intertemporal spillover effect occurs which is called "standing on shoulders"; when φ = 0, then the stock of innovations is no longer a factor of R&D; when φ < 0, then any new innovation reduces the productivity of future research and the depletion of technology opportunities occurs which is known as a "fishing out" effect (Jones 1995 b). 5 The mean

productivity index A t = 1 0
A jt dj, is then defined. It should be pointed out that the frontier quality and the mean quality increase at the same rate on the balanced-growth path because by nature

A t = Āt 1+κ , i.e. Ȧ A t = Ȧ Ā t
. The appearance of innovations in the laboratories follows Poisson's stochastic approach but it can be considered as smoothed out for the R&D sector [START_REF] Aghion | A Model of Growth Through Creative Destruction[END_REF].

Lastly, in the model's steady state, Ȧ Ā t (Equation ( 2)) is constant if and only if:

Ȧ Ā = l 1 -φ (3)
with l > 0 denoting the constant growth rate of the labour force such that L t = L 0 e lt . Equation (3) shows that the long-term growth rate under the semi-endogenous growth approach is exogenous (and strictly positive). The population growth rate l and the yield of research φ are the determinants of this growth rate. Conversely, in an endogenous growth model with φ = 1, Equation (2) becomes Ȧ Ā = δnL in steady-state conditions with n and L remaining constant. This growth rate is directly proportional to the resources allocated to R&D. Therefore, it is possible to evidence the existence of two long-term effects: that of the economic policy through the intensity of R&D (n) and a scale effect with the size of the labour force (L). Yet since Jones' review, it is believed that this scale effect must be disregarded because it contradicts the facts (an economy with twice the population of another does not necessarily growth twice as fast).6

This paper now sets out to answer the following questions. How will the factors of production be distributed in an optimal version of this economy? What is the objective of production? Under what conditions can the optimal path to semi-endogenous growth become sustainable in the strong sense of the word?

Maximising Well-Being and the Optimal Path to Sustainable Growth

The social planner maximises the well-being of the representative agent with the aim of determining the optimal values of per capital consumption (c t ), R&D intensity (n t ) and the polluting intensity of the economy (z t ). The state variables are the physical capital stock (K t ), the qualiy (A t ) and the pollution stock (S t ). ρ denotes the time preference rate. The planner's programme is written as follows:

                   max {(c t ,n t ,z t )} +∞ t=0 ∞ 0 e -ρt U (c t , S t )dt s.t./ Kt = Y t -L t c t Ȧt = δn t L t A φ t Ṡt = P t -ΘS t K t ≥ 0,A t ≥ 0,S t ≥ 0... K 0 , A 0 , S 0 known (4)
Lemma 1 The long-term optimal growth rates for the model's variables (written g o ) after having resolved the planner's programme (4) are as follows:

                   g A = l 1-φ > 0 g o c = γ(1-α)g A -l γ(1-α)+ ε+̟ 1+̟ g o z = -1 γ ε+̟ 1+̟ g o c + l g o S = 1-ε 1+̟ g o c n o = 1 1+Z o with Z o = (ε-1)g o c +ρ+l g A (5)
Proof. See Appendix A for resolution of the social optimum.

Next, we recall the fact that sustainable growth in the strong sense is defined by a strictly positive production growth rate (g o c > 0) and a strictly negative pollution stock growth rate (g o S < 0), with respect to sign and limit constraints.

Proposition 2 It is necessary and sufficient to combine all three of the following conditions to achieve optimal growth in the long term that is sustainable in the strong sense:

i. The parameter φ must be greater than a given strictly positive threshold:

g o c > 0 0 if and only if φ>1 -γ (1 -α)>0 (6)
ii. The intertemporal elasticity of substitution must be less than the unit:

g o S < 0 if and only if ε > 1 (7)
iii. The natural assimilation rate must not be too low:

P S > 0 if and only if Θ > ε -1 ̟ + 1 γ (1 -α) g A -l γ (1 -α) + ε+̟ 1+̟ (8)
Proof. See Appendix B on the existence of the optimal path. The first Condition ( 6) is specific to sustainable semi-endogenous growth: the parameter φ must be strictly positive and not too low, failing which the economy's growth which reduces its polluting intensity can be negative in the long term (see impact of z on Y in (1)). It can also be seen that the exhaustion of technology opportunities (fishing-out effect: φ < 0) is incompatible with the criterion of strong sustainability. The second Condition ( 7) is, however, similar to that of sustainable endogenous growth (Aghion andHowitt 1998, Grimaud and[START_REF] Grimaud | The Growth-Environment Trade-off : Horizontal versus Vertical Innovations[END_REF]. It corresponds to a marginal utility of strongly decreasing consumption so the representative household can choose to cut back on its current consumption to reach sustainable growth in the strong sense. The third Condition (8) can be summarised as follows: a higher long-term production growth rate can be accompanied by a reduced pollution stock if and only if the natural regeneration rate increases. 7 Under these three conditions, the level variables show coherent values, particularly the level of production which remains strictly positive in the long term despite a polluting intensity that tends asymptotically towards zero: its reduction is therefore offset by the continuous increase -at an acceptable rate -in the quality of intermediate goods.

Decentralised Economy

In a second-best economy such as the one we are investigating here, three instruments are needed to get as close as possible to the optimal target (like [START_REF] Grimaud | The Growth-Environment Trade-off : Horizontal versus Vertical Innovations[END_REF]): subsidising the purchase of intermediate goods (written τ ), taxing polluting emissions (h), and subsidising or taxing the cost of R&D (σ). The economy experiences market distortion and two external effects: the potential monopoly of producers over intermediate goods, polluting emissions from the final goods sector, and R&D activities. It should be pointed out that a fourth instrument is required to balance the Government's budget: a non-distortive lump sum (T ) for households to avoid resorting to debt.

R&D Sector

To integrate the best available quality Āt into the intermediate good j, each laboratory uses the quantity of labour consumed within the labour input to directly determine its probability for innovation ψ j . If successful, the R&D project is rewarded by a local monopoly rent resulting from its protection by a patent. The programme for maximising a laboratory j is written:

       max (n jt ) V jt ψ jt -(1 -σ t )
o S = 1-ε 1+̟ g o c .
data for the innovator. This parameter takes into account all the knowledge spillover in R&D. From an atomistic viewpoint, the R&D yields are constant because the laboratories see their marginal product as being equal to the mean product [START_REF] Segerstrom | Endogenous Growth Without Scale Effects[END_REF]. The programme leads to the following free-entry

condition: V jt = 1-σt ∼ β w t ∼
β s replaced to obtain the following:

V jt = 1 -σ t β (n t L t ) Ā1-φ t w t = V t (9)
Equation ( 9) shows that the value of innovation is the same for all innovators since they achieve an identical frontier quality in the case their project is successful.

Final Goods Sector

The producer's final goods programme involves maximising his instant profit 8 by determining the polluting intensity of his production, his labour demand and the desired quantity of intermediate goods, i.e. the following first-order conditions:

z t = [(γ + 1) h t ] -1 γ if h t > 1 γ+1 1 if h t ≤ 1 γ+1 w t = (1 -α) γ γ + 1 Y t (1 -n t ) L t p jt = α 1 -τ t γ γ + 1 z t [(1 -n t ) L t ] 1-α A jt x α-1 jt (10)
with x jt denoting the volume of the intermediate good j.

Intermediate Goods Sector

Resolving the programme of a monopolistic company j in the intermediate goods sector 9 leads to   

p jt = rtAjt α x jt = (1 -n t ) L t α 2 γ γ+1 zt (1-τt)rt 1 1-α = x t ,∀j
. It can be seen that x jt = x t , ∀j: this result makes sector aggregation possible. The following interest rate is obtained:

r t = α 2 1-τ γ γ+1
Yt Kt . The company's profit is therefore written:

Π jt = (1 -α) p jt x t (11)
With Equation ( 10), we can rewrite Equation ( 11) as a function of the frontier quality:

Π t A t = (1 + κ) (1 -α) α 1 -τ t γ γ + 1 Y t A t (12)

Households and the Governement

The representative consumer maximises his intertemporal utility function under a dynamic budget constraint without being able to influence the amount of emissions to which he is subjected. The consumer's programme is given by the following dynamic problem:

             max(c t ) +∞ t=0 ∞ 0 e -ρt c 1-ε t 1-ε - S 1+̟ t 1+̟ dt s.t./ Ẇt = w t L t + r t W t -L t c t + T t and lim t→+∞ e - ∞ 0 rsds W t ≥ 0 (13)
with Ẇt denoting the dynamic budget constraint, where W t represents the household wealth and T t , the tax or subsidy making it possible to balance the government budget which is written:

T t + σw t n t L t + τ r t K t = h t P t .
Resolving Problem (13) implies using the Hamiltonian:

H = e -ρt U (c, S) -χ Ẇ = e -ρt c 1-ε 1-ε -S 1+̟ 1+̟ -χ (wL + rW -C + T ).
The first-order conditions are:

∂H ∂c = 0 χ = -∂H ∂W ⇔ e -ρt c -ε = χ χ χ = -r
. The previous system leads to the Keynes-Ramsey rule: g c,t = rt-ρ-l ε . Calculation of the single interest rate for the market (r t ) follows on from this rule. 10

Determining the R&D Intensity and Growth Rates of the main Aggregates in a Steady State

The free-entry condition for invention based on which we will determine the R&D intensity and the interest rate in a steady state can be written:

V Āt = Π Āt r t + p Āt -V V t (14)
with Π Āt /V t = δ 12) and ( 9)). Equation ( 14) is common to all growth models with vertical differentiation [START_REF] Aghion | A Model of Growth Through Creative Destruction[END_REF].

α 1+κ κ (1-σ)(1-τ ) 1-nt nt g At and V t = (1 -α) γ γ+1 1-σt β nt 1-nt Yt g A t (Equations (
Lemma 3 The R&D intensity and the interest rate of a steady growth path in a decentralised economy are equivalent to (15) and ( 16) respectively:

n d ss = 1 Z d ss + 1 (15) with Z d ss = (1-σ)(1-τ ) αδ 1+κ κ (ε -1) + δ κ - ε-1 γ(1-α) g h -ρ l 1-φ
, and

r d ss = ε l 1 -φ - ε γ (1 -α) g h + ρ + l (16)
Equation ( 16) shows that the higher growth rate of the environmental tax, the lower the interest rate. Conversely, the higher the exogenous growth rate of innovation, the higher the interest rate. The cost of capital r d ss increases when the TFP rises at a faster rate but it drops when the pollution generated by production and thus by capital is taxed even more.

Equation ( 15) shows that the R&D intensity increases with the three economic instruments. Internalising externalities of an environmental and technological nature while resolving distortion therefore boosts R&D investments, n d ss .

10 The optimal programme for the representative consumer also leads to the following condition:lim t→+∞ e

- ∞ 0 rsds Wt = 0.
Lemma 4 The long-term growth rates in a decentralised economy (written g d ) are as follows:

             g A = l 1-φ g d c = g A -1 γ(1-α) g h g d S = g A -γ(1-α)+1 γ(1-α) g h + l g d z = -g h γ if g h > 0 (17)
The second and fourth lines of System (17) show that the pollution emitted by the final goods producer is taxed such that it is in his every interest to reduce the polluting intensity of his production in a continuous manner. Reducing z mechanically reduces the producer's level of actual output. 17)).

The 'growth and environment' double-dividend or win-win effect can therefore be defined as follows: the double dividend exists in a steady second-best economy when there is both an improvement in the environmental quality (i.e. g d S < 0) and a production growth rate great than that of the laissez-faire economy (no instruments).

Proposition 5 In the steady state of a semi-endogenous growth model with pollution, the 'growth and environment' double-dividend no longer operates since the growth rate of a laissez-faire economy is always greater than a second-best economy:

g d c (g h =0) > g d c (g h >0) .
Proof. In our model, g d c (g h ), g d c ′ < 0: see second line of System (17). Contrary to endogenous growth, the fact that an environmental tax increases the intensity of R&D (Equation (15): Lemma (3)) does not validate the Porter hypothesis in our model. The R&D intensity has no impact on the growth rate of innovations in the long term or on that of production. The long-term economic growth rate decreases with the increasing taxing of polluting emissions (second line of System (17)), regardless of the level of support for R&D (constant g A , first line of System ( 17)). In the long term, R&D subsidies thus only have an impact on the economy's level of production but not on its growth rate.

Optimal Economic Policy

This section determines the levels of three instruments which can be used to match the long-term growth rates of optimal economies and decentralised economies. Whenever possible, we will only refer to the impact of the parameter characteristic of semi-endogenous growth, φ, on the instruments; other parameters of the model and their impact have already been discussed in the literature.

Lemma 6 The optimal subsidy for purchasing intermediate goods, the subsidy (or tax) enabling R&D to use the optimal number of researchers (n o = n d ), and the growth of tax on emissions equating the pollution stock growth rates of optimal and decentralised economies (g o S = g d S ) can be written respectively:

                       τ o = 1 -α 1 -σ o = δ α 1-τ o 1+κ κ (ε-1)g A γ(1-α)-l g A γ(1-α)+ ε+̟ 1+̟ +ρ+l g A (ε-1)+ δ κ - ε-1 γ(1-α) g o h -ρ g A g o h = γ(1-α) ε+̟ 1+̟ g A +γ(1-α)l γ(1-α)+ ε+̟ 1+̟ g A = l 1-φ (18)
As expected, the optimal subsidy for purchasing intermediate goods (first line of System ( 18)) is equal to 1 -α [START_REF] Barro | Economic Growth[END_REF]. With this subsidy, the final goods producer pays p j α for the intermediate good j which is equal to the marginal cost. α denotes the mark-up rate applied by the intermediate goods company which is equal to the elasticity of substitution between intermediate goods. 11The second line of System (18) shows that the optimal instrument for R&D has an undefined sign: it is possible that the instrument be a tax (negative σ o ). This case arises when decentralised research efforts are excessive with respect to what is required by the Pareto optimal. Intuitively, an economy can over-invest in R&D due to the creative destruction inherent to the vertical differentiation model [START_REF] Grimaud | Pollution Permits and Sustainable Growth in a Schumpeterian Model[END_REF]. It enables the innovator to capture the entire rent flow. Yet this rent can be over-estimated. In this case, the creative destruction increases the private rate of return of R&D and positions it above the social rate of return. Proposition 7 In the long term, the optimal environmental policy is stricter in the sense that the tax on polluting emissions increases when R&D is easier. 18). The easier research (φ → 1) proves to be, the more innovations will be produced, which increases the growth rate and the pollution (System ( 17)). The growth rate of the incentive tax must increase to limit the latter.

Proof. g o h (φ), g o h ′ > 0: see third line of System (
Furthermore, we obtained the following environmental policy rule (like Grimaud and Ricci 2004):

h o t P o t /Y o t = 1/(1 + γ)
, which can be re-written as g o ht = g o Yt -g o Pt . This rule is in line with Proposal (7).

Conclusion

In a growth model with pollution and a function of the production of innovations with a non-unitary yield on the accumulable factor, we have demonstrated that the growth path can become sustainable in the strong sense of the word under certain conditions. To be positive, the long-term exogenous growth rate therefore requires a knowledge stock yield above a given, strictly positive threshold (our first proposal: Proposal (2)). This threshold corresponds to a low level of difficulty in research. We have also demonstrated that a double dividend is impossible since the level of support for innovation no longer has a positive impact on the long-term growth rate (our second proposal: Proposal (5)). The environmental policy has a negative impact on the long-term growth and becomes more severe when R&D is easier (our third and last proposal: Proposal ( 7)). The long-term efficiency of policies designed to reduce polluting emissions is therefore seriously called into question by the semi-endogeneity of growth. The 'growth and environment' double-dividend for endogenous growth configurations is no longer possible on a steady semi-endogenous growth path, but must we nevertheless consider that economic policy is futile if it does not increase growth in the long term? Most certainly not since the environmental policy will have achieved its objective, i.e. increasing social well-being. Sustainable semi-endogenous growth therefore provides us with the perfect opportunity to stress the importance of well-being as an ultimate economic objective.

Finally, we would like to mention a few avenues of research which are worth exploring in the future. Firstly, it could be worth presenting the transition simulations calculated by our model, particularly when studying the transitory effect of the double dividend. Even if there is no double dividend in the long term, nothing proves that it will not be operational beyond ten or so years, for example, during the transition to a steady state, in which case the implications of economic policy will be different. Furthermore, empirical studies should be used this time to check that the semi-endogenous approach to growth is more plausible than the fully-endogenous approach when describing polluting economies.

8 0 p 0 P 0 A

 000 i.e. Ytwt (1 -nt) Lt -1 jt (1 -τt) x jt djht 1 jt dj, with p jt denoting the price of the intermediate good j, x jt denoting the volume of the intermediate good j such that x jt = K jt A jt and the aggregate capital: Kt = 1 jt x jt dj. 9 ı.e. max Πt = p jt x jt -rtA jt x jt under (10) with rt denoting the cost of capital.

  w t n jt L t

s.t./ ψ jt = ∼ β (n jt L t ) , with V jt denoting the value of innovation, w t denotes the nominal wage, and ∼ β= β Ā1-φ t denotes exogenous 7 Condition (8) can be written Θ > -g o S and knowing that g

  The final goods sector therefore requires less labour and thus fewer intermediate goods. Concerning R&D, neither the reduced wage level (due to the reduced labour demand from the final goods sector), nor the reduced profit of intermediate goods producers (related to the reduced demand for intermediate goods by the final goods sector) can affect the growth rate of innovations g A , which is independent of all endogenous parameters (first line of System (

[START_REF] Eriksson | Population Density, Pollution and Growth[END_REF] take the specific stance that the damage resulting from pollution depends on the population density and the type of pollution.

[START_REF] Groth | A New-Growth Perspective on Non-Renewable Resources[END_REF] studies the dynamics of economies exploiting non-renewable resources by disregarding the issues of pollution.

[START_REF] Ricci | Channels of transmission of environmental policy to economic growth: A survey of the theory[END_REF] provides a literature review of the transmission channels of environmental policy with respect to economic growth.

We could also cite the numerical simulations byGrimaud, Lafforgue and Magne (2011) whose conclusions are similar enough, or even those of[START_REF] Henriet | A stylized applied energy-economy model for France[END_REF].

It should be pointed out that φ > 1 is disregarded otherwise growth becomes explosive.

Some papers endorsing endogenous growth without scale effects nonetheless recommend models that combine horizontal and vertical differentiations so economic policy has an impact on the long-term growth without this producing, however, any scale effect. It is worth referring to the canonical models by[START_REF] Aghion | Endogenous Growth Theory[END_REF],[START_REF] Peretto | Technological Change and Population Growth[END_REF] and the review by[START_REF] Jones | Growth: With or Without Scale Effects?[END_REF]. Lastly, it is worth citing the econometric estimations by[START_REF] Sedgley | Reinterpreting the Jones critique: A time series approach to testing and understanding idea driven growth models with transitional dynamics[END_REF] who set out to measure the parameter φ.

In this case, there is perfect substitutability between intermediate goods (Cobb-Douglas production function: Equation (1)), which reduces the mark-up rate and the market power of intermediate goods companies.

Appendices Appendix A: Optimal Equilibrium

The common Hamiltonian resulting from programme (4) is equivalent to: H = U (c, S)

. ϑt represents the implicit cost of capital (or shadow price), while µt denotes the implicit price of innovations, and ζt the implicit price of the pollution stock.

The first-order conditions are as follows:

Resolving this equation makes it possible to determine the optimal growth rates (5), knowing that in the long term gy = gc = g k (long-term condition associated with the capital variation equation) and gP = gS (long-term condition associated with the pollution stock variation equation).

Appendix B: The Existence of the Optimal Growth Path

The conditions -three in total -are as follows:

• Growth rates and all sign or limit constraints on variables must be met.

Proof. Conditions ( 6), ( 7) and ( 8) concern the validity of these constraints. With the first two Conditions ( 6) and ( 7), the following is obtained:

≤ gA; this inequality is always verified if gA > 0 (Equation ( 3)).

The remaining constraint P S > 0 results in Condition (8). Under conditions ( 6) and ( 7): Condition ( 8) -corresponding to the minimum natural regeneration rate -is positive.

• The intertemporal utility of consumers must be limited: limt→+∞ e -ρt U (ct, St) = 0. 6) and ( 7).

• The conditions of transversality must be verified: limt→+∞ e -ρt ϑtKt = 0 (i) limt→+∞ e -ρt µtAt = 0 (ii) limt→+∞ e -ρt ζtSt = 0 (iii)

Proof. For (i) : ϑtKt increases at the long-term rate (1 -ε) g o c -l < 0 therefore Condition (i) is verified. For (ii) : µtAt increases at the long-term rate (1 -ε) g o c < 0 therefore Condition (ii) is verified. For (iii) : -ρ + g ζ + gS < 0 is necessary, which is equivalent to ρ > 1-ε 1+̟ gc : (iii) is verified for ε > 1 (Condition ( 7)).


Amal Attouchi
email: attouchi@math.univ-paris13.fr

Gradient estimate and a Liouville theorem for a p-Laplacian evolution equation with a gradient nonlinearity

Keywords: 2010 Mathematics Subject Classification. 35K65, 35K92, 35B45, 35B53 35K55 . p-Laplacian, local gradient estimate, parabolic Liouville-type Theorem, gradient non-linearity

In this paper, we establish a local gradient estimate for a p-Lpalacian equation with a fast growing gradient nonlinearity. With this estimate, we can prove a parabolic Liouville theorem for ancient solutions satisfying some growth restriction near infinity.

Introduction and main result

In this paper, we are interested in qualitative properties of solutions of the non-linear degenerate parabolic equation

u t -∆ p u = |∇u| q , ( 1.1)
where ∆ p u = div(|∇u| p-2 ∇u), q > p -1 > 1.

The kind of result we are going to prove are gradient estimates for local solutions in time-space, and a Liouville type theorem for ancient solutions. In the last years, gradient estimates have played a key role in geometry and PDE since at least the early work of Bernstein. Gradient a priori estimates are fundamental for elliptic and parabolic equations, leading to Harnack inequalities, Liouville theorems, and compactness theorems for both linear and nonlinear PDE. For the corresponding elliptic equation of (1.1), gradient estimates were first considered by Lions [START_REF] Lions | Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre[END_REF] for the linear diffusion case p = 2. These estimates were based upon the Bernstein technique. Recently for the possibly degenerate elliptic equation with q > p -1 > 0, Bidaut-Véron, Huidobro, Véron [START_REF] Bidaut-Veron | Local and global properties of solutions of quasilinear Hamilton-Jacobi equations[END_REF] obtained a priori universal gradient estimate for equations on a domain Ω of R N and they extended their estimates to equations on complete non compact manifolds satisfying a lower bound estimate on the Ricci curvature. These estimates allowed them to derive some Liouville type theorems.

It is natural to look also for parabolic Liouville-type-theorems. In the linear diffusion case p = 2 and for q > 1, Souplet and Zhang [START_REF] Ph | Global solutions of inhomogeneous Hamilton-Jacobi equations[END_REF] obtained local gradient estimate for locally upper bounded solution of (1.1) (u ≤ M) of the form

|∇u| ≤ C(p, N, q) t -1 q + R -1 + R -1 q-1 (M + 1 -u) in B(x 0 , R) × (0, T ).
Relying on this estimate they proved that, under some growth condition at infinity, ancient solutions in the whole of R N are constant. Motivated by their result, we generalize the gradient estimate and Liouville theorem to the case 1 < p -1 < q. We also require that the solution is locally lower bounded. Using a Bernstein method, we have the following gradient estimate.

Theorem 1.1 Let q > p-1 > 1, x 0 ∈ R N and R, T > 0. We set Q T,R = B(x 0 , R)× (0, T ). Let u be a solution in L ∞ ((0, T ); W 1,∞ (B(x 0 , R)) of ∂ t u -∆ p u = |∇u| q in Q T,R . Suppose that |u| ≤ M for some constant M ≥ 1. Then, |∇u| ≤ C(p, N, q) t -1 q + R -1 + R -1 q-p+1 M in Q T, R 2 .
(1.2)

For the Cauchy-Dirichlet problem associated to (1.1), a gradient estimate involving the W 1,∞ norm of the initial data has been obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Global solutions of inhomogeneous Hamilton-Jacobi equations[END_REF]. In Theorem 1.1 we only use the local L ∞ norm of the solution but we get a weaker estimate regarding the exponent on the distance to the boundary R.

Recently, for the singular diffusion case 1 < p < 2 and for q = p, F. Wang [START_REF] Wang | Gradient estimates for the p-Laplace heat equation under the Ricci flow[END_REF] established gradient estimates similar to (1.2) for smooth, upper bounded, local solutions to (1.1) on a closed manifolds or on complete noncompact Riemannian manifolds evolving under a Ricci flow. These estimates are of the form:

|∇u| 1 -u (x, t) ≤ C(N, p) R -1 + t -1 p + K 2 p + K in Q T, R 2 . (1.3)
where K > 0 is a constant related to the Ricci flow and the sectional curvature of the manifold. These estimates allowed to the author to provide some Harnack inequalities for positive solutions of the following p-Laplace heat equation

|z| p-2 z t = ∆ p z. (1.4)
The estimates (1.3) have been obtained by deriving an equation for w = |∇v| p , v = f -1 (-u) and f (s) = e s/(p-1) -1. For q > p > 2, we take a different auxiliary function f , adapted to the degenerate diffusion case and to the fast growing gradient non-linearity.

As an application of the gradient estimate (1.2), we can state the following Liouville theorem for (1.1).

Theorem 1.2 Assume that q > p -1 > 1 and let σ = min 1, 1 q-p+1 . Assume that u ∈ L ∞ loc ((-∞, 0); W 1,∞ loc (R N )) is a weak solution of u t -∆ p u = |∇u| q , x ∈ R N , -∞ < t < 0, satisfying |u(x, t)| = o(|x| σ + |t| 1 q ), as |x| σ + |t| 1 q → ∞. (1.5)
Then u is constant.

Remark 1.1 The growth hypothesis (1.5) is important (see the example of the function u(x, t) = x 1 + t). However, we do not know if the exponents are sharp.

Besides the works mentioned above, there are few other studies on gradient estimates and nonlinear Liouville theorems for a parabolic type equation on noncompact Riemannian manifolds. In this case the proof mostly relies on two types of gradient estimates or a combination of them. These estimates are known as Hamilton gradient estimate (the estimate only involves ∇u and u) [START_REF] Hamilton | A matrix Harnack estimate for the heat equation[END_REF] and Li-Yau's gradient estimate (the estimate involves ∇u, u and u t ) [START_REF] Li | On the parabolic kernel of the Schr 'odinger operator[END_REF]. Let us also mention that the linear heat equation on noncompact manifolds was studied by Souplet and Zhang in [START_REF] Ph | Sharp gradient estimate and Yau's Liou ville theorem for the heat equation on noncompact manifolds[END_REF] where they obtained a local gradient estimate related to the elliptic Cheng-Yau estimate and Hamilton's estimate for the heat equation on compact manifolds. A Liouville theorem was also proved in [START_REF] Ph | Sharp gradient estimate and Yau's Liou ville theorem for the heat equation on noncompact manifolds[END_REF]. Hamilton-type gradient estimates were also used in [START_REF] Wang | Liouville theorems for the ancient solution of heat flows[END_REF][START_REF] Ma | Gradient estimate for the degenerate parabolic equation u t = ∆(F (u)) + H(u) on manifolds[END_REF][START_REF] Zhu | Hmilton's gradient estimate and Liouville theorems for porous medium equations on noncompact Riemannian manifolds[END_REF]. For q = p > 1, a nonlinear analogue of Li-Yau's estimate has been established in [START_REF] Kotschwar | Local gradient estimates of p-harmonic functions, 1/Hflow and an entropy formula[END_REF] for positive solutions of (1.1) on compact manifolds with nonnegative Ricci curvature. In [START_REF] Kotschwar | Local gradient estimates of p-harmonic functions, 1/Hflow and an entropy formula[END_REF], the gradient estimate was not used to get Liouville theorems but to obtain an entropy formula. Nevertheless, Liouville theorems should be obtained as a consequence of the obtained gradient estimate.

This paper is organized as follows: In Section 2, we provide the proof of the gradient estimate (1.2) and we prove Theorem 1.2. In Sections 3 we give the proof of a technical auxiliary lemma that appears in the proof of the gradient estimate.

Bernstein-type gradient estimate

The proof of Theorem 1.1 is based on the following technical lemma which is based on a Bernstein method. The most significant difficulty being the choice of the auxiliary function f and the estimates coming from the cut-off argument. Let us mention that for different suitable choice of f , gradient bounds global in space for the Cauchy problem associated to (1.1) have been obtained in [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]. First let us make precise that by local weak solution of (1.1) we mean a function

u ∈ C loc (Ω × (0, T )) ∩ L ∞ loc (0, T ; W 1,∞ loc (Ω))
where Ω is a smooth domain and such that the integral equality

Ω (u(x, t)ψ(x, t) -u(x, s)ψ(x, s)) dx + t s Ω -uψ t + |∇u| p-2 ∇u • ∇ψ dx dτ = t s Ω
|∇u| q ψ dx dτ holds for all 0 < s < t < T and for all for all testing function ψ ∈ C 1 (Ω × [0, T ]) such that ψ = 0 near ∂Ω × (0, T ). Now let α ∈ (0, 1) to be chosen later on. Set R ′ = 3R 4 . We select a cut-off function

η ∈ C 2 ( B(x o , R ′ )), 0 ≤ η ≤ 1, satisfying η = 0 for |x -x 0 | = R ′ and such that |∇η| ≤ CR -1 η α |D 2 η| + η -1 |∇η| 2 ≤ CR -2 η α for |x -x 0 | < R ′ , (2.1)
for some C = C(α) > 0 (see [START_REF] Ph | Sharp gradient estimate and Yau's Liou ville theorem for the heat equation on noncompact manifolds[END_REF] for the existence of such function).

Lemma 2.1 Assume that u is a local weak solution of (1.1) and that |u| ≤ M in Q T,R for some M > 1. We consider a C 3 smooth increasing function f satisfying f ′′ > 0, the following differential equation

f ′′ f ′ ′ + (p -1)(1 + N) f ′′ f ′ 2 = 0 (2.2)
and mapping

[0, 3] onto [-M, M]. Defining v = f -1 (-u)
, we set w = |∇v| 2 and z = ηw. Then at any point where |∇u| > 0, z satisfies the following differential inequality

L(z) ≤ -2(q -1)(f ′ ) q-2 f ′′ w q+2 2 η + C(p, N)(f ′ ) p-2 R -2 η α w p 2 + C(p, q)R -1 η α w p+1 2 (f ′ ) p-3 f ′′ + w q+1 2 (f ′ ) q-1 (2.3)
where

L(z) := ∂ t z -Az + H • ∇z (2.4)
with A is given by (3.4) H is given by (3.5).

The proof of lemma 2.1 is postponed to the the next section.

Proof of Theorem 1.1

Let u ∈ L ∞ loc ((0, ∞); W 1,∞ loc (Ω)
) be a local weak solution of (1.1). Since u and ∇u are locally bounded, using the result of Di Benedetto and Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic systems[END_REF][START_REF] Dibenedetto | Degenerate Parabolic Equations[END_REF], we get that ∇u is a locally Hölder continuous function. Thus z is a continuous function on

B(x 0 , R ′ ) × [0, T ] = Q, for any 0 < T . Therefore, unless z ≡ 0 in Q, z must reach a positive maximum at some point (x, t) ∈ B(x 0 , R ′ ) × [t 0 , T ]. Since z = 0 on ∂B R ′ × [0, T ], we deduce that x ∈ B R ′ . Since z(x, t) > 0, we have that |∇u| = f ′ (v)
γ = γ(p, N) = (p -1)(N + 1) + 1 (p -1)(N + 1) (2.5)
It is easy to see that f satisfies the differential equation (2.2) and f ′ , f ′′ > 0 and f maps 0, 3

1 γ -1 onto [-M, M]. Let us also note that γ ≥ 1 and γ -1 ≤ 1 p -1 ≤ 1.
By Lemma 2.1 we get that, in a small neighbourhood Q of (x, t), z satisfies

Lz ≤ -2(q -1)(f ′ ) q-2 f ′′ w q+2 2 η + C(p, N, α)(f ′ ) p-2 R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 (f ′ ) p-3 f ′′ + w q+1 2 (f ′ ) q-1 .
Hence

(f ′ ) 1-q Lz ≤ -2(q -1) f ′′ f ′ w q+2 2 η + C(p, N, α)(f ′ ) p-1-q R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 (f ′ ) p-q-1 f ′′ f ′ + w q+1 2
.

Since v ∈ 0, (3)

1 γ -1 , γ, M ≥ 1, we have 1 ≤ v + 1 ≤ (3) 1 γ ≤ 3 and hence 1 3(p -1)(N + 1) ≤ f ′′ f ′ ≤ 1 (p -1)(N + 1) ≤ 1 (2.6)
Using (2.6) together with the fact that 1 ≤ M ≤ f ′ and pq -1 < 0, we get that

(f ′ ) 1-q Lz ≤ - 2(q -1) 3(p -1)(N + 1) w q+2 2 η + C(N, p, α)R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 + w q+1 2
.

We take α = max q+1 q+2 , p+1 q+2 . Using the Young's inequality and recalling that η ≤ 1, then

• for the conjugate exponents r 1 = q+2 p , s 1 = q+2 q-p+2 we have that

C(N, p, α)R -2 η α w p 2 = η p q+2 w p 2 C(N, p, q, α)η α-p/(q+2) R -2 ≤ ε 1 (N, p, q)ηw q+2 2 + C(N, p, q, α)R -2(q+2) q-p+2 ,
• for the conjugate exponents r 2 = q+2 p+1 , s 2 = q+2 q-p+1 we have that

C(N, p, q, α)R -1 η α w p+1 2 = η p+1 q+2 w p+1 2 C(N, p, q, α)R -1 η α-p+1 q+2 ≤ ε 2 ηw q+2 2 + C(N, p, q, α)R -(q+2) q-p+1
• and finally for the conjugate exponent r 3 = q+2 q+1 , s 3 = (q + 2) we have that

C(N, p, q, α)R -1 η α w q+1 2 = η q+1 q+2 w q+1 2 C(N, p, q, α)R -1 η α-q+1 q+2 ≤ ε 3 ηw q+2 2 + C(N, p, q, α)R -(q+2) .
Choosing ε i in such way that

ε 1 + ε 2 + ε 3 = 1 4 2(q -1) 3(p -1)(N + 1)
, we get that

(f ′ ) 1-q Lz ≤ - (q -1) 2(p -1)(N + 1) w q+2 2 η + C(N, p, q, α)R -2(q+2) q-p+2 +C(N, p, q, α)R - (q+2)
q-p+1 + C(N, p, q, α)R -(q+2) .

(2.7)

Using the fact that 1 q-p+1 ≤ 2 q-p+2 ≤ 1 for q ≥ p, 1 ≤ 2 q-p+2 ≤ 1 q-p+1 for q ≤ p, we have that

(f ′ ) 1-q Lz ≤ - (q -1) 2(p -1)(N + 1) w q+2 2 η + C(N, p, q, α) R -q+2 q-p+1 + R -(q+2) .
Setting

A = A(R, p, q, N) := C(N, p, q) R -1 q-p+1 + R -1 2
and using that

(f ′ ) q-1 ≥ M q-1 ≥ 1, it follows that Lz ≤ - (q -1) -1)(N + 1) z q+2 2 in {(x, t) ∈ Q T,R ′ ; z(x, t) ≥ A} . (2.8)
Next for λ = λ(q, N, p) > 0 suitably chosen, the function

ψ(t) = λt -2 q satisfies ψ ′ (t) ≥ - (q -1) 4(p -1)(N + 1) ψ q+2 2 .
Now for t 0 ∈ (0, T ) fixed, we define z(t) := z(t + t 0 , x)ψ(t). It is easy to see that

Lz ≤ 0 in {(x, t) ∈ Q T -t 0 ,R ′ ; z(x, t) ≥ A} .
Since z(t) ≤ 0 for t > 0 sufficiently small, we deduce from the maximum principle that z(t) ≤ A, i.e. z(x, t

+ t 0 ) ≤ A + ψ(t) in Q T -t 0 ,R ′ .
Finally using that z = η|∇v| 2 , letting t 0 to 0, we get that

|∇v| ≤ C(N, p, q)(A + t -2 q ) 1/2 .
Using that

v + 1 = 2 - u M 1 γ with u M ≤ 1, we get ∇v = -1 γM 2 - u M 1-γ γ ∇u. It follows that |∇u| ≤ Mγ|∇v| ≤ C(N, p, q)(A + t -2 q ) 1/2 M in Q T, R 2
.

(2.9)

Here we used the fact that 2

-u M γ-1 γ ≤ 1. Hence we have |∇u| ≤ C(N, p, q) R -1 + R -1 q-p+1 + t -1 q M in Q T, R 2 .
and the proof of Theorem 1.1 is complete.

Proof of Theorem 1.2

Fix x 0 ∈ R N and t 0 ∈ (-∞, 0). Take R ≥ 1, T = R σq and set Q = B(0, R) × (0, T ). Now we consider the function U := u(x + x 0 , t + t 0 -T ). Using (1.5), we have that

|U| ≤ M R in Q, where M R := sup B(x 0 ,R)×(t 0 -T,t 0 ) |u| = o(T 1 q + R σ ) = o(R σ ), as R → ∞.
Applying Theorem 1.1 to U in Q, we get that

|∇u(x 0 , t 0 )| = |∇U(0, T )| ≤ C(N, p, q)R -σ M R
and the conclusion follows by sending R to +∞.

3 Proof of Lemma 2.1

Our proof consists of three steps.

Step 1: computations

Let f be a C 3 -function to be determined. We assume that f ′ , f ′′ > 0. We put v = f -1 (-u) and w = |∇v| 2 . By a straightforward computation, we have that v satisfies the following equation

∂ t v = (f ′ ) p-2 w p-2 2 ∆v + (p -2) D 2 v, ∇v, ∇v w + (p -1)(f ′ ) p-3 f ′′ w p 2 -(f ′ ) q-1 w q 2 =(f ′ ) p-2 w p-2 2 ∆v + (p -2) ∇w • ∇v 2w + (p -1)(f ′ ) p-3 f ′′ w p 2 -(f ′ ) q-1 w q 2 . (3.1)
For i = 1, ..., N, we set v i = ∂v ∂x i . In a neighbourhood Q := ω × (τ 1 , τ 2 ) of any point (x, t) ∈ Q T,R for which |∇u| = f ′ (v)|∇v| > 0, the equation is uniformly parabolic and hence differentiating (3.1) with respect to x i , we have

∂ t v i =(f ′ ) p-2 w p-2 2 ∆v i + p -2 2 ∇w i • ∇v + ∇w • ∇v i w - w i ∇w • ∇v w 2 + (p -2)(f ′ ) p-3 f ′′ v i w p-2 2 ∆v + (p -2) ∇w • ∇v 2w + p -2 2 (f ′ ) p-2 w i w p-4 2 ∆v + (p -2) ∇w • ∇v 2w (3.2)
+ (p -1)((f ′ ) p-3 f ′′ ) ′ v i w p 2 -(q -1)(f ′ ) q-2 f ′′ v i w q 2 + p(p -1) 2 (f ′ ) p-3 f ′′ w i w p-2 2 - q 2 (f ′ ) q-1 w i w q-2 2 .
Here and in all the manuscript, the variable v is omitted in the expression of

f ′ , f ′′ , f ′′ f ′ ′
, etc. The equalities are understood in a classical sense in Q. Multiplying (3.2) by 2v i , summing over i and using that

D 2 v, ∇v, ∇v = 1 2 ∇w • ∇v, ∆w = 2∇v • ∇∆v + 2|D 2 v| 2 , i 2(∇v i • ∇w)v i = |∇w| 2 , i (∇w i • ∇v)v i = D 2 w, ∇v, ∇v ,
we get that

∂ t w = |∇u| p-2 ∆w + (p -2)|∇u| p-4 D 2 w, ∇u, ∇u -2|∇u| p-2 |D 2 v| 2 + (p -2)(f ′ ) p-2 w p-4 2 ∆v (∇v • ∇w) + (p -2) 2 (f ′ ) p-2 w p-4 2 |∇w| 2 + (p -2)(p -4) 2 (f ′ ) p-2 w p-6 2 (∇v • ∇w) 2 (3.3) -q(f ′ ) q-1 w q-2 2 ∇w • ∇v + (p(p -1) + (p -2) 2 )(f ′ ) p-3 f ′′ w p-2 2 ∇w • ∇v + 2 (p -1)((f ′ ) p-3 f ′′ ) ′ w p+2 2 -(q -1)(f ′ ) q-2 f ′′ w q+2 2 + (p -2)(f ′ ) p-3 f ′′ w p 2 ∆v .
Here, when passing from (3.2) to (3.3), the terms have been transformed according to

L 1 t1 → L1 t1 + L1 t3 , L 1 t2 → L1 t2 , L 1 t3 → L2 t2 , L 1 t4 + L 3 t2 → L3 , L 2 t1 → L5 t3 , L 2 t2 → L4 t3 , L 3 t1 → L2 t1 , L 4 t1 → L5 t1 , L 4 t2 → L5 t2 , L 5 t1 → L4 t2 , L 5
H = (p -2)(f ′ ) p-2 w p-4 2 ∆v∇v + (p -2) 2 (f ′ ) p-2 w p-4 2 ∇w, + (p -2)(p -4) 2 (f ′ ) p-2 w p-6 2 (∇v • ∇w) ∇v -q(f ′ ) q-1 w q-2 2 ∇v (3.5) + (p(p -1) + (p -2) 2 )(f ′ ) p-3 f ′′ w p-2 2 ∇v N (w) = 2(p -1)((f ′ ) p-3 f ′′ ) ′ w p+2 2 -2(q -1)(f ′ ) q-2 f ′′ w q+2 2 + 2(p -2)(f ′ ) p-2 f ′′ f ′ w p 2 ∆v. (3.4)
Step 2: equation for z and useful estimates

We set z = ηw. Defining the operator

L(z) := ∂ t z -A(z) -H • ∇z,
we have that

Lz = ηLw + wLη -2|∇u| p-2 ∇η • ∇w -2(p -2)|∇u| p-4 (∇u • ∇η)(∇w • ∇u) = -2|∇u| p-2 ∇η • ∇w -2(p -2)|∇u| p-4 (∇u • ∇η)(∇w • ∇u) + ηN w + wLη -2|∇u| p-2 |D 2 v| 2 η.

Leading estimates

Recalling that f is increasing and that f ′′ > 0, we get the following estimates.

1. Estimate of ηN w ηN w ≤ 2(p -1)((f ′ ) p-3 f ′′ ) ′ w p+2 2 η -2(q -1)(f ′ ) q-2 f ′′ w q+2 2 η + (f ′ ) p-2 2 w p-2 2 |D 2 v| 2 η + 2N(p -1) 2 (f ′ ) p-2 f ′′ f ′ 2 w p+2 2 η. (3.7)
Here we used that

2(p -2)| f ′′ f ′ w∆v| ≤ 2N(p -1) 2 w 2 f ′′ f ′ 2 + |D 2 v| 2 2 .

Estimate of wL(η)

• Estimate of wA(η)

|wA(η)| ≤ (f ′ ) p-2 w p 2 ( √ N + (p -2))|D 2 η|. (3.8) • Estimate of |wH • ∇η| |wH • ∇η| ≤ (f ′ ) p-2 w p-2 2 C 1 (p, N, δ 1 )η -1 |∇η| 2 w + δ 1 [D 2 v| 2 η 1 + (f ′ ) p-2 w p-2 2 C 2 (p, N, δ 2 )η -1 |∇η| 2 w + δ 2 [D 2 v| 2 η 2 + (f ′ ) p-2 w p-2 2 C 3 (p, N, δ 3 )η -1 |∇η| 2 w + δ 3 [D 2 v| 2 η 3 +2(p -1) 2 (f ′ ) p-3 f ′′ w p+1 2 |∇η| + q(f ′ ) q-1 w q+1 2 |∇η|. (3.9)
(1) comes from an estimate via the Young's inequality of |(p -2)w∆v∇v • ∇η|.

Recalling that ∇w = (2D 2 v, ∇v), (2) comes from an estimate of (p-2) 2 w∇w • ∇η and (3) come from an estimate of (p-2)(p-4) 2 w (∇v • ∇w) (∇v • ∇η) . Finally recalling that ∇u = f ′ ∇v and choosing δ i in such way that -2 + δ 1 + δ 2 + δ 3 + δ 4 + δ 5 = -1 and then recalling the properties of the function η, we arrive at

L(z) ≤ 2(p -1)η (f ′ ) p-3 f ′′ ′ w p+2 2 + N(p -1)(f ′ ) p-2 f ′′ f ′ 2 w p+2 2
-2(q -1)(f ′ ) q-2 f ′′ w q+2 2 η + C(p, N, α)(f ′ ) p-2 R -2 w p 2 η α + C(p, q, α)η α R -1 w p+1 2 (f ′ ) p-3 f ′′ + w q+1 2 (f ′ ) q-1 .

Step 3: suitable choice for the function f

To get rid of the term

(f ′ ) p-3 f ′′ ′ w p+2 2 + N(p -1)(f ′ ) p-2 f ′′ f ′ 2 w p+2 2
(3.10)

= (f ′ ) p-2 w p+2 2 f ′′ f ′ ′ + (p -2) f ′′ f ′ 2 + (p -1)N f ′′ f ′ 2 (3.11) ≤ (f ′ ) p-2 w p+2 2 f ′′ f ′ ′ + (p -1)(N + 1) f ′′ f ′ 2 .
(3.12)

we shall take a function f satisfying the following differential equation

f ′′ f ′ ′ + (p -1)(1 + N) f ′′ f ′ 2 = 0. (3.13)
Hence we get that

L(z) ≤ -2(q -1)(f ′ ) q-2 f ′′ w q+2 2 η + C(p, N, α)(f ′ ) p-2 R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 (f ′ ) p-3 f ′′ + w q+1 2 (f ′ ) q-1 .

  |∇v| > 0 and hence we can use Lemma 2.1. Now let us take f (s) = M(s + 1) γ -2M where γ is given by

  t2 → L4 t1 , (with obvious labeling). Hence w satisfies ∂ t w -A(w) -H • ∇w = -2|∇u| p-2 |D 2 v| 2 + N (w) where A(w) = |∇u| p-2 ∆w + (p -2)|∇u| p-4 D 2 w, ∇u, ∇u ,

3 .-2 2 C 4 4 .

 3244 Estimate of 2|∇u| p-2 |∇η • ∇w|. Using the Young inequality, we have2|∇u| p-2 |∇η • ∇w| ≤ (f ′ ) p-2 w p(p, N, δ 4 )η -1 |∇η| 2 w + δ 4 |D 2 v| 2 η . Estimate of 2(p -2)(∇u • ∇η)(∇w • ∇u) |2(p -2)(∇u • ∇η)(∇w • ∇u)| ≤ (f ′ ) 2 w C 5 (N, p, δ 5 )η -1 |∇η| 2 w + |D 2 v| 2 η .