2 Notations, hypotheses and statements of the main results

Large time behavior of solutions to Neumann problem for a quasilinear second order degenerate parabolic equation in

domains with non compact boundary

D. Andreucci,

Dip. di Metodi e Modelli, Universit`a ”La Sapienza”

Via A. Scarpa 16, 00161 Roma, Italy

G. R. Cirmi, S. Leonardi, Dip. di Matematica, Citt`a Universitaria,

Viale A. Doria 6, 95125 Catania, Italy

A. F. Tedeev

Inst. Applied Mathematics, Academy of Sciences, R. Luxemburg st. 74, 340114 Donetsk, Ukraine

Appeared in Journal of Differential Equations, 174 (2001), pp.253–288.

Copyright c Academic Press 2001.

Abstract

We investigate the optimal rate of stabilization at large time of a solution to the Neumann problem











u^t =P^N

i=1

∂

∂xⁱ(aⁱ(x, t, ∇u)) − b(x, t, u), in Ω × (0, T ), T > 0 P^N

i=1aⁱ(x, t, ∇u)nⁱ= 0, on ∂Ω × (0, T ) u(x, 0) = u⁰(x) x ∈ Ω, u⁰(x) ≥ 0 in Ω

where Ω ⊂ R^N, N ≥ 2, is an unbounded domain with sufficiently smooth noncompact boundary ∂Ω satisfying certain isoperimetrical inequality and n = (nⁱ) is the outward normal to ∂Ω.

1 Introduction

In this paper we consider the following Neumann problem u_t =

N

X

i=1

∂

∂xi

(a_i(x, t, ∇u)) − b(x, t, u), in D = Ω × (0, +∞) (1)

N

X

i=1

ai(x, t, ∇u)ni = 0, on ∂Ω × (0, +∞) (2)

u(x, 0) = u₀(x) in Ω, u₀(x) ≥ 0 in Ω. (3) Here Ω ⊂ R^N, N ≥ 2, is an unbounded domain with sufficiently smooth noncompact boundary ∂Ω and n = (n_i) is the outward normal to ∂Ω.

The coefficients a_i(x, t, ξ), i = 1, 2, . . . , N and b(x, t, u) are Carathe`odory functions satisfying suitable growth conditions; moreover we assume that the following ellipticity condition holds

N

X

i=1

ai(x, t, ξ) ξi ≥ ν(x) ψ(t) |ξ|^m+1 a.e. x ∈ Ω t ∈]0, +∞[, ∀ξ ∈ R^N

where ν(x) and ψ(t) are nonnegative functions verifying additional conditions to be precised later on. The function b(x, t, u) is a lower order term playing the role of absorption.

A typical example of (1) is the following equation

u_t =

N

X

i=1

∂

∂x_i



|x|^θt^κ|∇u|^m−1∂u

∂x_i



− λ u^q, where 0 ≤ θ < m, 0 ≤ κ < 1, m > 1, λ ≥ 0 and q ≥ 1.

Our goal is to find the optimal bound of ku(., t)k_L^∞_(Ω) for t large, where u is a nonnegative solution of the above problem with initial datum belonging to L¹(Ω) or slowly decaying at infinity.

Moreover, when b = 0, m > 1 and u₀ is compactly supported, we establish a sharp bound of the interface.

The results presented here obviously hold for the Cauchy problem corresponding to the equation (1). Optimal bounds of maximum modulus of solution to Cauchy problem for nonstationary p-laplacian in the nonweighted case can be found in [14, 10, 15]. Analogous results for porous medium equation are contained in papers [1, 7, 17].

Nonweighted parabolic Neumann problems in domains with non compact boundary and with L¹- initial datum have been studied in [13] in the linear case and in [26, 5] in the non linear one.

For further literature concerning qualitative properties of solutions of parabolic equations the reader can refer for instance to Kalashnikov’s survey [16].

Concerning the existence of weak solutions in weighted spaces we quote, among others, the papers [24, 11, 9, 19].

Our approach relies on sharp energy estimates which are essentially based on a sharp form of weighted Gagliardo-Niremberg embedding result for a wide class of domains satisfying suitable isoperimetrical properties.

Let us briefly summarized the contents of the paper. After a section devoted to the notations and the statements of the main results, we state and prove an embedding theorem (see Section 3). Sections 4 and 5 contain the proofs of the optimal bounds of the maximum modulus of a nonnegative solutions of the problem (1)-(3) with initial data respectively in L¹ and in L^po∩ L^∞ with po > 1. In the last Section we prove that a solution of the above problem with m > 1, b ≡ 0 and u_o compactly supported, has the property of ”finite speed of propagation”.

2 Notations, hypotheses and statements of the main results

Let Ω ⊂ R^N, N ≥ 2, be an unbounded domain, containing the origin, with sufficiently smooth and noncompact boundary ∂Ω. We denote by x ≡ (x₁, x₂, . . . , x_N) a point in Ω .

As we have remarked in the Introduction, in order to prove the embedding theorem (see next section) we need some assumptions on the geometry of Ω; to this aim we introduce the function

L(V ) = inf {meas_{N −1}(∂Q ∩ Ω) , Q ⊂ Ω open with lipschitz boundary, meas_NQ = V } and we give the following

Definition 2.1 Ω belongs to the class B₁(g) if there exists a positive, nondecreasing function g ∈ C(0, +∞) such that V^1−1/N

g(V ) is nondecreasing and

L(V ) ≥ g(V ) ∀V > 0. (1)

The above definition implies the existence of two positive constants γ₁, γ₂ such that L(V ) ≥ γ₁V^{N −1N} for V small enough ,

and

L(V ) ≥ γ₂ for V sufficiently large .

The first estimate gives us back the classical isoperimetrical inequality in the case of bounded domains with lipschitz boundary (see [21] pag. 301), while the second one characterizes domains which ”do not contract at infinity”.

Let us note moreover that examples of domain which do not satisfy condition (1) for small V can be found in [21] at pages 10 and 164.

Given R > 0, let

Ω_R= Ω ∩ⁿx ∈ R^N : |x| < R^o and

V (R) = meas_NΩ_R. Let us denote by R the inverse function of V (R).

Definition 2.2 Ω belongs to the class B₂(g) if Ω ∈ B₁(g) and there exists a constant c₀ > 0 such that

R(V ) ≥ c₀ V

g(V ) ∀V > 0. (2)

Domains belonging to classes similar to B₁(g), B₂(g) where considered by Gushchin [12], [13]

and subsequent papers.

It is easy to prove that if Ω ∈ B₁(g) then R(V ) ≤ N V

g(V ) ∀V > 0. (3)

Thus, assuming Ω ∈ B₂(g) essentially amounts to requiring that the volume V (ρ) is equivalent to ρg(V (ρ)). As a matter of fact (2) and (3) are equivalent to

1/N ρg(V (ρ)) ≤ V (ρ) ≤ 1/c₀ρg(V (ρ)) ∀ρ > 0. (4) Moreover from (4) it follows that |Ω| = +∞, otherwise we would get a contradiction letting ρ → +∞ in (4).

An example of domain of class B₂(g) (and then of class B₁(g)) is the paraboloid–like domain Ω^h=ⁿx ∈ R^N : |x⁰| < x^h_N^o

where |x⁰| = (x²₁+ . . . + x²_{N −1})^1/2, X_N > 1, 0 ≤ h ≤ 1 (note that Ω⁰ is a cylinder and Ω¹ is a cone). In this case (see [5])

g(V ) = γ min^V^{N −1N} , V^η, η = h(N − 1)

h(N − 1) + 1 ≤ N − 1 N .

Let now ν(x) be a nonnegative function in Ω and ˜ν(s) be the decreasing rearrangement of _ν(x)¹ . We assume that

ν(x) ∈ L^∞_loc(Ω), (5)

R→+∞lim

sup_Ω2R\ΩRν(x)

R^m = 0 (6)

1

ν(x) ∈ L^α(Ω), 1 + 1

α < m + 1 < N , α ≥ N

m + 1 (7)

∃κ1 ∈



0,m + 1 N



such that hκ1(s) = s^κ1ν(s) is non decreasing in ]0, +∞[.˜ (8) Let also ψ :]0, +∞[→ R be a monotone nondecreasing function such that

ψ ∈ L¹(0, T ) ∀ T > 0. (9)

Set

ψ(t) =˜ Z _t

0

ψ(s)ds.

Assumptions (5), (7) and (9) are classical in the theory of weighted parabolic equations (see [22]); technical assumption (8) implies that ˜ν(s) have power–like behavior and it is necessary, at least for power–like weight (see [8]), moreover hypothesis (6) will be used to obtain a mass estimate (see corollary 4.1 later on).

Let a_i(x, t, ξ), i = 1, 2, . . . , N be Carath´eodory functions in Ω×R^{N +1}such that the following structural assumptions are satisfied a.e. (x, t) ∈ Ω × R, ∀ ξ, η ∈ R^N:

a_i(x, t, 0) = 0 i = 1, 2, . . . , N, (10)

N

X

i=1

(a_i(x, t, ξ) − a_i(x, t, η)) (ξ_i− η_i) ≥ ν(x)ψ(t)|ξ − η|^m+1, (11)

N

X

i=1

|a_i(x, t, ξ) − a_i(x, t, η)| ≤ ν(x)ψ(t) (|ξ| + |η|)^m−1|ξ − η| (12) where m > 1 .

Let b(x, t, χ) be a Carath´eodory function in Ω × R² such that the following assumptions are satisfied a.e. (x, t) ∈ Ω × R, ∀ χ, ˜χ ∈ R:

b(x, t, 0) = 0, (13)

(b(x, t, χ) − b(x, t, ˜χ)) (χ − ˜χ) ≥ σ|χ − ˜χ|^q+1 (14)

where q > 1 and σ is a positive constant.

The previous assumptions are classical in the theory of parabolic equations with general coefficients and are satisfied, for example, if we take

a_i(x, t, ξ) = ν(x)ψ(t)|ξ|^m−1ξ_i i = 1, 2, . . . , N, b(x, t, χ) = |χ|^q−1χ.

In order to give the definition of weak solution of the problem (1)-(3) we have to specify the functional setting we shall use.

Let p, γ ≥ 1; then W_p,γ^1,0(ν, Ω) is the space of functions u for which is finite the norm

kuk_W1,0 p,γ(ν,Ω)=

Z

Ω

|u|^γdx

1/γ

+

Z

Ω

ν(x)|∇ u|^pdx

1/p

; W_p,γ^1,0(νψ, D_T) is the space of functions u for which is finite the norm

kuk_W1,0

p,γ(νψ,DT)=

Z

DT

|u|^γdxdt

1/γ

+

Z

DT

ν(x)ψ(t)|∇ u|^pdxdt

1/p

; W_p,γ^1,1(νψ, D_T) is the space of functions u for which is finite the norm

kuk_W1,1

p,γ(νψ,DT) =

Z

D_T (|u|^γ + |u_t|^γ) dx

1/γ

+

Z

D_T ν(x)ψ(t)|∇ u|^pdx

1/p

.

Due to assumptions (5), (7) and (9) the above weighted Sobolev spaces are Banach spaces (see [22]).

Now, we are in position to give the definition of weak solution of problem (1)-(3).

Let D_T = Ω × (0, T ), T > 0, u₀ ∈ L^p0(Ω) ∩ L^∞(Ω), u₀≥ 0 a.e. in Ω and p₀≥ 1.

Definition 2.3 Let 1 ≤ p₀ ≤ 2. A weak solution of the problem (1)-(3) in D_T is a nonnegative function u ∈ W_m+1,2^1,0 (νψ, D_T) ∩ L^∞(D_T) such that holds

Z

DT

"

−uv_t+

N

X

i=1

a_i(x, t, ∇u)v_xi+ b(x, t, u)v

#

dxdt = Z

Ωu₀(x)v(x, 0)dx (15) for any v ∈ W_m+1,2^1,1 (νψ, D_T) such that v(x, T ) = 0.

A weak solution of the problem (1)–(3) in D is a weak solution of the problem (1)–(3) in D_T for any T > 0.

Definition 2.4 Let p₀ > 2. A weak solution of the problem (1)-(3) is a nonnegative func- tion u ∈ W_m+1,p^1,0 _o(νψ, D_T) ∩ L^∞(D_T) such that the identity (15) is satisfied for any v ∈ W^1,1

m+1,p⁰₀(νψ, D_T) such that v(x, T ) = 0.

A weak solution of the problem (1)–(3) in D is a weak solution of the problem (1)–(3) in D_T for any T > 0.

Under the hypotheses (5), (7), (9), (10)- (14) the existence of a u of the problem (1)-(3) follows from the results of [9, 11, 18, 19, 20, 23].

Now, let us denote by J⁻¹ the inverse function of J(V ) = V^m−1

 V g(V )

m+1

˜

ν(V ), (16)

The following theorem concerns with the large time behavior of a nonnegative solution of the problem (1)-(3) with initial datum in L¹.

Theorem 2.1 Let Ω ∈ B₁(g). Assume that hypotheses (5), . . . , (14) hold and let u(x, t) be a solution of the problem (1)-(3) in D_T and u₀ ∈ L¹(Ω) ∩ L^∞(Ω).

Then there exist two positive constants C₁, Γ such that for any t > 0 the following estimate is true

ku(·, t)k_L^∞_(Ω)≤ C₁min





ku₀k_L1(Ω)

J^−1Γ ˜ψ(t)ku₀k^m−1_L1(Ω)

, t^−q−11



. (17)

Remark 2.1 We point out that, using approximation arguments, the above theorem still holds under the assumption u₀ ∈ L¹(Ω).

Moreover, when Ω ∈ B₂(g) the result of previous theorem can be sharpened (see Theorem 4.1).

If we take Ω ≡ Ω^h, ν(x) = |x|^θ, 0 ≤ θ < m, m + 1

N > θ

h(N − 1) + 1, ψ(t) = t^κ, 0 ≤ κ ≤ 1, u₀ ∈ L¹(Ω^h) ∩ L^∞(Ω^h), u₀ ≥ 0 then Theorem 2.1 implies that for all t > 1

ku(·, t)k_L^∞_(Ω) ≤ C(N, m)ku₀k

m+1−θ K

L¹ t^−λ, if λ > 1

q − 1 (18)

while

ku(·, t)k_L^∞_(Ω)≤ C(N, m)t^−q−11 if λ < 1

q − 1 (19)

where

λ = (1 + κ)[1 + (N − 1)h]

K

and

K = K(h, θ) = (m − 1)[1 + (N − 1)h] + m + 1 − θ.

We notice that the number q^∗ defined by the relationship λ = 1

q^∗− 1 plays the role of critical exponent for the problem (1)-(3). Moreover, we note that K(1, 0) = (m − 1)N + m + 1 is the well known Barenblatt’s exponent and the expression (18) when θ = 0, κ = 0, h = 1 is the same obtained in [14] .

In the case κ = 0 and h = 1 formula (18) was given in [25] for a solution to the Cauchy problem, while for κ = 0, θ = 0 it was proven in [5].

Let us also remark that (18) provides sharp dependence of maximum modulus of a solution on the parameters of the problem, i.e. on ν(x), ψ(t), m and the geometry of Ω^h when b ≡ 0, see Corollary 6.1.

Now let us set

J_p0(s) = s^m−1p0

 s g(s)

m+1

˜

ν(s), ∀ s > 0. (20)

If the initial datum belongs to L^p0(Ω) ∩ L^∞(Ω), p₀ > 1 we can prove the following

Theorem 2.2 Let Ω ∈ B₁(g). Assume that hypotheses (5, . . . , (14) hold and let u(x, t) be a solution of the problem (1)-(3) in DT and u0 ∈ L^p0(Ω) ∩ L^∞(Ω), p0 > 1.

Then, there exist two positive constants C₂, Λ such that for any t > 0 ku(·, t)k_L^∞_(Ω)≤ C₂ ku₀k_L1(Ω_{2 ˜R(t)})

J⁻¹



Γ ˜ψ(t)ku₀k^m−1_L1(Ω_{2 ˜}R(t))

 (21)

where ˜R(t) is defined by the relationship ku₀k_L1(Ω_{2 ˜R(t)})

J⁻¹



Γ ˜ψ(t)ku₀k^m−1_L1(Ω

2 ˜R(t))

 = ku₀k_Lp0(Ω\ΩR˜(t))

Jp⁻¹0



Λ ˜ψ(t)ku0k^m−1_Lp0(Ω\Ω

R(t)˜ )

1/p0. (22)

Remark 2.2 When Ω belongs to the class B₂(g) the result of the previous theorem can be sharp- ened (see Theorem 5.1). Moreover in the particular case Ω ≡ Ω^h, ν(x) = |x|^θ with 0 ≤ θ < m,

m + 1

N > θ

h(N − 1) + 1 ψ(t) = t^κ, u₀(x) = (1 + |x|)^−β, 0 < β < 1 + (N − 1)h (see also Remark 5.2), for any t > 1 we have

ku(·, t)k_L^∞_(Ω)≤ γ₁t^−λβ. (23)

where

λ = 1 + κ

m + 1 − θ + β(m − 1).

Let us note that, if κ = θ = 0, (23) can be rewritten as ku(·, t)k_L^∞_(Ω) ≤ γ₁t^{−m+1+β(m−1)β} and the above estimate reduces to the results of [26, 15].

Let us denote

Z(t) = inf{ρ > 0 : supp u(·, t) ⊂ Ω_ρ}.

Obviously, Z(t) gives a measure of the speed of propagation of the support of u. When m > 1 and the initial datum has compact support we shall prove the property of finite speed of propagation for a solution of problem (1)-(3) without absorption term . As a matter of fact in Section 6 we prove the following

Theorem 2.3 Let the hypotheses of theorem 2.1 be satisfied and b ≡ 0. Let supp u_o ⊂ B_R0 Then for all t > 0

Z(t) ≤ C˜ ₃^Z(0) + G˜ ₀(J⁻¹( ˜ψ(t)ku₀k^m−1_L1(Ω)))^ (24) where

Z(t) =˜ Z(t)

(sup_Ω2Z(t)ν(x))^m+11 and

G₀(s) = s

g(s)(˜ν(s))^m+11 .

3 An embedding result

The description of the geometrical characteristics of the domain via isoperimetrical prop- erties allows us to use naturally the symmetrization approach to prove an embedding result.

This approach seems to be the most suitable for domains with noncompact boundary.

The following embedding lemma, which has interest in itself, will be crucial in the proofs of theorems 2.1 and 2.2; the nondegenerate case has been considered in [27].

For the sake of simplicity, from now on we will always denote by c a positive constant, depending only on the data, which may vary from line to line.

Lemma 3.1 Let Ω ∈ B₁(g) and ν(x) satisfy the condition (5) and 1

ν(x)^1/(p−1) ∈ L¹_loc(Ω). (1)

Assume moreover that there exist constants θ, γ₀ > 0, with 1 < θ < p < N , such that Z s

0 ν(τ )˜ ^p−θθ dτ ≤ γ₀s˜ν(s)^p−θθ ∀ s > 0. (2)

Then for any u ∈ W_p,p^1,0(ν, Ω) ∩ L^β(Ω), 0 < β < q ≤ N θ

N − θ the following inequality holds

I ≥ ¯γ E

p

qq

G₁

 E

q q−β

β /E

β

qq−β

, (3)

where

I = Z

Ων(x)|∇u|^pdx, E_γ = Z

Ω|u|^γdx, G₁(s) = s^pq−1

 s g(s)

p

˜ ν(s), provided G₁ is increasing.

If θ ≥ β, ¯γ depends only on θ, γ_o, N . Proof.

We first prove the theorem in the case θ < q ≤ N θ N − θ.

Following the paper [3], we can construct for any s ∈ (0, +∞) a measurable set D(s) ⊂ Ω such that

i) measND(s) = s,

ii) s₁ < s₂⇒ D(s₁) ⊂ D(s₂),

iii) D(s) = {x ∈ Ω : |u(x)| > r} for s = µ(r),

where µ(r) = measN{|u(x)| > r} is the distribution function.

Moreover there exists ν(s) > 0, s > 0, such that Z

D(s)

1 ν(x)^p−11

dx = Z s

0

1 ν(τ )^p−11

dτ. (4)

Let us denote by u^∗(s) the decreasing rearrangement of u(x) i.e.

u^∗(s) = inf {τ > 0 : µ(τ ) < s} , and observe that if θ < q is well known that

E_q= Z +∞

0 [u^∗(s)]^qds ≤ c

Z _+∞

0 [u^∗(s)]^θs^θq−1ds

q/θ

. (5)

For any fixed k > 0, to be chosen later on, we have Z _+∞

0 [u^∗(s)]^θs^θq−1ds = Z _µ(k)

0 [u^∗(s)]^θs^θq−1ds +

Z +∞

µ(k)

[u^∗(s)]^θs^θq−1ds ≡ A₁+ A₂.

(6)

In order to estimate A₁ we integrate between 0 and µ(k) the identity d

dσ

 u^∗(σ)^θ

Z σ

0 s^θq−1ds



= θ(u^∗(σ))^θ−1du^∗(σ) dσ

Z σ 0

s^θq−1ds + (u^∗(σ))^θσ^θq−1,

then we use Young and Chebychev inequalities and known properties of rearrangements so that A₁ ≤ c

"

Z _µ(k)

0



−du^∗(σ) dσ

θ

σ^θq+θ−1dσ + µ(k)^θ/q−θ/βE_β^θ/β

#

≡ c[A₃+ A₄].

(7)

Discriminating the cases θ < β and θ ≥ β, one can easily prove that

A₂ ≤ c µ(k)^θ/q−θ/βE_β^θ/β (8)

(the constant c in (8) equals 1 if θ ≥ β).

Applying H¨older’s inequality we can estimate A3 getting

A₃≤

Z _µ(k)

0



−du^∗(σ) dσ

p

(g(σ))^pν(σ)dσ

!θ/p

 Z _µ(k)

0

σ^{(θq+θ−1)p−θp} (g(σ))^p−θpθ ν^p−θθ (σ)

dσ





1−θ/p

. (9)

Let us prove now that

Z _+∞

0



−du^∗(σ) dσ

p

(g(σ))^pν(σ)dσ ≤ I. (10)

Working as in [3] we get 1

h Z

{τ <|u|≤τ +h}

|∇u|dx

≤ 1

h Z

{τ <|u|≤τ +h}

ν(x)|∇u|^pdx

!1/p

1 h

Z

{τ <|u|≤τ +h}

ν(x)^−p−11 dx

!^p−1_p

∀ h > 0.

Letting h → 0 it follows

− d dτ

Z

{|u|>τ }

|∇u|dx

≤ − d

dτ Z

{τ <|u|}ν(x)|∇u|^pdx

!1/p

− d dτ

Z

{τ <|u|}ν(x)^−p−11 dx

!^p−1_p .

(11)

By the definition of ν(s) we have

− d dτ

Z

{τ <|u|}ν(x)^−p−11 dx = −µ⁰(τ ) 1

[ν(µ(τ ))]^p−11 . (12)

On the other hand by the isoperimetrical property of Ω and the Rishel -Fleming formula we deduce

g(µ(τ )) ≤ L(µ(τ )) ≤ − d dτ

Z

{τ <|u|}

|∇u|dx, so that, from (11) and (12), it turns out

(g(µ(τ )))^pν(µ(τ ))



−dµ(τ ) dτ

−(p−1)

≤ − d dτ

Z

{τ <|u|}ν(x)|∇u|^pdx.

Integrating in ]0, +∞[ the above inequality and using the identity dµ(τ )

dτ = 1

d

dτ(u^∗(µ(τ ))) we obtain (10).

Now, by virtue of Lemma 2.2 in [3] there exists a sequence {ν_n} such that

 1 ν_n

∗

→

 1 ν(x)

∗

a.e. (13)

and

Z _µ(k)

0

ds (ν(s))^p−θθ

ds = lim

n→+∞

Z _µ(k)

0

ds (ν_n(s))^p−θθ

. (14)

Thus from (13) and (14) and our hypotheses on the weight ν we infer Z _µ(k)

0

ds (ν(s))^p−θθ

ds ≤ γ₀µ(k)(˜ν(µ(k)))^p−θθ . (15) The monotonicity of the function

s^{(θq+θ−1)p−θp} (g(s))^p−θpθ

, along with inequalities (9), (10) and (15) implies

A₃ ≤ cI^θp(µ(k))^θq+θ−θp

(g(µ(k)))^θ (˜ν(µ(k)))^θ/p. (16)

From (5), (7), (8) and (16) we deduce

E_q ≤ c



I^θp(µ(k))^{θq+θ−θ/p}

(g(µ(k)))^θ ((˜ν(µ(k)))^θ/p+ µ(k)^θ/q−θ/βE^θ/β_β





q/θ

≤ c

"

I^1/p(µ(k))^1/q+1−1/p

g(µ(k)) ((˜ν(µ(k)))^1/p+ µ(k)^1q−β1E_β^1/β

#q

= c^h(I G₁(µ(k)))^1p + (µ(k))^1/q−1/βE_β^1/βiq.

(17)

Now we choose k such that

I^1/p(G₁(µ(k))^1/p= µ(k)^1/q−1/βE_β^1/β or, equivalently,

µ(k) = Φ⁻¹



 E_β^p/β

I



 where Φ(s) = ^G1(s)

s^p/q−p/β and Φ⁻¹(s) is its inverse function.

Substituting the above value in (17) we complete the proof in the case q > θ.

To achieve the complete proof we argue in the following way; let q₁ such that β < q₁ ≤ θ and choose q > θ; note

q₁ = qq₁− β

q − β + βq − q₁ q − β. Applying H¨older’s inequality we deduce

E_q1 ≤ E

q1−βq−β

q E

q−q1q−β

β . (18)

The thesis now follows immediately from the first part of the theorem and monotonicity of right-hand side of (3) with respect to E_q.

Remark 3.1 The assumption (1) and the one on the monotonicity of G1 will be fulfilled in the instances of application of Lemma 3.1, as a consequence of (7), (8) and our choice of the parameters p, q, β, θ.

Remark 3.2 Assume that ν(x) satisfy conditions (8), then Z _s

0

˜

ν(τ )^m+1N dτ ≤ γ₀s˜ν(s)^m+1N ∀ s > 0 (19) with γ₀ = ¹

1−κ1 N m+1

.

Remark 3.3 If Ω = Ω¹ and ν(x) ≡ 1 then inequality (3) is the classical Nirenberg–Gagliardo inequality.

4 Proof of Theorem 2.1

Before proving theorem 2.1 let us premise some auxiliary results useful in the sequel.

Proposition 4.1 Let u be a weak solution of the problem (1)-(3). Assume that hypotheses (5), (7) and (9) hold.

Let

supp u₀ ⊂ B_Ro.

Then for any t > 0 there exists a positive constant c, depending on ku(·, t)k_∞ too, such that the following estimate holds

H_R(t) ≤ e ku₀k²_2,Ωexp







−c





(R − R_o)^m+1 tψ(t)η(R)

!_m¹









∀R > R_o (1)

where

HR(t) = Z

Ω\ΩR

u²dx + Z _t

0

Z

Ω\ΩR

ν(x)ψ(τ )|∇u|^m+1dxdτ.

and

η(R) = sup

Ω2R\ΩR

ν(x).

Proof.

Let R > R_o, 0 < ρ ≤ R and

ξ(x) =











0 if |x| ≤ R

|x| − R

ρ if R < |x| < R + ρ 1 if |x| ≥ R + ρ

Choosing ξ^m+1(|x|)u as test function in the weak formulation of problem (1)-(3) we obtain:

1 2

Z

Ω

u²ξ^m+1dx + Z t

0

Z

Ω N

X

i=1

a_iu_xiξ^m+1dxdτ + Z t

0

Z

Ω

bξ^m+1udxdτ =

− (m + 1) Z _t

0

Z

Ω N

X

i=1

a_iξ_xiξ^mudxdτ

Using Young’s inequality and growth conditions in the right hand side of the previous in- equality we have

1 2

Z

Ωu²ξ^m+1dx + Z t

0

Z

Ων(x)ψ(τ )|∇u|^m+1ξ^m+1dxdτ

≤ cε^m+1m Z _t

0

Z

Ω

ν(x)ψ(τ )|∇u|^m+1ξ^m+1dxdτ + cε^−(m+1)

Z t 0

Z

Ων(x)ψ(τ )|u|^m+1|∇ξ|^m+1dxdτ.

Hence, for ε > 0 sufficiently small, we obtain H_R+ρ(t) ≤ cη(R)

ρ^m+1 Z _t

0

ψ(τ )(M (τ ))^m−1H_R(τ )dτ, (2) where

M (t) = sup

Ω

u(x, t).

By the maximum principle (its formal proof can be readily carried out as in [28], Proposition 2) there exists a constant Mo> 0 such that

M (t) ≤ M_o ∀t ∈ (0, T ) and, from (2), we obtain

H_R+ρ(t) ≤ cM_o^m−1η(R) ρ^m+1

Z _t

0

ψ(τ )H_R(τ )dτ. (3)

Let us absorb, for the sake of simplicity, M_o in the generic constant c.

Let us prove, by induction, the following inequality H_Ro+kρ(t) ≤ c^kku₀k²₂ (tψ(t))^k

ρ^(m+1)kk![η(R)]^k. (4)

Taking u as test function in the weak formulation of problem (1)-(3), easily follows

H_Ro(t) ≤ cku₀k²₂ (5)

that proves (4) for k = 0.

Let us assume that inequality (4) holds for some integer k > 0. By virtue of (3) and using (4), we can obtain

H_Ro+(k+1)ρ(t) ≤ cη(R) ρ^m+1

Z t 0

ψ(τ )H_Ro+kρ(τ )dτ ≤

c^k+1ku₀k²₂ (tψ(t))^k+1

ρ^(m+1)(k+1)(k + 1)![η(R)]^k+1. Then, inequality (4) holds for any integer k > 0.

Let k ≥ 1. Choosing in (4) ρ = R − R_o

k we obtain H_R(t) ≤ c^kku₀k²₂ k^(m+1)k[tψ(t)]^k

(R − R_o)^(m+1)kk![η(R)]^k.

¿From this inequality, using also Stirling’s formula it follows H_R(t) ≤ ku₀k²₂exp

(

−k log c(R − R_o)^m+1 e tψ(t)η(R)k^m

)

. (6)

Now, if

c(R − R_o)^m+1

e tψ(t)η(R) ≤ exp(1)

the estimate (1) easily follows from (5). Otherwise, we can obtain (1) from (6) taking as k the integer part of

(c(R − R_o)^m+1 e²tψ(t)η(R)

)_m¹ .

Corollary 4.1 (Mass estimate) Let u be a weak solution of the problem (1)-(3). Assume that hypotheses (5), (6), (7), (9) hold and supp u₀ ⊂ B_Ro.

Then, ∀t > 0

Z

Ω

u(x, t)dx ≤ Z

Ω

u₀(x)dx. (7)

Proof.

Let R > 0, t > 0 and u(x, τ ) be a solution of problem (1)- (3).

Taking

ξ_R(x) =











0 if x ∈ Ω_R

|x| − R

R if x ∈ Ω_2R\ Ω_R 1 if x ∈ Ω_2R

as test function in the weak formulation of problem (1)-(3). we have Z t

0

Z

Ωu_τξ_Rdxdτ + Z t

0

Z

Ω N

X

i=1

a_i(x, τ, ∇u)(ξ_R)_xidxdτ +

Z t 0

Z

Ωb(x, τ, u)ξ_Rdxdτ = 0

(8)

Using the growth condition (12) and Holder’s inequality (with exponents m + 1 and ^m+1_m ), it follows

Z t 0

Z

Ω N

X

i=1

ai(x, τ, ∇u)(ξR)xidxdτ     

≤ c

R(H_R(t))^m+1m Z t

0

Z

Ω_2R\Ω_Rν(x)ψ(τ )dxdτ

!_m+1¹

≤

c^H_R(t)R^Nm

m

m+1tψ(τ )η(R) R^m+1

_m+1¹

. (9)

By mean of Proposition (4.1) we obtain the following estimate

H_R(t)R^Nm ≤ cku₀k²_2,Ωexp







−γ

"

(R − R_o)^m+1 tψ(t)η(R)

#_m¹





·

·

"

R^m+1 tψ(t)η(R)

#^N_m

tψ(t)η(R) R^m

^N_m .

(10)

Therefore, from (9), (10) and by virtue of hypothesis (6) we obtain

R→+∞lim Z _t

0

Z

Ω N

X

i=1

ai(x, τ, ∇u)(ξR)x_idxdτ = 0.

Letting R → +∞ in (8), easily follows (7).

The next lemma will be crucial in the proof of Theorem 2.1.

Let us denote by γ the constant involved in (3) with θ = ^{N (m+1)}_{N +m+1} and γ₀ = 1/(1 − κ₁N/(m + 1)). Moreover, for any r ≥ 1 we set

Γ(r) = ¯γ(m − 1)(r + 1)

m + 1 r + m

m+1

. (11)

Lemma 4.1 Let Ω ∈ B₁(g) and u(x, t) be a weak solution of problem (1)-(3) in D_T. Suppose that hypotheses (5)-(9) are satisfied. Then, for any integer r ≥ 1 the following inequalities hold:

Z

Ω

u^r+1dx

_r+1¹

≤ ku_ok_1,Ω

J⁻¹(Γ(r) ˜ψ(t)ku_ok^m−1_1,Ω )^

r r+1

, (12)

Z

Ω

u^r+1dx

_r+1¹

≤ c^r+1r ku_ok

1 r+1

1,Ωt^{−r+1r q−11} (13)

where c is a positive constant, independent of r, J(s) is the function defined in (16).

Proof.

Let r ≥ 1; multiplying by u^r both sides of equation (1) and integrating on Ω, we get d

dt Z

Ωu^r+1dx = −r(r + 1) Z

Ω N

X

i=1

ai(x, t, ∇u)uxiu^r−1dx − (r + 1) Z

Ωb(x, t, u)u^rdx. (14)

Using the ellipticity and growth conditions, together with the identity

|∇u|^m+1u^r−1=

m + 1 m + r

m+1

|∇u^r+mm+1|^m+1 we obtain

d dt

Z

Ωu^r+1dx ≤ −r(r + 1)

m + 1 r + m

m+1

ψ(t) Z

Ων(x)|∇u^r+mm+1|^m+1, (15) d

dt Z

Ω

u^r+1dx ≤ −c(r + 1) Z

Ω

u^q+rdx. (16)

Now, let us estimate the right–hand side of (15) by applying Lemma 3.1(with p = m + 1, q = ^m+1_m+r(r + 1), θ = ^{N (m+1)}_{N +m+1}, λ = ^(m+1)(r+1)_m+r , β = ^m+1_r+m) to the function u^r+mm+1 and by using also the mass estimate (7). Thus it follows

dE_r+1

dt ≤ −r Γ(r) m − 1

ψ(t)[E_r+1(t)]^m+rr+1 G₁

 ku_ok

r+1

1,Ωr /E_r+1(t)^1r

. (17)

If we set

w(t) = ku_ok

r+1

1r

E

1

r+1r (t) we can rewrite (17) as follows

w^m−2G˜₁(w)dw ≥ Γ(r)

m − 1ψ(t)ku_ok^m−1_1,Ω dt. (18) where

G˜₁(s) =

 s g(s)

m+1

˜ ν(s).

Integrating the last inequality between w(0) and w(t) we obtain J(w) ≥ Γ(r) ˜ψ(t)ku_ok^m−1_1,Ω

which may be rewritten as follows ku_ok

r+1 r

1,Ω

E_r+1(t)^1r ≥ J^−1Γ(r) ˜ψ(t)ku_ok^m−1_1,Ω ^. (19)

¿From the last inequality immediately follows (12).

Let us prove (13).

Applying the Holder’s inequality and the mass estimate (7) we get Z

Ω

u^r+1dx ≤ ku_ok

q−1 q+r−1

1,Ω

Z

Ω

u^q+rdx

_q+r−1^r .

Using this inequality we can estimate the right hand side of (16) from above obtaining d

dt Z

Ωu^r+1dx ≤ −c(r + 1)kuok⁻

q−1 r

1,Ω

Z

Ωu^r+1dx

^q+r−1_r

and (13) easily follows integrating the above formula.

Proof of the Theorem 2.1 First of all we notice that letting r → +∞ in (13) we get ku(·, t)k_L^∞_(Ω)≤ ct^−q−11 , ∀t > 0. (20) Let

p_k= (m + 1)^k− 1

m, k = 1, 2, ...

Then p_k+ m

m + 1 = p_k−1+ 1.

Let us denote

E_k(t) = Z

Ω

u(x, t)^pk+1dx,

E˜_k(t) = ku_ok^p₁^k+1A^p_k^k+1J^−1Γ ˜ψ(t)ku_ok^m−1₁ ^−pk where

Γ = (m − 1)¯γ 2^m+1 , A_k=

k

Y

i=1

θ_op^θ_i^1

1 pi+1 , with θ₀, θ₁ > 1 constants to be chosen later on.

Our goal will be to prove that for any t > 0 and k ≥ 1 it holds

E˜_k(t) ≥ E_k(t). (21)

In fact, once inequality (21) is achieved, since

k

Y

i=1

θop^θ_i^1

1

pi+1 ≤ exp

+∞

X

i=1

1

p_i+ 1log (θop^θ_i¹)

! ,

letting k → ∞ we will obtain

ku(·, t)k_L^∞_(Ω) ≤ ckuok1

J^−1Γ ˜ψ(t)kuok^m−1₁ ^−1. (22) Comparing estimates (20) and (22) we will get (17).

We now proceed by induction on k: the validity of (21), for k = 1, follows from Lemma 4.1, since Γ(p1) > Γ and A1 > 1.

Rewriting (15) for r = p_k+1 and manipulating therein the constants we can deduce dE_k+1

dt ≤ − 1

p^m−1_k+1 ψ(t)J_k(t) (23)

where

J_k(t) = Z

Ων(x)|∇u(x, t)^pk+1|^m+1dx.

¿From the interpolation Lemma 3.1 it follows

J_k(t) ≥ ¯γ E

m−1 pk+1−pk+1

k+1 (t)

E

m−1 pk+1−pk

k (t) ˜G₁ E

pk+1+1 pk+1−pk

k (t)/E

pk+1−pkpk+1

k+1 (t)

!. (24)

Now we set

w(z) =

"

z^{N −1N} g(z)

#m+1

˜ ν(z),

λ_k= m − 1

p_k+1− p_k + 1 + p_k+ 1 p_k+1− p_k

m + 1 N , β_k= m − 1

p_k+1− p_k + p_k+1+ 1 p_k+1− p_k

m + 1 N . Working as in [5], from (24) we deduce

J_k(t) ≥ ¯γ (E_k+1(t))^λk (E_k(t))^βkw E

pk+1+1 pk+1−pk

k (t)/E

pk+1 pk+1−pk

k+1 (t)

!. (25)

With the help of H¨older’s inequality, mass estimate (7) and the inductive hypothesis we infer J_k(t) ≥ ¯γ (E_k+1(t))^λk

( ˜E_k(t))^βkw ku₀k

pk+1+1 pk+1

1 /E

1 pk+1

k+1 (t)

!.

Therefore from (23) we get dE_k+1

dt ≤ − γ¯

p^m−1_k+1 ψ(t) E_k+1^λk ( ˜E_k(t))^βkw ku₀k

pk+1+1 pk+1

1 /E_k+1(t)

1 pk+1

!. (26)

If we set

f_k(t) = ku₀k

pk+1+1 pk+1

1

E_k+1(t)

1 pk+1

, from (26) it turns out

w(f_k(t))(f_k(t))^{pk+1λk−pk+1−1}df_k≥ ¯γ

p^m_k+1ψ(t)ku₀k^(p₁ ^{k+1+1)(λk−1)}

( ˜E_k(t))^βk dt. (27) We notice that assumption (8) and the monotonicity property of ^s1−1/N_g(s) (recall the definition of the class B₁(g) ) imply the monotonicity of the function w(s)s^κ1; moreover there exists κ₂ > 0 such that the function _J−^sκ21(s) is non decreasing (we can take κ₂ = m − 1 + (m + 1)/N − κ₁).

These two facts give Z f_k(t)

fk(0)

w(s)s^{pk+1(λk−1)−1}ds ≤ w(f_k(t))(f_k(t))^{(λk−1)pk+1}

(λ_k− 1)p_k+1− κ₁ , (28) Z _t

0

ψ(τ )( ˜E_k(τ ))^−βkdτ

≥ A^−β_k ^k(pk+1)ku₀k^−(p₁ ^k+1)βk J⁻¹(a(t))
^pkβk ( ˜ψ(t))^pkβkκ2

Z t 0

ψ(τ )( ˜ψ(τ ))^pkβkκ2dτ where

a(t) = Γ ˜ψ(t)ku₀k^m−1₁ . Moreover we have

Z _t

0

ψ(τ )( ˜ψ(τ ))^pkβkκ2dτ = ( ˜ψ(t))^κ2βkpk+1 κ₂β_kp_k+ 1 and also

Z t

0 ψ(τ )( ˜E_k(τ ))^−βkdτ ≥ A^−β_k ^k(pk+1)

1 + κ₂p_kβ_kku₀k^−β₁ ^k(pk+1)J⁻¹(a(t))^pkβkψ(t).˜ (29)

Integrating (27) and using inequalities (28) e (29) we have w(f_k(t))(f_k(t))^{(λk−1)pk+1}

≥ γ¯ p^m_k+1

(λ_k− 1)p_k+1− κ1

1 + κ₂β_kp_k ku₀k^m−1₁ A^−β_k ^k(pk+1)J⁻¹(a(t))^βkpkψ(t).˜ (30) Let ϕ(z) = w(z)z^{pk+1(λk−1)}. Then from (30) it turns out

f_k(t) ≥ ϕ⁻¹[c(m, N )A^−β_k ^k(pk+1)p^−m_k+1^J⁻¹(a(t))^βkpka(t)] (31) where

c(m, N ) = m − 1 2^m+1

m + 1

N + κ₂(m + 1 < 1.

Since the function

s

1

pk+1(λk−1)/ϕ⁻¹(s) is non decreasing, it follows

ϕ⁻¹(δs) ≥ δ

1

pk+1(λk−1)ϕ⁻¹(s), ∀ 0 < δ < 1.

Therefore, from (31), after calculations we obtain E_k+1(t) ≤ ku_ok^p₁^k+1+1

[J₋₁(a(t))]^pk+1



A_k^h(1/σ)^1/Np^m/N_k+1 ^i1/(pk+1+1)

pk+1+1

and from here the estimate (21) choosing

θ_o= (1/σ)^1/N, θ₁> 1.

If Ω ∈ B₂(g) then the result of theorem 2.1 can be sharpened as the following theorem shows.

Theorem 4.1 Let Ω ∈ B2(g) and hypotheses of Theorem (2.1) be satisfied.

Assume the function

R(s) s^N1 be nondecreasing in ]0, +∞[.

Then, there exist two positive constants C3, Γ such that for any t > 0 the following estimate holds

ku(·, t)k_∞,Ω≤ C₃min ku_ok_1,Ω

P⁻¹(Γ ˜ψ(t)ku_ok^m−1_1,Ω ), t^−q−11

!

where P⁻¹(s) is the inverse function of

P(s) = s^m−1R^m+1(s)˜ν(s) (32)