A nonlinear parabolic equation with drift term

Nonlinear Analysis 177 (2018) 397–412

Contents lists available at ScienceDirect

Nonlinear Analysis www.elsevier.com/locate/na

A nonlinear parabolic equation with drift term Fernando Farroni, Gioconda Moscariello* Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia 80126 Napoli, Italy

article

info

Article history: Received 16 April 2018 Accepted 19 April 2018 Communicated by Enzo Mitidieri

abstract We consider weak solutions to a nonlinear evolution diffusion–convection equation with an unbounded convection field. We provide existence and uniqueness results as well as the asymptotic behaviour of solutions. © 2018 Elsevier Ltd. All rights reserved.

To Carlo on the occasion of his 70th birthday MSC: 35K55 35K61 35K65 35K92 Keywords: Nonlinear parabolic equations Existence Asymptotic behaviour

1. Introduction In this paper we study weak solutions to a nonlinear evolution diffusion–convection equation with an unbounded convection field. The general form of our problem reads as follows ⎧ in ΩT , ⎨ ut − div [A(x, t, ∇u) + B(x, t, u)] = − div F u=0 on ∂Ω × (0, T ), (1.1) ⎩ u(x, 0) = u0 (x) in Ω , where Ω is a regular bounded domain of RN with N ⩾ 3 and T > 0. Hereafter ΩT stands for Ω × (0, T ). For the data of the problem we assume that F ∈ L2 (ΩT )

*

and

u0 ∈ L2 (Ω ).

Corresponding author. E-mail addresses: [email protected] (F. Farroni), [email protected] (G. Moscariello).

https://doi.org/10.1016/j.na.2018.04.021 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

(1.2)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

398

Here A = A(x, t, ξ) : ΩT × RN → RN is a Carath´eodory function satisfying the following conditions |A(x, t, ξ)| ⩽ β|ξ| + g(x, t)

for some β > 0 and g ∈ L2 (ΩT ), 2

⟨A(x, t, ξ) − A(x, t, η), ξ − η⟩ ⩾ α|ξ − η|

for some α > 0

(1.3) (1.4)

for a.e. (x, t) ∈ ΩT and for any ξ, η ∈ RN . Moreover, we assume that B = B(x, t, s) : ΩT × R → RN is a Carath´eodory function verifying the following properties |B(x, t, s) − B (x, t, s′ )| ⩽ b(x, t) |s − s′ |

(1.5)

B(x, t, 0) = 0

(1.6)

for a.e. x ∈ Ω , for any t ∈ (0, T ), for any s, s′ ∈ R and for some suitable nonnegative function b such that ( ) b ∈ L∞ 0, T ; LN,∞ (Ω ) .

(1.7)

In the linear homogeneous case, equation in (1.1) is also known as the Fokker–Planck equation and describes the evolution of some Brownian motion. If the convection term b = b(x, t) is bounded in ΩT , problem (1.1) has been studied by many authors (see e.g. [3] and the references therein). In many applications, the boundedness of the convective field is not available as underlined in [5] in the case of the diffusion model for semiconductor devices. Many definitions of solutions have been introduced for problem (1.1) under consideration as the renormalized solution (see e.g. [16]) and entropy solutions (see e.g. [2]). Here, we assume that conditions (1.5)–(1.7) hold true and we are interested in the existence and uniqueness of a weak solution, according to the following definition. Definition 1.1. We say that ( ) ( ) u ∈ L2 0, T ; W01,2 (Ω ) ∩ L∞ 0, T ; L2 (Ω ) is a weak solution to problem (1.1) if ∫ ∫ − u ∂t φ dx dt + ΩT

⟨A(x, t, ∇u) + B(x, t, u), ∇φ⟩ dx dt

ΩT

∫ ΩT

(1.8)

∫ ⟨F, ∇φ⟩ dx dt +

=

u0 φ(0) dx Ω

∞

for all φ ∈ C (ΩT ) with supp φ ⊂⊂ Ω × [0, T ). The only assumptions (1.5)–(1.7) do not infer in general the existence of such a solution. Indeed, if Ω is the unit ball of RN with N ⩾ 3 then we can see that the problem ] } ( ) {[ ⎧ ( ) ⎪ x 1 x N −2−δ ⎪ ⎪ ut − ∆u − div δ 2+ 1 − |x| x u = − div ⎪ N −δ ⎪ 2−N +δ ⎨ |x| |x| (1.9) in Ω × (0, T ), ⎪ ⎪ ⎪ ⎪ u=0 on ∂Ω × (0, T ), ⎪ ⎩ u(0) = 0 in Ω , does not admit a solution in the sense of Definition 1.1 if N2 − 1 < δ ⩽ N − 1 with δ ̸= N − 2. For further details see Example A.2 in the Appendix. Nevertheless, we can find a proper subset X(ΩT ) of the space ( ) L∞ 0, T ; LN,∞ (Ω ) such that if the conduction term b = b(x, t) belongs to X(ΩT ) then problem (1.1) is uniquely solvable. More precisely, we prove the following result.

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

399

Theorem 1.1 (Existence and Uniqueness). Let us assume that conditions (1.2)–(1.7) are fulfilled. There ( ) exists a proper convex subset X(ΩT ) of the space L∞ 0, T ; LN,∞ (Ω ) such that, if b ∈ X(ΩT ) ( ) ( ) then problem (1.1) admits a unique solution u ∈ L2 0, T ; W01,2 (Ω ) ∩ C [0, T ]; L2 (Ω ) . ( ) For the definition of X(ΩT ) see Section 2.2. The set X(ΩT ) is larger than L∞ 0, T ; LN,q (Ω ) whenever N ⩽ q < ∞. We see that b belongs to X(ΩT ) essentially if it can be decomposed as the sum of a bounded ( ) function and a L∞ 0, T ; LN,∞ (Ω ) function having a control on its norm. In the stationary case, a result comparable to Theorem 1.1 can be found in [9,10]. We also deal with the time behaviour of the solution to problem (1.1). Indeed, in Section 4 we give estimates on time of the L2 -norm of the solution in terms of the data. Besides the presence of the lower order term, if T = +∞ the L2 -norm of the solution decays exponentially fast, as estimate (4.8) reveals. In a forthcoming paper, we will consider pointwise estimates of the gradient in the spirit of [11,12]. 2. Notation and preliminary results 2.1. Lorentz spaces Let Ω be a bounded domain of RN . For any p, q ∈ (1, ∞), the Lorentz space Lp,q (Ω ) consists of all measurable functions g defined on Ω for which the quantity )1/q ( ∫ ∞ q q−1 p dh ∥g∥p,q = p |Ωh | h 0

is finite, where Ωh = {x ∈ Ω : |g(x)| > h} for any h > 0. Here and in what follows |E| stands for the Lebesgue measure of a measurable subset E of RN . Note that ∥ · ∥p,q is equivalent to a norm and Lp,q (Ω ) becomes a Banach space when endowed with it (see [15]). For p = q, the Lorentz space Lp,p (Ω ) reduces to the Lebesgue space Lp (Ω ). On the other hand, for q = ∞, the class Lp,∞ (Ω ) consists of all measurable functions g defined on Ω for which the quantity ∫ 1 −1 ∥g∥p,∞ = sup |E| p |g| dx E⊂Ω p,∞

is finite. The class L

E

(Ω ) coincides with the Marcinkiewicz class weak-Lp . Moreover, if we set 1

[[g]]p,∞ = sup h|Ωh | p h>0

one has p−1 1

p1+ p

∥g∥p,∞ ⩽ [[g]]p,∞ ⩽ ∥g∥p,∞

(see Lemma A.2 in [1]). For the Lorentz spaces the following inclusions hold Lr (Ω ) ⊂ Lp,q (Ω ) ⊂ Lp,r (Ω ) ⊂ Lp,∞ (Ω ) ⊂ Lq (Ω ) whenever 1 ⩽ q < p < r ⩽ ∞. Moreover, for 1 < p < ∞, 1 ⩽ q ⩽ ∞ and ′ ′ f ∈ Lp,q (Ω ) and g ∈ Lp ,q (Ω ), we have the H¨ older-type inequality ∫ |f (x)g(x)| dx ⩽ ∥f ∥p,q ∥f g∥p′ ,q′ . Ω

1 p

+

1 p′

= 1,

1 q

+

1 q′

= 1, if

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

400

It is well known that L∞ (Ω ) is not a dense subspace of Lp,∞ (Ω ). The distance to L∞ (Ω ) in Lp,∞ (Ω ) is defined as (see [4]) distLp,∞ (Ω) (f, L∞ (Ω )) =

inf

g∈L∞ (Ω)

∥f − g∥Lp,∞ (Ω) .

We conclude this section by recalling the Sobolev embedding theorem in the setting of Lorentz spaces. Theorem 2.1. Let us assume that 1 < p < N and 1 ⩽ q ⩽ p. Then, any function g ∈ W01,1 (Ω ) satisfying ∗ p and |∇g| ∈ Lp,q (Ω ) belongs to Lp ,q (Ω ) where p∗ = NN−p ∥g∥p∗ ,q ⩽ SN,p ∥∇g∥p,q where SN,p =

−1/N p ωN N −p

(2.1)

and ωN stands for the measure of the unit ball in RN .

2.2. The space L∞ (0, T ; Lp,∞ (Ω )) Let [0, T ] be a bounded interval of R which is provided with the usual Lebesgue measure denoted by dt and let Ω be a bounded open subset of RN . Given p ∈ (1, ∞), the class L∞ (0, T ; Lp,∞ (Ω )) is the space of measurable functions f : Ω × [0, T ] → R such that ∥f ∥L∞ (0,T ;Lp,∞ (Ω)) := ess sup ∥f (·, t)∥Lp,∞ (Ω) 0
is finite. For any given δ ⩾ 0, we consider the subset Xδ (ΩT ) of L∞ (0, T ; Lp,∞ (Ω )) defined as Xδ (ΩT ) := {f ∈ L∞ (0, T ; Lp,∞ (Ω )) : distL∞ (0,T ;Lp,∞ (Ω)) (f, L∞ (0, T ; L∞ (Ω ))) ⩽ δ}. In other words, Xδ (ΩT ) consists of all those functions f ∈ L∞ (0, T ; Lp,∞ (Ω )) such that there exists g ∈ L∞ (0, T ; L∞ (Ω )) such that ∥f − g∥L∞ (0,T ;Lp,∞ (Ω)) ⩽ δ. We remark that X0 (ΩT ) is the closure of L∞ (0, T ; L∞ (Ω )) in L∞ (0, T ; Lp,∞ (Ω )) and L∞ (0, T ; Lp,q (Ω )) ⊂ X(ΩT ) for p ⩽ q < ∞. To find a formula for the distance to L∞ (0, T ; L∞ (Ω )) in L∞ (0, T ; Lp,∞ (Ω )), we consider, for κ > 0, the truncation operator at level ±κ, namely s min{|s|, κ}. Tκ (s) = |s| Then distL∞ (0,T ;L∞ (Ω)) (f, L∞ (0, T ; L∞ (Ω ))) = lim ∥f − Tκ (f )∥L∞ (0,T ;Lp,∞ )(Ω) . κ→∞

∞

∞

Indeed, for every g ∈ L (0, T ; L (Ω )) and for every κ ⩾ ∥g∥L∞ (0,T ;L∞ (Ω)) we have for a.e. (x, t) ∈ ΩT |f (x, t) − g(x, t)| ⩾ |f (x, t) − Tκ f (x, t)|. The following lemma then follows. Lemma 2.2. Let δ ⩾ 0 be given. Then, f ∈ Xδ (ΩT ) if and only if lim ∥f − Tκ (f )∥L∞ (0,T ;Lp,∞ (Ω)) ⩽ δ.

κ→∞

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

401

2.3. Some useful asymptotic estimates We recall an abstract result whose application will be essential in the study of the time behaviour of the solution to (1.1). Proposition 2.3. Let t0 ⩾ 0 and T ∈ (t0 , +∞]. Assume that ϕ = ϕ(t) is a continuous and nonnegative function defined in [t0 , T ) verifying ∫ t2 ∫ t2 ϕ(t2 ) − ϕ(t1 ) + M ϕ(t) dt ⩽ g(t) dt t1

t1

for every t0 ⩽ t1 < t2 < T , where M is a positive constant and g is a nonnegative function belonging to L1 ([t0 , T )). Then, for every t ⩾ t0 we get ∫ t −M (t−t0 ) ϕ(t) ⩽ ϕ(t0 )e + g(s) ds. t0

Moreover, if T = +∞ and g belongs to L1 ([t0 , +∞)) there exists t1 > t0 such that ∫ t M g(s) ds ϕ(t) ⩽ Λe− 2 (t) + t/2

for every t ⩾ t1 , where ∫

+∞

Λ = ϕ(t0 ) +

g(s) ds. t0

The proof of the Proposition above is considered in [14] to study stability results in the spirit of [6,7]. 3. The main results 3.1. Existence and uniqueness of solutions Before we give the proof of Theorem 1.1, we consider the following auxiliary parabolic problem ⎧ vt − div [A(x, t, ∇v) + (1 − θ(x, t)) B(x, t, v)] = − div(F − θ(x, t)B(x, t, w)) ⎪ ⎪ ⎨ in ΩT ≡ Ω × (0, T ), ⎪ v = 0 on ∂Ω × (0, T ), ⎪ ⎩ v(0) = u0 (x) in Ω ,

(3.1)

where θ(x, t) =

TM (b(x, t)) b(x, t)

(3.2)

where TM is the usual truncation operator at level ±M . We simply denote by X(ΩT ) the set Xδ (ΩT ) as in Section 2.2 where ( ) α 1 1/N N 0<δ⩽ = ωN − 1 α. (3.3) 2SN,2 2 2 We assume that b ∈ X(ΩT ).

(3.4)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

402

We follow the strategy of [8–10]. Precisely, assumptions (3.3) and (3.4) allow us to choose M > 0 is chosen in such a way to ensure that sup ∥b(x, t) − TM (b(x, t))∥LN,∞ (Ω) ⩽

0
α . 2SN,2

(3.5)

We start by proving that the operator A : L2 (0, T ; W01,2 (Ω )) → L2 (0, T ; W −1,2 (Ω )) ) ( defined for u, v ∈ L2 0, T ; W01,2 (Ω ) as follows T

∫ ⟨Au, v⟩ :=

∫ ⟨A(x, t, ∇u) + (1 − θ(x, t)) B(x, t, u), ∇v⟩ dx,

dt Ω

0

is coercive because of condition (3.5). ( ) Lemma 3.1. Assume that conditions (3.3) and (3.4) hold true. Then, for each u, v ∈ L2 0, T ; W01,2 (Ω ) , one has ∫ α T 2 (3.6) ⟨Au − Av, u − v⟩ ⩾ ∥∇u − ∇v∥L2 (Ω) dt. 2 0 In particular, ⟨Au, u⟩ ⩾

α 2

∫ 0

T

2

∥∇u∥L2 (Ω) dt.

(3.7)

Proof . We use (1.5), (1.6) and the definition of θ given by (3.2) to get ⟨Au − Av, u − v⟩ ∫ T ∫ = dt ⟨A(x, t, ∇u) − A(x, t, ∇v) + (1 − θ(x, t)) B(x, t, u) − B(x, t, v), ∇(u − v)⟩ dx Ω

0

∫

T

⩾α

∫

∫

2

|∇u − ∇v| dx −

dt Ω

0

T

|b(x, t) − TM (b(x, t))| · |u − v| · |∇u − ∇v| dx.

dt 0

(3.8)

∫ Ω

Next, we use H¨ older inequality in the setting of Lorentz spaces, Sobolev inequality (2.1) and (3.5) to obtain ⟨Au − Av, u − v⟩ ∫ T ∫ 2 dt |∇u − ∇v| dx ⩾α Ω

0 T

∫ − 0 T

∫ ⩾α

∥b(x, t) − TM (b(x, t)) ∥LN,∞ (Ω) · ∥u − v∥L2∗ ,2 (Ω) · ∥∇u − ∇v∥L2 (Ω) dt ∫

2

|∇u − ∇v| dx

dt Ω

0

∫ − SN,2 0

α ⩾ 2

∫ 0

T

T

∥b(x, t) − TM (b(x, t)) ∥LN,∞ (Ω) · ∥∇u − ∇v∥2L2 (Ω) dt 2

∥∇u − ∇v∥L2 (Ω) dt

which proves (3.6). □

(3.9)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

403

Proof of Theorem 1.1. First, with the help of Schauder’s fixed point theorem, we establish the existence of a local weak solution of (1.1). Next, we will prove that this solution may be extended to an arbitrary large interval which leads to a global weak solution. Let T∗ ∈ (0, T ] be arbitrary chosen. We set { } ( ) M(T∗ ) := w ∈ L2 0, T∗ ; L2 (Ω ) : ∥w∥L2 (0,T∗ ;L2 (Ω)) ⩽ 1 . Then, appealing to the( classical theory operators, for every w ∈ M(T∗ ) there exists a unique ) of monotone ( ) 1,2 2 weak solution v ∈ L 0, T∗ ; W0 (Ω ) ∩ C [0, T∗ ); L2 (Ω ) to problem (3.1). Then, we define a mapping ( ) K : M(T∗ ) → L2 0, T∗ ; L2 (Ω ) by K(w) =: v is the unique solution to problem (3.1). STEP 1: We claim that ∥K(w)∥2L2 (0,T

)⩽1 ⩽C

2 ∗ ;L (Ω)

) ∥K(w)∥2 2 ( 1,2 L 0,T∗ ;W0 (Ω)

holds for all w ∈ M(T∗ ) and for some suitable constant C > 0. Testing by φ = vχ[0,t) in the equation in (3.1) with t ∈ (0, T∗ ), the integration by parts yields ∫ t ∫ 1 ∥v(t)∥2L2 (Ω) + ds ⟨A(x, s, ∇v) + (1 − θ(x, s)) B(x, s, v), ∇v⟩dx 2 0 Ω ∫ t ∫ 1 = ∥u0 ∥2L2 (Ω) + ds ⟨F − θ(x, s)B(x, s, w), ∇v⟩dx 2 0 Ω

(3.10) (3.11)

(3.12)

for any t ∈ (0, T∗ ). We deduce that ∫ t ∫ 1 ∥v(t)∥2L2 (Ω) + ds ⟨A(x, s, ∇v) − A(x, t, 0) dx 2 0 Ω ∫ t ∫ + ds (1 − θ(x, s)) (B(x, s, v) − B(x, t, 0)) , ∇v⟩ dx Ω

0

∫ t ∫ 1 ds ⟨F − θ(x, s)B(x, s, w), ∇v⟩dx = ∥u0 ∥2L2 (Ω) + 2 0 Ω ∫ t ∫ − ds ⟨A(x, s, 0), ∇v⟩dx. 0

(3.13)

Ω

Using (3.6), we have ∫ 1 α t 2 ∥v(t)∥2L2 (Ω) + ∥∇v∥L2 (Ω) ds 2 2 0 ∫ t ∫ 1 2 ⩽ ∥u0 ∥L2 (Ω) + ds ⟨F − θ(x, s)B(x, s, w), ∇v⟩dx 2 0 Ω ∫ t ∫ − ds ⟨A(x, s, 0), ∇v⟩dx. 0

(3.14)

Ω

We deduce by (1.3) and by H¨ older’s inequality we obtain ∫ t 1 α 2 ∥v(t)∥2L2 (Ω) + ∥∇v∥L2 (Ω) ds 2 2 0 ∫ t( ) 1 ⩽ ∥u0 ∥2L2 (Ω) + ∥F ∥L2 (Ω) + ∥θ(x, s)B(x, s, w)∥L2 (Ω) + ∥g(x, s)∥L2 (Ω) ∥∇v∥L2 (Ω) ds 2 0

(3.15)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

404

we use Young’s inequality and we get ∫ α t 1 2 2 ∥∇v∥L2 (Ω) ds ∥v(t)∥L2 (Ω) + 2 2 0 ∫ )2 1 1 t( ⩽ ∥u0 ∥2L2 (Ω) + ∥F ∥L2 (Ω) + ∥θ(x, s)B(x, s, w)∥L2 (Ω) + ∥g(x, s)∥L2 (Ω) ds 2 α 0 ∫ t α + ∥∇v∥2L2 (Ω) ds. 4 0

(3.16)

Now, in (3.16) we are able to reabsorb the latter term by the left-hand side, so we manage (3.16) and we obtain ∫ 1 α t 2 ∥∇v∥L2 (Ω) ds ∥v(t)∥2L2 (Ω) + 2 4 0 (3.17) ∫ ) 1 2 t( 2 2 2 2 ⩽ ∥u0 ∥L2 (Ω) + ∥F ∥L2 (Ω) + ∥θ(x, s)B(x, s, w)∥L2 (Ω) + ∥g(x, s)∥L2 (Ω) ds. 2 α 0 Since ∥θ(x, s)B(x, s, w)∥2L2 (Ω) ⩽ M 2 ∥w∥2L2 (Ω) one has ∥v(t)∥2L2 (Ω) + ⩽

∥u0 ∥2L2 (Ω)

⩽ γ1 +

α 4

t

∫

2 + α

2

∥∇v∥L2 (Ω) ds

0

T∗

∫

(

0

) ∥F ∥2L2 (Ω) + M 2 ∥w∥2L2 (Ω) + ∥g(x, s)∥2L2 (Ω) ds

(3.18)

γ2 ∥w∥2L2 (0,T ;L2 (Ω)) ∗

where γ1 := ∥u0 ∥2L2 (Ω) + γ2 :=

) 2( ∥F ∥2L2 (Ω ) + ∥g∥2L2 (Ω ) T T α

(3.19)

2 2 M α

(3.20)

and ∥v∥2L∞ (0,T

∗

+ ;L2 (Ω))

α 4

∫ 0

T∗

2

∥∇v∥L2 (Ω) dt ⩽ γ1 + γ2 ∥w∥2L2 (0,T

From (3.18) we deduce, integrating over (0, T∗ ), ∫ T∗ ∥v(t)∥2L2 (Ω) dt ⩽ γ1 T∗ + γ2 T∗ ∥w∥2L2 (0,T

)

2 ∗ ;L (Ω)

0

while, from (3.21), we deduce ∫ 0

T∗

2

).

2 ∗ ;L (Ω)

∥∇v∥L2 (Ω) dt ⩽

4γ2 4γ1 + ∥w∥2L2 (0,T ;L2 (Ω)) . ∗ α α

(3.21)

(3.22)

(3.23)

Thus, setting T∗ ⩽

1 min 2

{

1 1 , , 2T γ1 γ2

}

we finally prove that (3.10) and (3.11) hold for any w ∈ M(T∗ ). In particular, K maps M(T∗ ) into itself. STEP 2: Next, we prove that K : M(T∗ ) → M(T∗ ) is continuous and compact. For the distributional (time) derivative vt ≡ v ′ of v we have ( ) v ′ ∈ L2 0, T∗ ; W −1,2 (Ω ) .

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

405

Indeed, 2

∥v ′ ∥2L2 (0,T

∗ ;W −1,2 (Ω)

2

2

2

) ⩽ β ∥∇v∥L2 (ΩT∗ ) + ∥g∥L2 (ΩT∗ ) + ∥b(x, t)∥L∞ (0,T∗ ;) · ∥∇v∥L2 (ΩT∗ ) 2

2

+ ∥F ∥L2 (ΩT ) + M 2 ∥w∥L2 (ΩT ) (3.24) ∗ ∗ ) 4(γ1 + γ2 ) ( 2 2 2 β + ∥b(x, t)∥L∞ (0,T∗ ;) + ∥g∥L2 (ΩT ) + M 2 + ∥F ∥L2 (ΩT ) . ⩽ ∗ ∗ α By (3.10) and (3.11), we are in a position to apply Aubin–Lions Lemma; thus K (M(T∗ )) is relatively compact in L2 (0, T∗ ; L2 (Ω )). Now, we prove that K is continuous. Let {wn }n∈N be a sequence in M(T∗ ) which converges in 2 L (0, T∗ ; L2 (Ω )) to a function w ∈ M(T∗ ) as n → ∞. By the compactness of K(M(T∗ )) and relation (3.11), there exists a subsequence {wnk }n∈N of {wn }n∈N such that in L2 (0, T∗ ; L2 (Ω ))

K(wnk ) → v

2

K(wnk ) ⇀ v

in L

(0, T∗ ; W01,2 (Ω ))

(3.25) (3.26)

as n → ∞. Moreover, if we let vnk := K(wnk ), thanks to the assumptions in L2 (Ω × (0, T∗ ))n

˜ A(·, ∇vnk ) ⇀ A

B(x, t, vnk ) → B(x, t, v)

2

in L2 (Ω × (0, T∗ )).

(3.27) (3.28)

By the definition of K, the functions vnk satisfy the following integral identity ∫ T∗ ∫ ∫ T∗ ∫ − dt vnk ∂t φdx + dt ⟨A(x, t, ∇vnk ) + (1 − θ(x, t))B(x, t, vnk ), ∇φ⟩ dx Ω

0

∫

T∗

∫

dt

=

u0 φ(0)dx + Ω

0

Ω

0

∫

T∗

(3.29)

∫ dt

0

Ω

⟨F + θ(x, t)B(x, t, wnk ), ∇φ⟩ dx

for any φ ∈ C0∞ (Ω × [0, T∗ )). Then, with the aid of (3.25), (3.26), (3.27) and (3.28), by passing to the limit in (3.29) as k → ∞ we have ∫ T∗ ∫ ∫ T∗ ∫ ˜ + (1 − θ(x, t))B(x, t, v), ∇φ⟩ dx − dt v∂t φdx + dt ⟨A 0 Ω 0 Ω (3.30) ∫ ∫ ∫ ∫ T∗

T∗

dt

= 0

u0 φ(0)dx + Ω

⟨F + θ(x, t)B(x, t, w), ∇φ⟩ dx

dt 0

Ω

for any φ ∈ C0∞ (Ω × [0, T∗ )). It is not hard to prove (see Lemma A.1 in the Appendix) that ˜ t) = A(x, t, ∇v) A(x,

a.e. in Ω × (0, T∗ )

(3.31)

then v = K(w) and, noticing that this limit is unique, from (3.25)–(3.26) we get as k → ∞ K(wnk ) → K(w)

in L2 (Ω × (0, T∗ ))

thus K is continuous. STEP 3: Now, applying Schauder’s fixed point theorem one finds a function vT∗ ∈ M(T∗ ) such that K(vT∗ ) = vT∗ , that is to say that vT∗ is a weak solution to problem (1.1) in the cylinder Ω × (0, T∗ ). Testing Eq. (1.1) by vT∗ and integrating over the interval (0, T∗ ), using integration by parts, we get ∫ T∗ ∫ 1 2 ∥vT∗ (T∗ )∥L2 (Ω) + dt ⟨A(x, t, ∇vT∗ ) + (1 − θ(x, t)) B(x, t, vT∗ ), ∇vT∗ ⟩dx 2 0 Ω (3.32) ∫ T∗ ∫ 1 2 = ∥u0 ∥L2 (Ω) + dt ⟨F − θ(x, t)B(x, t, vT∗ ), ∇vT∗ ⟩dx 2 0 Ω

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

406

and so by (3.9) and (3.10) 1 1 ∥vT∗ (T∗ )∥2L2 (Ω) ⩽ ∥u0 ∥2L2 (Ω) + c(α) 2 2

T

∫

(

0

) ∥F ∥2L2 (Ω) + ∥g(x, t)∥2L2 (Ω) dt + c(α)M 2 .

Therefore, we are able to extend vT∗ to a weak solution of problem (1.1) in Ω × [0, T ]. STEP 4: uniqueness of the solution. ) ( ( ) 2 Let u, v ∈ L 0, T ; W01,2 (Ω ) ∩ C [0, T ]; L2 (Ω ) be solutions to problem (1.1). Then, by using w = u − v as a test function in the equation appearing in (1.1) and subtracting we get ∫ ∫ 1 α t 2 ds ∥w(t)∥2L2 (Ω) + |∇w| dx 2 2 0 Ω ∫ t ∫ ds θ(x, s)⟨B(x, s, v) − B(x, s, u), ∇w⟩ dx ⩽ ∫

t

∫ |TM (b(x, s))| · |u − v| · |∇w| dx.

ds

+ 0

(3.33)

Ω

0

Ω

Now, by (1.5), (1.6) and Young’s inequality ∫ ∫ 1 α t 2 ∥w(t)∥2L2 (Ω) + ds |∇w| dx 2 4 0 Ω ∫ t ∫ 2 2 ⩽ C(α) ds |TM (b(x, t))| · |w| dx

(3.34)

Ω

0

and so 1 ∥w(t)∥2L2 (Ω) ⩽ CM 2 2

∫

t

(3.35)

∥w(s)∥L2 (Ω) ds. 0

Then, by the Gronwall inequality, since w(0) = 0 ∥w(t)∥L2 (Ω) = 0

for a.e. t ∈ (0, T )

and therefore u = v a.e. in ΩT . □

4. Time behaviour of the solution We begin with the following lemma (see also [2,13]). Lemma 4.1. The solution u of (1.1) satisfies sup |{x ∈ Ω : |u(x, t)| > k}| ⩽ c(α)

∥u0 ∥2L2 (Ω) + ∥F ∥2L2 (Ω

+ ∥g∥2L2 (Ω

log2 (1 + k)

t∈(0,T )

for each k > 0. Proof . We fix T∗ ∈ (0, T ) and we choose φ=

T)

u χ(0,T∗ ) 1 + |u|

T)

+ ∥b∥2L2 (Ω

T)

(4.1)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

407

as a test function for Eq. (1.1). We observe that ∇φ =

∇u χ(0,T∗ ) . (1 + |u|)2

Therefore we get ∫

T∗

∫

∫

dt

T∗

ut φ dx + Ω

0

∫

[A(x, t, ∇u)∇φ + B(x, t, u)∇φ] dx

dt Ω

0

T∗

(4.2)

∫ F ∇φ dx.

dt

=

∫

Ω

0

We rearrange the equation above to get ] ∫ [∫ u(x,T∗ ) ∫ T∗ ∫ r dr dx + dt [A(x, t, ∇u) − A(x, t, 0)] ∇φ dx 1 + |r| Ω 0 0 Ω ] ∫ T∗ ∫ ∫ [∫ u0 (x) r dr dx = dt F ∇φ dx + 1 + |r| 0 Ω Ω 0 ∫ T∗ ∫ − dt [A(x, t, 0)∇φ + B(x, t, u)∇φ] dx. 0

(4.3)

Ω

A direct computation gives us ∫ 0

s

r dr = |s| − log(1 + |s|) 1 + |r|

and, since ew − 1 − w ⩾ 12 w2 for any w ⩾ 0, we have ∫ s r 1 dr ⩾ log2 (1 + |s|). 2 0 1 + |r| From (4.3) we have 1 2

T∗

2

|∇u| dx (1 + |u|)2 Ω 0 Ω )1/2 (∫ )1/2 ∫ T∗ (∫ 2 2 |∇u| |F | ⩽ dx dx dt + ∥u0 ∥22 2 2 0 Ω (1 + |u|) Ω (1 + |u|) )1/2 (∫ )1/2 ∫ T∗ (∫ 2 2 |g| |∇u| + dx dx dt 2 2 0 Ω (1 + |u|) Ω (1 + |u|) ∫ T∗ ∫ dx + b(x, t)|u||∇u| dt. (1 + |u|)2 0 Ω

∫

log2 (1 + |u(x, T∗ )|) dx + α

∫

∫

dt

We observe that, since N > 2, we have b(·, t) ∈ L2 (Ω ); from the fact that inequality gives us T∗

|u| 1+|u|

⩽ 1 and

(4.4)

1 1+|u|

2

|∇u| dx (1 + |u|)2 Ω 0 Ω ∫ T∗ ∫ ∫ 2 ( ) α T∗ 3 |∇u| ⩽ ∥F ∥22 + ∥g∥22 + ∥b(x, t)∥22 dt + dt dx + ∥u0 ∥22 2 2α 0 2 0 (1 + |u|) Ω

1 2

∫

log2 (1 + |u(x, T∗ )|) dx + α

∫

∫

⩽ 1, Young’s

dt

(4.5)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

408

therefore we have 1 2

T∗

2

|∇u| dx 2 Ω 0 Ω (1 + |u|) [∫ ] ∫ T∗ ∫ T∗ T∗ 2 2 2 2 ⩽ c2 (α) ∥F ∥2 dt + ∥u0 ∥2 + ∥g∥2 dt + ∥b(x, t)∥2 dt .

∫

log2 (1 + |u(x, T∗ )|) dx + c1 (α)

0

∫

∫

dt

0

(4.6)

0

This estimate in turn implies (4.1). □ We are in position to state a result concerning the L2 -norm of the solution to problem (1.1). Theorem 4.2 (Behaviour of the Solution). Let all the assumptions of Theorem 1.1 be fulfilled. There exist positive constants M and C0 such that the following estimate ∫ t ∥u(t)∥2L2 (Ω) ⩽ ∥u0 ∥2L2 (Ω) e−M t + C0 h(s) ds (4.7) 0

holds for every t > 0. Here h(t) = ∥F (t)∥2L2 (Ω) + ∥g(t)∥2L2 (Ω) + ∥b(t)∥2L2 (Ω) . Moreover, if h ∈ L1 ((t0 , +∞)) for some t0 ⩾ 0, then the following estimate ∫ ( ) M ∥u(t)∥2L2 (Ω) ⩽ ∥u0 ∥2L2 (Ω) + ∥h∥L1 ((t0 ,+∞)) e− 2 t + C0

t

h(s) ds

(4.8)

t/2

holds for every t ⩾ t0 . Proof . First, we rewrite equation (1.1) as follows ut − div [A(x, t, ∇u) + (1 − θ(x, t))B(x, t, u)] = − div(F − θ(x, t))B(x, t, u) and we use the solution u itself as a test function. Arguing as before, for every t1 , t2 ∈ (0, T ) with t1 < t2 we obtain ∫ t2 ∫ t2 ( ) 1 1 ∥u(t2 )∥22 − ∥u(t1 )∥22 + c1 (α) ∥∇u∥22 dt ⩽ c2 (α) ∥F ∥22 + ∥g∥22 dt 2 2 t1 t1 (4.9) ∫ t2 ∫ + dt TM (b(x, t)) |u| · |∇u| dx. t1

Ω

The latter term can be estimated as follows ∫ ∫ ∫ TM (b(x, t)) |u| · |∇u| dx = TM (b(x, t)) |u| · |∇u| dx + |u|⩽k

Ω

∫ ⩽k

TM (b(x, t)) |u| · |∇u| dx

|u|>k

∫ |u| · |∇u| dx

b(x, t)|∇u| dx + M |u|⩽k

|u|>k

(4.10) c1 (α) 1/N 2 ∥∇u∥2 + M |Ek | ∥u∥2∗ ,2 ∥∇u∥2 ⩽ + 4 c1 (α) 1/N ⩽ c5 (α, k)∥b∥22 + ∥∇u∥22 + M |Ek | ∥∇u∥22 4 where we have defined Ek := {|u| > k}. We use (4.1) to observe that |Ek | → 0 as k → ∞. Therefore, if 1/N k > 0 is sufficiently large, we have M |Ek | ⩽ c2 4(α) , which means that we can reabsorb by the left-hand side the latter two terms in (4.9). We finally get ∫ t2 ∫ t2 ( ) 1 1 ∥u(t2 )∥22 − ∥u(t1 )∥22 + c6 ∥∇u∥22 dt ⩽ c7 ∥F ∥22 + ∥g∥22 + ∥b∥22 dt. (4.11) 2 2 t1 t1 c5 (α, k)∥b∥22

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

409

Finally, by means of Poincar´e inequality, we have 1 1 ∥u(t2 )∥22 − ∥u(t1 )∥22 + c8 2 2

∫

t2

∥u∥22

t2

∫

(

dt ⩽ c7

t1

) ∥F ∥22 + ∥g∥22 + ∥b∥22 dt.

(4.12)

t1

The assertion is a consequence of Proposition 2.3 whenever ϕ(t) = ∥u(t)∥2L2 (Ω) . □ Acknowledgement The Authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Appendix ˜ be as in (3.27). Then (3.31) holds Lemma A.1. Let all assumptions of Theorem 1.1 be fulfilled and let A true. Proof . Since (3.27) and (3.28) hold, we are allowed to pass to the limit in (3.29) and we get T∗

∫ −

∫

∫

dt

v∂t φdx + Ω

0 T∗

∫

˜ t) + (1 − θ(x, t))B(x, t, v), ∇φ⟩ dx ⟨A(x,

dt Ω

∫ u0 φ(0)dx +

Ω

0

∫

0

∫ dt

=

T∗

T∗

(A.1)

∫ ⟨F + θ(x, t)B(x, t, w), ∇φ⟩ dx

dt 0

Ω

for any φ ∈ C0∞ (Ω ×[0, T∗ )). In particular, (A.1) holds if we pick v in place of φ as a test function. Therefore, we get ∫ T∗ ∫ 1 ˜ t) + (1 − θ(x, t))B(x, t, v), ∇v⟩ dx dt ⟨A(x, − ∥v(T∗ )∥22 + 2 0 Ω ∫ T∗ ∫ = dt ⟨F + θ(x, t)B(x, t, w), ∇v⟩ dx.

(A.2)

Ω

0

On the other hand, (3.29) holds if we pick vnk in place of φ as a test function. Therefore, we get ∫ T∗ ∫ 1 − ∥vnk (T∗ )∥22 + dt ⟨A(x, t, ∇vnk ) + (1 − θ(x, t))B(x, t, vnk ), ∇vnk ⟩ dx 2 0 Ω ∫ T∗ ∫ = dt ⟨F + θ(x, t)B(x, t, wnk ), ∇vnk ⟩ dx.

(A.3)

Ω

0

We pass to the limit in the latter relation and, with the aid of (A.2) we obtain ∫

T∗

∫ dt

lim

k→∞

Ω

0

∫ ⟨A(x, t, ∇vnk ), ∇vnk ⟩ dx =

T∗

∫ ˜ t), ∇v⟩ dx. ⟨A(x,

dt 0

(A.4)

Ω

( ) For our purposes, it will be sufficient to prove that for each ψ ∈ L2 0, T∗ ; W01,2 (Ω ) one has ∫ ˜ t) − A(x, t, ∇ψ), ∇v − ∇ψ⟩ dx dt ⩾ 0. ⟨A(x, Ω×(0,T∗ )

(A.5)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

410

If we set for simplicity vk = vnk , we have ∫ ˜ t) − A(x, t, ∇ψ), ∇v − ∇ψ⟩ dx dt ⟨A(x, Ω×(0,T∗ )

∫ ˜ t) − A(x, t, ∇vk ), ∇v − ∇ψ⟩ dx dt ⟨A(x,

= Ω×(0,T∗ )

∫

∫ ⟨A(x, t, ∇vk ), ∇v⟩ dx dt −

+ Ω×(0,T∗ )

⟨A(x, t, ∇vk ), ∇vk ⟩ dx dt

(A.6)

Ω×(0,T∗ )

∫ −

⟨A(x, t, ∇ψ), ∇v − ∇vk ⟩ dx dt Ω×(0,T∗ )

∫ ⟨A(x, t, ∇vk ) − A(x, t, ∇ψ), ∇vk − ∇ψ⟩ dx dt.

+ Ω×(0,T∗ )

According to (1.4) we have ∫ ⟨A(x, t, ∇vk ) − A(x, t, ∇ψ), ∇vk − ∇ψ⟩ dx dt ⩾ 0

for each k ⩾ 1

(A.7)

Ω×(0,T∗ )

and, according to (3.27) and (3.26) respectively we have ∫ ˜ t) − A(x, t, ∇vk ), ∇v − ∇ψ⟩ dx dt → 0 ⟨A(x,

as k → ∞

Ω×(0,T∗ )

∫ ⟨A(x, t, ∇ψ), ∇v − ∇vk ⟩ dx dt → 0

as k → ∞.

Ω×(0,T∗ )

On the other hand, according to (3.27) and (A.4) we have ∫ ∫ ⟨A(x, t, ∇vk ), ∇v⟩ dx dt − ⟨A(x, t, ∇vk ), ∇vk ⟩ dx dt → 0 Ω×(0,T∗ )

as k → ∞.

Ω×(0,T∗ )

Thus, taking into account (A.6) and (A.7) and passing to the limit as k → ∞, we conclude that (A.5) holds true. □ Example A.2. Let Ω be the unit ball of RN (which will be also denoted by B1 ) with N ⩾ 3. We fix T > 0 and N − 1 < δ ⩽ N − 1. 2

(A.8)

Let us consider for moment the further assumption that δ ̸= N − 2. We claim that the problem ⎧ {[ ] } ( ) ( ) ⎪ x 1 x ⎪ N −2−δ ⎪ u − ∆u − div δ 2+ 1 − |x| x u = − div ⎪ ⎪ N −δ ⎨ t 2−N +δ |x| |x| in Ω × (0, T ), ⎪ ⎪ ⎪ u = 0 on ∂Ω × (0, T ), ⎪ ⎪ ⎩ u(0) = 0 in Ω , does not admit a solution in the sense of Definition 1.1.

(A.9)

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

As a preliminary step, let us consider the function v defined as follows ⎧ ( ) 1 2−N +δ ⎨ 1 − |x| for δ ̸= N − 2 2−N +δ v(x) = ⎩ − log|x| for δ = N − 2.

411

(A.10)

A direct calculation gives us ∇v(x) = −

x N −δ

|x|

therefore, condition (A.8) on the parameter δ gives that v actually belongs to W01,2 (Ω ). We also emphasize that v is a nontrivial solution to problem ⎧ ⎨ −∆v + δ x · ∇v = 0 in Ω , 2 (A.11) |x| ⎩ v=0 on ∂Ω . Suppose that a solution to (A.9) exists. If it is the case, we may test equation (A.9) with the following function φ(x, t) = v(x)e−t χ(t1 ,t2 ) (t) where 0 < t1 < t2 < T . Appealing to the fact that v is a solution to (A.11) we immediately get ∫ t2 ∫ 2 dt e−t |∇v| dx = 0 t1

(A.12)

(A.13)

Ω

which leads to a contradiction. Following the same lines as before, one can show that, if δ =N −2 then the problem ⎧ {[ ] } ( ) ⎪ x x x ⎪ ⎪ u − ∆u − div (N − 2) 2 + G(x) u = − div ⎪ 2 ⎪ ⎨ t |x| |x| |x| in Ω × (0, T ), ⎪ ⎪ ⎪ u = 0 on ∂Ω × (0, T ), ⎪ ⎪ ⎩ u(0) = 0 in Ω , does not admit a solution in the sense of Definition 1.1, where { |x| log|x| for x ∈ Ω \ {0}, G(x) = 0 for x = 0

(A.14)

(A.15)

which is still an L∞ function in the unit ball. □ Remark A.3. A rigorous argument to prove that (A.9) does not admit a solution can be obtained by testing with φk (x, t) = v(x)e−t ηk (t)

(A.16)

in place of the function in (A.12). Here v is the same as in (A.10) while, for k ⩾ 1 large enough, we denote by ηk a cut-off function with the following properties ηk ∈ C0∞ (R) 0 ⩽ ηk (t) ⩽ 1 ηk (t) = 1 ηk (t) = 0

for every t ∈ R for every t ∈ [t1 , t2 ] ] [ 1 1 . for every t ∈ R \ t1 − , t2 + k k

F. Farroni, G. Moscariello / Nonlinear Analysis 177 (2018) 397–412

412

So, if we test equation in (A.9) with φk we obtain ∫ T ∫ ∫ 2 −t ηk (t)e dt |∇v| dx = 0

Ω

ηk′ (t)e−t u(x, t)v(x) dxdt

ΩT

then, passing to the limit as k → ∞ we obtain the same contradiction as in (A.13). Remark A.4. As a matter of fact, the conductive term b = b(x, t) of both examples can be decomposed as follows b(x, t) =

δ + G(x) |x|

where G ∈ L∞ and bδ (x) :=

δ ∈ LN,∞ (B1 ) |x|

1/N

with ∥bδ ∥LN,∞ (B1 ) = distLN,∞ (B1 ) (bδ , L∞ (B1 )) = δωN .

References

[1] P. B´ enilan, H. Brezis, M. Crandall, A semilinear equation in L1 (RN ), Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 2 (4) (1975) 523–555. [2] L. Boccardo, L. Orsina, A. Porretta, Some noncoercive parabolic equations with lower order terms in divergence form, J. Evol. Equ. 3 (3) (2003) 407–418. Dedicated to Philippe B´ enilan. [3] P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, A. Porretta, Long time average of mean field games, Netw. Heterog. Media 7 (2) (2012) 279–301. [4] M. Carozza, C. Sbordone, The distance to l∞ in some function spaces and applications, Differential Integral Equations 10 (4) (1997) 599–607. [5] W. Fang, K. Ito, Weak solutions for diffusion-convection equations, Appl. Math. Lett. 13 (3) (2000) 69–75. [6] F. Farroni, L. Greco, G. Moscariello, Estimates for p–Laplace type equation in a limiting case, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25 (4) (2014) 445–448. [7] F. Farroni, L. Greco, G. Moscariello, Stability for p-Laplace type equation in a borderline case, Nonlinear Anal. 116 (2015) 100–111. [8] F. Giannetti, L. Greco, G. Moscariello, Linear elliptic equations with lower–order terms, Differential Integral Equations 26 (5–6) (2013) 623–638. [9] L. Greco, G. Moscariello, G. Zecca, An obstacle problem for noncoercive operators, Abstr. Appl. Anal. (2015) 890289, 8 pp. [10] L. Greco, G. Moscariello, G. Zecca, Very weak solutions to elliptic equations with singular convection term, J. Math. Anal. Appl. 457 (2) (2018) 1376–1387. [11] T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal. 212 (3) (2014) 727–780. [12] T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc. (JEMS) 16 (4) (2014) 835–892. [13] G. Moscariello, Existence and uniqueness for elliptic equations with lower-order terms, Adv. Calc. Var. 4 (4) (2011) 421–444. [14] G. Moscariello, M.M. Porzio, Quantitative asymptotic estimates for evolution problems, Nonlinear Anal. 154 (2017) 225– 240. [15] R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963) 129–142. [16] A. Porretta, Weak solutions to Fokker–Planck equations and mean field games, Arch. Ration. Mech. Anal. 216 (1) (2015) 1–62.