Qualitative results for parabolic equations involving the p –Laplacian under dynamical boundary

North-Western European Journal of Mathematics

W N

M

E J

Qualitative results for parabolic equations involving the p –Laplacian under dynamical boundary

conditions

Joachim von Below¹ Mabel Cuesta¹ Ga¨elle Pincet Mailly¹

Received: October 7, 2016/Accepted: June 22, 2017/Online: March 21, 2018

Abstract

We discuss comparison principles, the asymptotic behaviour, and the occur- rence of blow up phenomena for nonlinear parabolic problems involving the p–Laplacian operator of the form











∂_tu=∆_pu+f(t, x, u) inΩ for t >0, σ ∂_tu+|∇u|^p−2∂_νu= 0 on∂Ω for t >0,

u(0,·) =u₀ inΩ,

whereΩis a bounded domain ofR^N with Lipschitz boundary, and where

∆_pu:= div

|∇u|^p−2∇u

is the p–Laplacian operator for p >1. As for the dynamical time lateral boundary conditionσ ∂_tu+|∇u|^p−2∂_νu= 0the coefficientσ is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearityf(·,·, u) =λ|u|^q−2uwill be investigated by means of the evolution of associated norms.

Keywords: Nonlinear degenerate parabolic problems,p–Laplacian, dynamical bound- ary conditions, blow up, comparison principles.

msc: Primary 35K92, 35B40, 35K10; Secondary 35B44, 35K57, 35K65, 35R45..

1 Introduction

This paper deals with the behaviour of solutions of nonlinear parabolic problems of the form

(Pσ ,f)











∂_tu=∆_pu+f(t, x, u) inΩ for t >0, σ ∂_tu+|∇u|^p−2∂_νu= 0 on∂Ω for t >0,

1LMPA Joseph Liouville ULCO, Universit´es Lille Nord de France, 50, rue F. Buisson, CS 80699, F–62228 Calais

where Ωis a bounded domain ofR^N,N ≥1, with Lipschitz boundary, and where

∆_pu:= div

|∇u|^p−2∇u

is the well knownp–Laplacian operator defined inW^1,p(Ω)in a weak setting in the usual way for any real number p >1. Another distinctive feature in the present context is the dynamical boundary condition imposed on the time lateral boundary relating the outer normal derivative to the time derivative. For the sake of simplicity, the dynamical coefficientσ is assumed to be a nonnegative constant.

Of particular interest will be the Cauchy problem

(Pσ ,f_λ,q,u₀)















∂_tu=∆_pu+λ|u|^q−2u in Ω for t >0, σ ∂_tu+|∇u|^p−2∂_νu= 0 on∂Ω for t >0,

u(0,·) =u₀ in Ω,

where λis a real parameter andq >1.

The classical heat equation, i.e. p= 2, and reaction–diffusion equations under dynamical boundary conditions have been intensively studied onL^q-spaces or spaces of continuous functions, see e.g.² and the references therein. We refer also to³, where Escher proves that the heat equation generates a strongly continuous analytic semigroup onL^q(Ω)×W^1−1/q,q(∂Ω)forq > N for more general quasilinear equations.

Some special cases of (P_{σ ,f}) for p >1 have been considered recently by Gal⁴, Gal and Warma⁵ and Showalter⁶, where the generation of the correspondingC⁰- semigroups is shown. In fact, generation of C⁰-semigroups in L²(Ω) of the p- Laplacian heat equation under Dirichlet, Neumann and Robin boundary conditions was already studied by J.L. Lions⁷. Boundedness and higher H¨older–regularity have been extensively treated by DiBenedetto⁸in the homogeneous case, i.e. f ≡0.

2Bandle, Below, and Reichel, 2006, “Parabolic problems with dynamical boundary conditions:

eigenvalue expansions and blow up”;

Below and De Coster, 2000, “A qualitative theory for parabolic problems under dynamical boundary conditions”;

Below and Pincet Mailly, 2003, “Blow up for reaction diffusion equations under dynamical boundary conditions”;

Hintermann, 1989, “Evolution equations with dynamic boundary conditions”;

Pincet, 2001, “EDP sous des conditions de bords dynamiques”;

V´azquez and Vitillaro, 2009, “On the Laplace equation with dynamical boundary conditions of reactive-diffusive type”.

3Escher, 1993, “Quasilinear parabolic systems with dynamical boundary conditions”.

4Gal, 2012, “On a class of degenerate parabolic equations with dynamic boundary conditions”.

5Gal and Warma, 2010, “Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions”.

6Showalter, 1997, Monotone operators in Banach space and nonlinear partial differential equations.

7Lions, 1969,Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires.

8DiBenedetto, 1993,Degenerate parabolic equations.

1. Introduction

Recently, Cipriani and Grillo⁹ have shown generation of C⁰–semigroups for the p–Laplacian under Dirichlet boundary conditions onL^q–spaces and have obtained some ultracontractivity properties of the associated semigroups.

Beyond the approaches to local existence and higher regularity of weak solutions, the existence or exclusion of global solutions, as well as the occurrence of blow up phenomena for problem(P_{σ ,f})are of particular interest. In the works¹⁰ the authors dealt with the linear principal part, i.e. p= 2, and with different nonlinearitiesf. A main aim in the present paper is to generalize some of the results from these references to thep–Laplacian heat equation under dynamical boundary conditions.

We also prove some ultra conductivity bounds of the solutions for problem (P_{σ ,0}) that seem not to be available in the literature yet. Moreover, the absence of general boundedness result for weak solutions of (Pσ ,f) is one of the major difficulties in establishing the qualitative properties that we present in this work. Unfortunately, the existing results as e.g. the aforementioned ones¹¹ do not apply to the equations considered here. More recent existence results by Li and You, see¹² and the references therein, can be applied to some of the problems(Pσ ,f). However, as it stands, the aim of the present paper is not to deal with existence results, but to detail qualitative properties of solutions.

The present paper is organized as follows. In Section 2 the notion of weak solutions and of upper and lower weak solutions is made precise. Moreover, weak comparison principles are shown under a generalized one–sided Lipschitz condition imposed tof given in Definition 4 and involving a Lipschitz constant that can depend on the solutions. In particular, it will be shown in Theorem 2 that a weak lower solutionu₁ of(P_{σ ,f ,u}1

0)and a weak upper solution u₂ of(P_{σ ,f ,u}2

0)whose initial data satisfy u₀¹≤u₀² a.e. inΩ and on∂Ω, maintain the same inequality for t >0. We also compare solutions under different boundary conditions, namely homogeneous Dirichlet boundary conditions vs. dynamical ones, see e.g. Theorem 4.

Sections 3 and 4 are devoted to the evolution of associated norms and energy functionals. For time independent nonlinearitiesf under a specific Lipschitz condition (11), the energy of weak solutionsuof(P_{σ ,f ,u0}), E_F(u) =¹_pk∇uk^p_p−R

Ω

Ru

0 f(·, z)dz dx

9Cipriani and Grillo, 2001, “Uniform bounds for solutions to quasilinear parabolic equations”.

10Bandle, Below, and Reichel, 2006, “Parabolic problems with dynamical boundary conditions:

eigenvalue expansions and blow up”;

Below and Pincet Mailly, 2003, “Blow up for reaction diffusion equations under dynamical boundary conditions”;

Pincet, 2001, “EDP sous des conditions de bords dynamiques”.

11Escher, 1993, “Quasilinear parabolic systems with dynamical boundary conditions”;

Gal, 2012, “On a class of degenerate parabolic equations with dynamic boundary conditions”;

Gal and Warma, 2010, “Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions”.

12K. Li and You, 2013, “Uniform attractors for the non-autonomousp-Laplacian equations with dynamic flux boundary conditions”.

will be shown to fulfil an identity of the form

−d

dtE_F(u(t,·)) =k∂_tu(t,·)k²

2+σk∂_tu|_∂Ω(t,·)k²

2,∂Ω. (12)

For the special casef =fλ,q=λ|u|^q−2uwithλ <0, the evolution of theL²–norms inΩ and on∂Ω, can be completely controlled leading tolimt→∞ku(t,·)k_X2= 0. In particular, forp <2, the solutions vanish to0in finite time, see Theorem 2.

In Section 5 the behaviour in the large of weak solutions u of (Pσ ,f_λ,q,u₀)with λ≤0is investigated. Forλ <0andp >2, it turns out that an analogue to Berryman and Holland’s asymptotic result for the porous media equation¹³ holds, i.e. there is a sequence of time steps(t_n)_n∈Ntending to ∞such that

nlim→∞

k(1 + (p−2)t_n)^p2−2u(t_n,·)−wk_X2= 0, (Theorem 6) where w∈W^1,p(Ω) is the solution of the elliptic equation −∆_pw−λ|w|^p−2w=w under the Robin–Steklov boundary condition stemming from the dynamical one.

Another distinctive asymptotic result in this section deals with the solutions for the homogeneous problem(Pσ ,0,u₀)and says that

tlim→∞u(t,·) = R

Ωu₀dx+σH

∂Ωu₀dρ

|Ω|+σ|∂Ω| (Theorem 7)

inW^1,p(Ω). Note that this asymptotic formula is exactly the same as for the classical Laplacian¹⁴, i.e. p= 2, established via Fourier expansion of the initial data, that, however, does not apply forp,2. For initial data belonging toW^1,p(Ω)this holds also with respect to theL^∞–norm, see Theorem 8, and will be very useful in order to guarantee strict positivity a.e. in finite time, e.g. when showing the exclusion of global existence, see Theorem 10 without using a strong parabolic minimum principle.

It turns out thatλ= 0 plays the same role under dynamical boundary conditions as the first eigenvalueλ₁ does for homogeneous Dirichlet boundary conditions as it has been shown in¹⁵. For short, forλ≤0the weak solutions are bounded for each t >0, while forλ >0blow up occurs forq=p >2and forq >max{2, p}. Closing this section, we deduce global existence close to an equilibrium fulfillingf⁰(B)<0in the general autonomous case∂tu=∆_pu+f(u), see Theorem 9.

Section 6 deals with the occurrence of blow up phenomena with respect to the L^∞–norm. First, we generalize the result on the non existence of global solutions from¹⁶ to thep–Laplacian with nonlinearitiesf(t, x, u) =m(t, x)g(u)under the same

13Berryman and Holland, 1980, “Stability of the separable solution for fast diffusion”.

14Bandle, Below, and Reichel, 2006, “Parabolic problems with dynamical boundary conditions:

eigenvalue expansions and blow up”.

15Y. Li and Xie, 2003, “Blow-up forp-Laplacian parabolic equations”.

16Bandle, Below, and Reichel, 2006, “Parabolic problems with dynamical boundary conditions:

eigenvalue expansions and blow up”.

1. Introduction

assumptions as in¹⁷, see Theorem 10. As for Problem (P_{σ ,fλ,q,u0}) with λ >0 and q >2, it remains to determine the behaviour in the large. In fact, for p=q and for initial energy ¹_pR

Ω(|∇u0|^p− |u0|^p)dx <0 it will be shown that solutions blow up in finite time, see Theorem 11. For q >max{2, p} we first adopt a technique developed in¹⁸ in order to establish an upper bound for the blow up time under homogeneous Dirichlet boundary conditions for initial data0,u0∈W₀^1,p(Ω)∩L^∞(Ω) with energy ¹_pR

Ω|∇u₀|^pdx−^λ

q

R

Ω|u₀|^qdx≤0, see Theorem 12. Then the comparison techniques from Section 2 apply in order to establish the occurrence of blow up under dynamical boundary conditions too, see Theorems 13 and Corollary 5 for nonlinearities satisfyingf(·,·, z)≥λ|z|^q−2z. In the latter one it will be shown that for initial datau₀∈ W₀^1,p(Ω)∩L^∞(Ω; [0,∞))with nonpositive energy theL²–norm of the weak solution of(P_{σ ,f ,u0})blows up at the latest at timeT with

Tmax(u)≤T ≤ q

λ(q−2)(q−p)|Ω|^q

−2 2

Z

Ω

|u0|²dx

!²

−q 2

, (Corollary 5)

whereTmax(u)is the maximal existence time with respect to theL^∞–norm. Finally, we present an optimal upper bound for the blow up time under the Neumann bound- ary condition based on the evolution of theL¹–norm.

We close this introduction with some notations. We shall denote by dρ the restriction to ∂Ω of the (N −1)–dimensional Hausdorff measure, which coincides with the usual Lebesgue hyper-surface measure, since∂Ωis supposed to be Lipschitz.

Moreover, we shall denote byν=ν(x)its outer normal vector field atx∈∂Ωdefined ρ-a.e. in ∂Ω.

The Lebesgue norm ofL^q(Ω)will be denoted byk · k_q, and the Lebesgue norm of L^q(∂Ω, ρ)byk · k_q,∂Ω, forq∈[1,∞]. The scalar product ofL²(Ω)will be denoted by h·,·iand the scalar product ofL²(∂Ω, ρ)will be denoted byh·,·i₀ :

hu, vi= Z

Ω

uv dx, hu, vi₀= I

∂Ω

uv dρ.

The conjugate of any r ∈[1,∞] will be denoted by r⁰. The critical Sobolev exponent for the embedding W^1,p(Ω),→L^q(Ω) will be denoted by p^∗ := _N^pN−p if 1< p < N, and byp^∗=∞otherwise. It is worth noting that

W^1,p(Ω),→L²(Ω)⇐⇒p≥p0:= 2N N+ 2.

17Ibid.

18Ball, 1977, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations”;

Below and Pincet Mailly, 2003, “Blow up for reaction diffusion equations under dynamical boundary conditions”.

Thetraceu|_∂Ω of any functionu∈W^1,p(Ω)is well defined since∂Ωis regular enough.

We recall that, ifγ denotes the trace operator, thenγ(W^1,p(Ω)) =W^1−1/p,p(∂Ω, ρ). Moreover, the trace operator W^1,p(Ω)→ L^q(∂Ω, ρ) is continuous if and only if 1≤q≤p∗ ifp,N and for1≤q <∞ifp=N. Recall thatp∗:=^p(N_N−⁻p¹⁾ if1< p < N, and thatp∗=∞ifp≥N. Note that forq= 2, the trace operator is well–defined and continuous under the following condition:

W^1,p(Ω)→L²(∂Ω, ρ) ⇐⇒ p≥p₁:= 2N N+ 1.

Finally, for a reflexive Banach space(V ,k · k_V)andr∈[1,∞), the classical Bochner spaceL^r((0, T);V)will be endowed with the norm

kuk_Lr((0,T);V):=

Z T 0

kuk^r

Vdt

!1/r

.

2 Weak solutions, comparison principles and uniqueness results

Throughout this paper, unless otherwise stated, we shall assume thatp > p₁andp,2.

The case1< p < p₁is a bit more involved, as one has to work with functions belonging to W^1,p(Ω)∩L²(Ω) having L²(∂Ω, ρ)–trace instead of W^1,p(Ω). The casep=p₁ should also be treated separately, as the Sobolev embeddingW^1,p(Ω)→L^p∗(∂Ω, ρ) is not compact bearing in mind that herep∗= 2.

SetX^q=L^q(Ω)×L^q(∂Ω, ρ), for1≤q≤ ∞, and U = (u, ϕ)∈ X^q, kUk_Xq:=

kuk^q_q+σkϕk^q

q,∂Ω

1/q

, and forq= 2andU= (u, ϕ), V = (v, ψ)∈ X²

hU , Vi_X2:=hu, vi+σhϕ, ψi₀.

Identifying each elementu∈W^1,p(Ω)with the vectorU= (u, u|_∂Ω), the spaceW^1,p(Ω) can be regarded as a subspace ofX^s for any1≤s < p^∗. Moreover,X^s andL^s(Ω, dτ) can be identified in a natural way fors≥1, where the measuredτ=dx^|_Ω⊕dρ^|_∂Ω is defined for any measurable setA⊂Ωbyτ(A) =|A|+ρ(A∩∂Ω). We will agree that Ais measurable ifA∩Ω is Lebesgue measurable andA∩∂Ωis measurable with respect to the(N−1)-Hausdorff measure ρ.

For anyT >0, let us denoteΩ_T = (0, T)×Ω.Let us recall the definition of a local weak solution of the evolution problem(Pσ ,f). Letf :Ω_T×R→Rbe a Carath´eodory function.

2. Weak solutions, comparison principles and uniqueness results

Definition 1 – A functionu:Ω_T →Ris called aweak solution of problem(P_{σ ,f})if (i) u∈L^p((0, T);W^1,p(Ω))∩C([0, T];X²),

(ii) ∂_tu∈L²((0, T);L²(Ω));∂_tu|_∂Ω∈L²((0, T);L²(∂Ω, ρ)), (iii) f˜:=f(·,·, u(·,·))∈ L²((0, T);L²(Ω)),

(iv) for anyϕ∈W^1,p(Ω)and for almost allt∈[0, T]it holds

h∂_tu, ϕi+h|∇u|^p−2∇u,∇ϕi − hf(t, x, u), ϕi+σh∂_tu|_∂Ω, ϕi₀= 0.

Clearly, in (ii) the notation∂_tu(resp. ∂_tu|_∂Ω) stands for the weak temporal derivative ofu(resp. ofu|_∂Ω).

By writing u∈ X^q we mean thatu:Ω→R is such thatu^|_Ω∈L^q(Ω)and also u^|_∂Ω∈L^q(∂Ω, ρ).

Definition 2 – For givenu₀∈ X² a function u:Ω_T →R is called a weak solution of the Cauchy problem(P_{σ ,f ,u0})

(Pσ ,f ,u₀)















∂_tu=∆_pu+f(t, x, u) inΩ for t >0, σ ∂_tu+|∇u|^p−2∂_νu= 0 on∂Ω for t >0,

u(0,·) =u₀ inΩ,

ifu is a weak solution of(P_{σ ,f})in the sense of Definition 1 and if, in addition, u(0,·) =u₀ τ–a.e. inΩ.

Let us also recall the definition of upper and lower solution for our evolution problem.

Writing(u, ϕ)≤(v, ψ)for functions belonging toX^s means thatu≤v a.e. inΩand thatϕ≤ψ ρ-a.e. on∂Ω.

Definition 3 – For given u0∈ X², a function u:Ω_T →R satisfying (i)-(iii) from Definition 1 is called aweak lower solution of (P_{σ ,f ,u0})if for all ϕ∈W^1,p(Ω)with ϕ≥0, and for almost allt∈[0, T]











h∂tu, ϕi+h|∇u|^p−2∇u,∇ϕi − hf(t, x, u), ϕi+σh∂tu, ϕi₀≤0, u(0,·)≤u₀ τ–a.e. inΩ,

in its interval of existence. Similarly, a weak upper solution is defined by reversing the last two inequalities.

In order to compare two different solutions of the evolution equations, it is necessary to require some Lipschitz condition onf with respect to the variable uand with respect to the pair (u1, u2)∈ L²(ΩT)². This is the motivation of the following definition.

Definition 4 – A functionf is said to satisfy the following one-sided Lipschitz condi- tion for the pair(u₁, u₂)∈L²(Ω_T)² if

∃l∈L¹((0, T); [0,∞)) such that for a.a. (t, x)∈Ω_T, [f(t, x, u1(t, x))−f(t, x, u2(t, x))−l(t)(u1(t, x)−u2(t, x))]

u1(t, x)−u2(t, x)+

≤0.

(1) Observe that no order betweenu₁ andu₂ is assumed here and that (1) is in fact a condition for the set of(t, x)∈Ω_T for whichu₁(t, x)≥u₂(t, x). At the end of this section, we shall give some special cases where (1) is satisfied, c.f. Remark 3. Let us prove the following comparison result.

Theorem 1 – Let u₀¹, u²₀∈ X² be given, letu1 be a weak lower solution of (P_{σ ,f ,u}¹

0) and letu2 be a weak upper solution of(P_{σ ,f ,u}²

0). Assume that the pair(u1, u2)satisfies (1) and let us denotev:=u₁−u₂,v₀:=u₀¹−u₀². Then

(i) kv⁺(t,·)k²

X2≤e^2L(t)kv₀⁺k²

X2 for allt∈[0, T], withL(t) :=R_t

0l(s)ds.

(ii) Moreover, there exists a constant C=C(p, N , T ,klk₁)>0such that

k∇v⁺k^p

L^p(Ω_T)≤















Ckv⁺₀k²

X2 ifp≥2,

Ckv⁺₀k^p

X2

k∇u1k_Lp(Ω_T)+k∇u2k_Lp(Ω_T)

p(1−^p

2)

ifp <2.

Proof. (i) Fixingt∈[0, T]and takingϕ=v⁺(t,·)as a test function in the differential inequalities satisfied byu₁ andu₂, subtracting them and integrating over Ωyield

h∂_tv, v⁺i+σh∂_tv, v⁺i₀≤

−h|∇u₁|^p−2∇u₁− |∇u₂|^p−2∇u₂,∇(u₁−u₂)⁺i+hf(t, x, u₁)−f(t, x, u₂), v⁺i. (2) Here Conditions (i)-(iv) from Definition 1 assure that all the above integrals are finite. By convexity of the functionz7→ |z|^p in R^N the first term of the r.h.s. is negative, while the second term is bounded from above byl(t)hv(t,·), v⁺(t,·)iusing (1). Moreover, we deduce for almost allt that

h∂_tv, v⁺i=1 2

d dtkv⁺k²

2,

h∂tv, v⁺i₀=1 2

d dtkv⁺k²

2,∂Ω.

2. Weak solutions, comparison principles and uniqueness results

These identities and the existence of the derivatives on the r.h.s. are justified, see e.g.¹⁹. It follows that

d dtkv⁺k²

2+σ d dtkv⁺k²

2,∂Ω≤2l(t)kv⁺k²

2,

and therefore, by the Bellman–Gronwall Inequality²⁰ kv⁺(t,·)k²

X2≤e^2L(t)kv⁺₀k²

X2

for allt∈[0, T].

(ii) We apply the following inequality from²¹ : ∃C=C(p, N)>0∀z₁, z₂∈R^N :











h|z₁|^p−2z₁− |z₂|^p−2z₂, z₁−z₂i

R^N≥C|z₁−z₂|^p

RN, ifp≥2, h|z₁|^p−2z₁− |z₂|^p−2z₂, z₁−z₂i

RN≥C(|z₁|

RN+|z₂|

RN)^p−2|z₁−z₂|²

RN, if1< p≤2 In order to bound from above the first term of the r.h.s. of (2) by Ck∇v⁺(t,·)k^p_p if p≥2 or by CR

Ω(|∇u₁(t, x)|+|∇u₂(t, x)|)^p−2|∇v⁺(t, x)|²dx if p <2. Therefore we deduce for almost allt that

1 2

d

dtkv⁺(t,·)k²

X2+Ca(t)≤l(t)kv⁺(t,·)k²

X2, where we abbreviate

a(t) =











k∇v⁺(t,·)k^p_p, ifp≥2, R

Ω(|∇u₁(t,·)|+|∇u₂(t,·)|)^p−2|∇v⁺(t,·)|²dx, if1< p≤2.

Integrating over[0, T]and using the estimate (i) we obtain 1

2(kv⁺(T ,·)k²

X2− kv⁺(0,·)k²

X2) +C ZT

0

a(t)dt≤ klk₁e^2L(T)kv₀⁺k²

X2

i.e.

Z_T

0

a(t)dt≤ 1 C

klk₁e^2L(T)+1 2

kv⁺₀k²

X2.

Ifp≥2, then the statement (ii) is clear. Considerp <2. First, H¨older’s inequality leads to

k∇v⁺k^p_p≤a(t)^p2 Z

Ω

(|∇u1|+|∇u2|)^pdx

!1−^p

2

≤2^{(p−1)(1−p2)}a(t)^p/2

k∇u₁k^p_p+k∇u₂k^p_p1−^p

2,

since1< p <2. Then, we use H¨older’s inequality with respect to the dependence on t and obtain

k∇v⁺k^p

L^p(Ω_T)≤2^{(p−1)(1−p2)} ZT

0

a(t)dt

!p/2

k∇u₁k^p

L^p(Ω_T)+k∇u₂k^p

L^p(Ω_T)

1−^p

2

≤2^{(p−1)(1−p2)} ZT

0

a(t)dt

!p/2

k∇u₁k_Lp(Ω

T)+k∇u₂k_Lp(Ω

T)

p(1−^p

2)

.

Immediate consequences of the previous theorem are given by the following corollaries.

Corollary 1 – Let f₁, f₂:Ω_T →R be two measurable functions such thatf₁≤f₂ a.a.

(t, x)∈Ω_T. Letu¹₀, u₀²∈ X² be given, letu₁ be a weak lower solution of(P_{σ ,f}

1,u₀¹)and letu2 be a weak upper solution of (P_{σ ,f}

2,u²₀). Then the conclusions (i) and (ii) hold withl≡0.

Corollary 2 – Letu₀∈ X² be given. Let u₁ and u₂ be two weak solutions of (P_{σ ,f ,u0}).

Assume that the pair (u₁, u₂)satisfies the Lipschitz condition:

∃l∈L¹(0, T), l≥0, such that for a.a. (t, x)∈Ω_T, [f(t, x, u1(t, x))−f(t, x, u2(t, x))−l(t)(u1(t, x)−u2(t, x))]

u1(t, x)−u2(t, x)

≤0.

(3) Then for allt∈[0, T],u₁(t,·) =u₂(t,·)a.e. in Ω.

Note that (3) implies (1) for both (u1−u2)⁺ and (u1−u2)⁻, which means that uniqueness is plain. For the Lipschitz case we can state the following

Corollary 3 – Assume that f satisfies the following Lipschitz condition (4) with respect to the variableu:

∃l∈L¹(0, T), l≥0, such that for a.a. (t, x)∈Ω_T, for all(u1, u2)∈R², [f(t, x, u₁)−f(t, x, u₂)−l(t)(u₁−u₂)]

u₁−u₂

≤0. (4)

Let u₀∈ X². Let u_n⁰

n∈N be a sequence in X² such that lim_n→∞ku_n⁰−u₀k_X2 = 0. Let u_n be a weak solution of (P_{σ ,f ,u}0

n) and u a weak solution of (P_{σ ,f ,u0}). Then u_n(t,·)→u(t,·) inX² andu_n→u inL^p((0, T);W^1,p(Ω)).

19Showalter, 1997, Monotone operators in Banach space and nonlinear partial differential equations, Chapter III, Proposition 1.2.

20Bellman, 1953,Stability theory of differential equations.

21Simon, 1978,R´egularit´e de la solution d’une ´equation non lin´eaire dansR^N.

2. Weak solutions, comparison principles and uniqueness results

Proof. The convergence inX² follows readily from (i) of Theorem 1. From (ii), if p >2, we have directly that there exists a constantC=C(p, N , T ,klk₁)>0such that

k∇u_n− ∇uk^p

L^p(Ω_T)≤Cku_n⁰−u₀k²

X2. Ifp <2we have

k∇un− ∇uk^p

L^p(Ω_T)≤Cku_n⁰−u0k^p

X2

k∇unk_Lp(Ω_T)+k∇uk_Lp((Ω_T)

p(1−^p

2)

.

Sinceku⁰_n−u0k_X2→0, the latter inequality yields that the sequencek∇unk_Lp(Ω_T) is bounded and thereforek∇u_n− ∇uk^p

L^p(Ω_T)→0.

Another consequence of Theorem 1 is the following standard comparison principle.

Theorem 2 (Weak comparison principle with dynamical boundary condition) – Let u₀¹, u₀² ∈ X² satisfy u¹₀≤u₀² τ–a.e. inΩ. Letu1 be a weak lower solution of(P_{σ ,f ,u}¹

0) and letu2 be a weak upper solution of(P_{σ ,f ,u}²

0). Assume that the pair(u1, u2)satisfies (1). Then

u1(t,·)≤u2(t,·) τ–a.e. inΩ. Proof. By hypothesis

k(u1(0,·)−u2(0,·))⁺k₂=k(u₀¹−u²₀)⁺k₂= 0 and

k(u₁(0,·)−u₂(0,·))⁺k²

2,∂Ω=k(u₀¹−u₀²)⁺k_2,∂Ω= 0.

Hence, from Theorem 1 (i) it follows that(u1(t,·)−u2(t,·))⁺= 0 a.e. inΩ, which

permits to conclude.

In Section 5 we need to compare a weak solution of a parabolic problem with dynamical boundary condition with a solution of the analogous parabolic problem under homogeneous Dirichlet boundary condition for the forcing term

f(t, x, u) =fλ,q(u) :=λ|u|^q−2u

with λ >0andq >max{2, p}. Let us consider here a more general parabolic problem for thep-Laplacian under Dirichlet boundary condition











∂tv=∆_pv+f(t, x, v) in Ω for t >0,

v= 0 on∂Ω for t >0,

v(0,·) =v₀ in Ω,

(5)

for an initial value v0∈L²(Ω). A functionv : [0, T]×Ω→R is called a weak solution of (5) ifv∈L^p((0, T);W₀^1,p(Ω))∩C([0, T];L²(Ω)),∂_tv∈L²(Ω_T), v(0,·) =v₀ a.e. inΩ and for any test functionϕ∈W₀^1,p(Ω)we have

h∂_tv, ϕi+h|∇v|^p−2∇v,∇ϕi − hf(t, x, v), ϕi= 0

for a.a. t∈(0, T). Correspondingly, we define upper and lower solutions for Problem (5). For the sake of completeness let us state the following Weak Comparison Principle for Problem (5), though the result is partially known, namely for the special case f ≡0, c.f.²².

Theorem 3 (Weak Comparison Principle with Dirichlet boundary condition) – Letv₁ be a lower weak solution of (5) with initial datav¹₀∈L²(Ω)and let v₂ be an upper weak solution of (5) with initial data v₀² ∈L²(Ω), satisfying v₀¹ ≤v₀² a.e. in Ω.

Assume further that the pair(v₁, v₂)satisfies the one-sided Lipschitz condition (1).

Thenv₁(t,·)≤v₂(t,·)a.e. in Ω, for allt∈[0, T].

Proof. Fixt∈[0, T]. The choice ofϕ≡(v₁(t,·)−v₂(t,·))⁺ leads to h∂_t(v₁−v₂),(v₁−v₂)⁺i=−

Z

Ω

(|∇v₁|^p−2∇v₁− |∇v₂|^p−2∇v₂)· ∇(v₁−v₂)⁺dx +

Z

Ω

(f(t, x, v₁)−f(t, x, v₂))(v₁−v₂)⁺dx.

Thus, by convexity of the function z∈R^N7→ |z|^p and by Condition (1), h∂_t(v₁−v₂),(v₁−v₂)⁺i ≤l(t)k(v₁−v₂)⁺k²

2. As in the proof of the Theorem 2 we can conclude

2 Zs

0

l(t)k(v₁(t,·)−v₂(t,·))⁺k²

2dt≥ Zs

0

d

dtk(v₁(t,·)−v₂(t,·))⁺k²

2dt= k(v₁(s,·)−v₂(s,·))⁺k²

2− k(v₁(0,·)−v₂(0,·))⁺k²

2. By the Bellman–Gronwall Inequality and that (v₁(0,·)−v₂(0,·))⁺= (v₀¹−v₀²)⁺= 0 a.e. inΩ, it follows thatk(v1(s,·)−v2(s,·))⁺k²

2≤e^2L(s)k(v1(0,·)−v2(0,·))⁺k²

2= 0for all

s∈[0, T], which permits to conclude.

22DiBenedetto, 1993,Degenerate parabolic equations.

2. Weak solutions, comparison principles and uniqueness results

As a consequence of the previous theorem we have

Proposition 1 (Weak Maximum Principle with Dirichlet boundary condition) – Let v be a weak upper solution of (5) with initial data v₀∈L²(Ω)satisfying v₀≥0a.e.

inΩ. Assume further that f(·,·,0)≥0 and that the pair(0, v)satisfies the one-sided Lipschitz condition (1). Then v satisfiesv(t,·)≥0a.e. in Ω, for allt∈[0, T]. Finally, we compare solutions of parabolic problems under dynamical boundary conditions and nonnegative solutions of parabolic problems under Dirichlet boundary conditions. For that purpose, the solutionv of (5) is required to fulfil v(t,·)∈C¹(Ω), since we shall need some estimates of the gradient of the solutionv(t,·)on∂Ω. See also the Remark 1 below about the regularity of weak solutions of Problem (5).

Theorem 4 – Assume that ∂Ω is of class C². Let u be a weak lower solution of Problem (P_{σ ,f ,u0}) with initial datau₀∈ X². Let v be the weak solution of Problem (5) with initial data v0∈L²(Ω), satisfyingv0≥0 a.e. inΩ. Assume that

(i) f(·,·,0)≥0,

(ii) the pair (0, v) satisfies the one-sided Lipschitz condition (1), (iii) the pair(v, u)satisfies the one-sided Lipschitz condition (1),

(iv) for allt∈(0, T),v(t,·)∈C¹(Ω).

If, in addition,

u0≥v0≥0 a.e. in Ω, u0≥0 ρ–a.e. in∂Ω, (6) thenu(t,·)≥v(t,·)a.e. in Ω, for allt∈[0, T].

Proof. By Theorem 3 and hypotheses (i) and (ii), the solutionv is nonnegative in [0, T]×Ω. Sincev(t,·)∈W₀^1,p(Ω)∩C(Ω), and since∂Ωis of classC¹, the solutionv has to vanish on∂Ωfor allt∈(0, T). Thus,∂νv(t,·)≤0on ∂Ωfor allt∈(0, T), and multiplying the differential equation of Problem (5) by any nonnegative function ϕ∈W^1,p(Ω)and integrating overΩyield

h∂tv, ϕi=−h|∇v|^p−2∇v,∇ϕi+h|∇v|^p−2∂νv, ϕi₀+hf(t, x, v), ϕi

≤ −h|∇v|^p−2∇v,∇ϕi+hf(t, x, v), ϕi. (7) Although the test function ϕ used in the weak formulation of Problem (5) must belong toW₀^1,p(Ω), the above integration by parts is justified e.g. by Lemma A.1

from²³. Hence,∂_tv(t,·)≡0on∂Ω, (7), and the weak formulation of Problem(P_{σ ,f ,u0}) with the test function w(t,·) = (v(t,·)−u(t,·))⁺ yield that

Z_s

(h∂tw, wi+σh∂tw, wi₀)dt≤ Z_s

h|∇v|^p−2∇v− |∇u|^p−2∇u,∇widt +

Z s

hf(t, x, v)−f(t, x, u), widt.

for any0< < s < T. By convexity of the functionz7→ |z|^p in R^N, the first integral on the r.h.s. is seen to be positive, which implies in turn that

1 2

Zs

d

dtkw(t,·)k²

X2dt≤ Z s

hf(t, x, v)−f(t, x, u), widt and

kw(s,·)k²

X2− kw(,·)k²

X2≤2 Zs

Z

Ω

|f(t, x, v)−f(t, x, u)|w dx

! dt

≤2 Z s

l(t)kw(t,·)k²₂dt. (8) Asw(,·) =u⁻(,·)ρ-a.e. in ∂Ω,

lim→0

kw(,·)k²

X2=kw(0,·)k²₂+σku₀⁻k²

2,∂Ω= 0 by (6). Thus (8) reduces to

kw(s,·)k²₂≤2 Z s

0

l(t)kw(t,·)k²₂dt.

As the functionl(t)is integrable and nonnegative, the Bellman–Gronwall Inequality yields

∀s∈[0, T] :kw(s,·)k²

2= 0.

Thus, we can conclude thatv(s,·)≤u(s,·)a.e. inΩfor alls∈[0, T].

Remark 1 – Standard regularity results (c.f. Lieberman (1993, Theorem 01)) imply that a weak solution v of Problem (5) satisfies v ∈C¹((0, T]×Ω) provided that v∈L^∞((, T)×Ω) for all∈(0, T)and that ∂Ωis of class C^1,α for some0< α <1.

23Cuesta and Tak´aˇc, 2000, “A strong comparison principle for positive solutions of degenerate elliptic equations”.

3. An energy identity

Remark 2 – In order to assure the global boundedness in the large of any weak solution v of Problem (5), that is, v ∈L^∞((, T)×Ω), one should impose to f a growth condition with respect to theu-variable, say e.g.

|f(t, x, v)| ≤A|v|^q−1 f or a.a.(t, x)∈(, T)×Ω,∀v∈R, withq < p^N+2_N (c.f²⁴).

Remark 3 – In the special casef(t, x, u) =f_λ,q(u), withq >1

(i) Conditions (1) and (3) are trivially satisfied for any pair of functions(u₁, u₂)if λ≤0.

(ii) Ifλ >0andq >2, the one-sided Lipschitz condition (1) is satisfied for any pair and(u₁, u₂)∈L¹((0, T);L^∞(Ω))² because

f_λ,q(u₁(t,·))−f_λ,q(u₂(t,·))

≤l(t)|u₁(t,·)−u₂(t,·)| withl(t) := (q−1)λmax{ku1(t,·)k_∞,ku2(t,·)k_∞}^q−2.

(iii) Clearly, forq <2Condition (1) does not hold, and Theorem 2 does not apply.

In general, the solutions of(P_{σ ,fλ,q,u0})are even not unique. Take e.g. σ= 0and q=³₂, then the continuum of nonnegative solutions ofz˙=λz¹² underz(0) = 0 furnishes also non unique solutions of(P0,f_λ,3

2,0).

3 An energy identity

In this section we show an energy estimate for solutions of Problem(P_{σ ,f ,u0})for time independent nonlinearities:

∀(t, x, u)∈Ω_T ×R :f(t, x, u) =f(x, u). (9)

For any weak solutionuof Problem(Pσ ,f ,u₀)we introduce the energyEF by EF(u(t,·)) =1

pk∇u(t,·)k^p_p− Z

Ω

F(x, u(t, x))dx (10)

with

F(x, s) :=

Zs 0

f(x, z)dz.

24DiBenedetto, 1993,Degenerate parabolic equations, Theorem 3.2 Chapter V.

The following energy identityfor weak solutions of (P_{σ ,f ,u0})will be shown as in the casef ≡0, where the classical theory of maximal monotone operators²⁵ has been applied.

Theorem 5 – Letu:Ω_T →R be a weak solution of Problem(P_{σ ,f ,u0}) withu₀∈ X². Assume thatf satisfies (9) and that there exists a constantl≥0such that

[f(x, u(t, x))−f(x, u(s, x))−l(u(t, x)−u(s, x))] (u(t, x)−u(s, x))≤0 (11) for a.a. t, s∈(0, T]and for a.a. x∈Ω. Then for a.a. t∈(0, T], the time derivative ofEF(u(t,·)) exists and satisfies the identity

−d

dtE_F(u(t,·)) =k∂_tu(t,·)k²

2+σk∂_tu|_∂Ω(t,·)k²

2,∂Ω. (12)

For simplicity we shall abbreviate

h∂tu(t,·), ϕi_X2:=h∂tu(t,·), ϕi+σh∂tu^|_∂Ω(t,·), ϕi₀, for anyϕ∈W^1,p(Ω), and

k∂tuk²

X2:=k∂tu(t,·)k²

2+σk∂tu^|_∂Ω(t,·)k²

2,∂Ω.

Note that we do not state here that∂_tu|_∂Ω(t,·) is the trace of ∂_tu(t,·), since this function is assumed only to belong toL²(Ω)!

Proof. Recall the following basic convexity inequality, valid for anyp >1and any x, y∈R^N:

p|x|^p−2x·(y−x)≤ |y|^p− |x|^p. (13)

Fixt∈(0, T)andh >0small and chooseϕ=u(t+h,·)−u(t,·)in the weak formulation of(P_{σ ,f ,u0})at timet. Then, using (13)

h∂_tu(t,·), u(t+h,·)−u(t,·)i_X2

=−h|∇u(t,·)|^p−2∇u(t,·),∇u(t+h,·)− ∇u(t,·)i+hf(·, u(t,·)), u(t+h,·)−u(t,·)i

≥ −1

p(k∇u(t+h,·)k^p_p− k∇u(t,·)k^p_p) +hf(·, u(t,·)), u(t+h,·)−u(t,·)i. (14)

25Br´ezis, 1973,Op´erateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Theorem 3.2.