2. The stationary problem

https://doi.org/10.1051/cocv/2019023 www.esaim-cocv.org

ASYMPTOTIC LIMIT OF LINEAR PARABOLIC EQUATIONS WITH SPATIO-TEMPORAL DEGENERATED POTENTIALS

Pablo ` Alvarez-Caudevilla, Matthieu Bonnivard and Antoine Lemenant

Abstract. In this paper, we observe how the heat equation in a noncylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case [Alvarez-Caudevilla and Lemenant,Adv. Differ.

Equ. 15(2010) 649-688]. We provide a strong convergence result for the solution by use of energetic methods and Γ-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

Mathematics Subject Classification.35A05, 35A15 Received April 9, 2018. Accepted April 7, 2019.

1. Introduction

For Ω⊂R^N open andT >0, we define the cylinderQT = Ω×(0, T). Letλ >0 be a positive real parameter.

For f_λ∈L²(Q_T),g_λ ∈H₀¹(Ω) anda:Q_T →R⁺ a bounded measurable function, we consider the solutionu_λ of the parabolic problem

(Pλ)







∂tu−∆u+λa(x, t)u=fλ in QT,

u= 0 on ∂Ω×(0, T),

u(x,0) =gλ(x) in Ω.

Since (Pλ) is a classical parabolic problem, existence and regularity of solutions follow from standard theory well developed in the literature (see Sect.3). In particular, under our assumptions,u∈L²(0, T;H₀¹(Ω)) is continuous in time with values inL²(Ω) (thus the initial conditionu(x,0) =g_λ(x) is well defined inL²(Ω)) and the equation is satisfied in a weak sense (see Sect.3for an exact formulation).

Keywords and phrases:Parabolic problems, Gamma-convergence, energetic methods, variational methods, partial differential equations.

Universite Paris Diderot, Paris, France.

* Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, SMAI 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we are interested in the limit of uλ when λ→+∞. In particular, we assume spatial and temporal degeneracies for the potentiala, which means that

O_a := Int {(x, t)∈Q_T : a(x, t) = 0}

6=∅. (1.1)

We also assume that∂O_a has zero Lebesgue measure.

In order to describe the results of this paper, let us start with elementary observations. Assume that, whenλ goes to +∞,fλ converges tof andgλ converges tog, for instance inL². Assume also thatuλconverges weakly in L²(QT) to someu∈L²(QT).

Under those assumptions it is not very difficult to get the followinga prioribound using the equation in (P_λ) (see Lem.5.1)

λ Z

QT

au²_λdxdt≤C. (1.2)

This shows that u_λ converges strongly to 0 in any set of the form{a(x, t)> ε}, for anyε >0.

Then, multiplying the equation in (Pλ) by anyϕ∈C₀^∞(Oa) we get, after some integration by parts (in this paper we shall denote∇ for∇x,i.e.the gradient in space),

Z

Q_T

uλ∂tϕ− Z

Q_T

uλ∆ϕ= Z

Q_T

fλϕ.

Passing to the limit, we obtain that ∂_tu−∆u=f in D⁰(O_a). Under some suitable extra assumptions on the potential a, we will actually be able to prove that the limitusatisfies the following more precise problem:

(P_∞)















u∈L²(0, T;H₀¹(Ω)), u⁰∈L²(0, T;L²(Ω)) u= 0 a.e. inQT\Oa

R

Q_T(u⁰v+∇u∇v) =R

Q_Tf v,

for allv∈L²(0, T;H₀¹(Ω)) s.t. v= 0 a.e. in QT \Oa

u(x,0) =g(x) in Ω.

Problem (P∞), which arises here naturally as the limit problem associated with the family of problems (Pλ), is a nonstandard Cauchy–Dirichlet problem for the heat equation sinceO_a may, in general, not be cylindrical.

This type of heat equation in a noncylindrical domain appears in many applications, and different approaches have been developed recently to solve problems related to (P_∞) (see for e.g. [4–7, 9, 13] and the references therein). As a byproduct to our work, we have obtained an existence and uniqueness result for the problem (P_∞) (see Cor.5.5).

Furthermore, in this paper we study in more detail the convergence of uλ, whenλgoes to infinity. Our first result gives a sufficient condition on the potential a, for which the convergence ofuλ to uis stronger than a weakL² convergence. Indeed, assuming a monotonicity condition on the potentiala, and using purely energetic and variational methods, we obtain that the convergence holds strongly inL²(0, T;H¹(Ω)); see Section5. Our approach can be seen as the continuation of a previous work [1], where the stationary problem was studied using the theory of Γ-convergence, as well as in [2] using a different analysis.

Here is our first main result.

Theorem 1.1. For allλ >0, letuλ be the solution of(Pλ)withfλ∈L²(Ω×(0, T))andgλ∈H₀¹(Ω). Assume that a: Ω×[0, T]→R⁺ is a Lipschitz function which satisfies

∂_ta(x, t)≤0 a.e. inQ_T. (1.3)

Assume also that the initial conditiongλ satisfies sup

λ>0

λ

Z

Ω

a(x,0)gλ(x)²dx

<+∞, converges weakly to g inL²(Ω), and thatfλ converges weakly tof in L²(QT).

Thenu_λ converges strongly to uin L²(0, T;H¹(Ω)), whereuis the unique solution of(P_∞).

Remark 1.2. In particular, condition (1.3) implies that the family of sets Ω_a(t)⊂Ω, defined for t > 0 by Ωa(t) :={x∈Ω,(x, t)∈Oa}, is increasing in time for the inclusion. In that case, by a slight abuse of terminology, we will often write simply that Oa is increasing in time (for the inclusion).

Our second result is a quantitative convergence ofu_λto 0, outsideO_a(in other words, away from the vanishing region), with very general assumptions ona(only continuous andOa 6=∅), but in the special case whenfλ= 0 in QT \Oa. This is obtained using an adaptation of an argument due to Simon [14], and proves thatuλ decays exponentially fast to 0 with respect to λin the region QT \Oa. Compared to the standard bound (1.2), this results expresses that u_λ goes to 0 much faster than one could expect. We also take the opportunity of this paper to write a similar estimate for the stationary problem (see Lem. 6.1in Sect.6).

Theorem 1.3. For all λ >0, let uλ be the solution of (Pλ) with fλ∈L²(QT) and gλ∈H₀¹(Oa∩ {t= 0}).

Assume that fλ= 0 inQT \Oa. Leta: Ω×[0, T]→R⁺ be a continuous function for which Oa is nonempty.

For everyε >0, defineA_ε:={(x, t);dist((x, t), O_a)> ε}. Then, for everyε >0, there exists a constant C >0 such that

sup

λ>0

λe^cε

√λZ

Aε

u²_λdx

≤C,

where cε:=εmin_{(x,t)∈Aε/2}a(x, t).

The convergence of weak solutions of (Pλ) was already observed in [8] as a starting point for a more detailed analysis about the associated semigroup. This was then used in [8] to analyze the asymptotic behaviour of a nonlinear periodic-parabolic problem of logistic type (firstly analyzed by Hess [12]) where the equation is the following, also considered before in [9],

∂tu−∆u=µu−a(x, t)u^p, (1.4)

used in some models of population dynamics. A possible link between our problem (P_λ) and the nonlinear equation (1.4) is coming from the fact that asymptotic limit of the principal eigenvalue for the linear parabolic operator∂t−∆ +λa(x, t) plays a role in the dynamical behaviour of nonlinear logistic equation (cf.[3,8,9,11]).

We thus believe that the results and techniques developed in the present paper could possibly be used in the study of more general equations such as (1.4).

Furthermore, another possible application of our results could be for numerical purposes. Indeed, for the ones who would be interested by computing a numerical solution of the noncylindrical limiting problem (P_∞), one could use the cylindrical problem (Pλ) for a largeλ, much easier to computeviastandard methods. The strong

convergence stated in Theorem 1.1, together with the exponential rate of convergence stated in Theorem1.3, gives some good estimates about the difference between those two different solutions.

2. The stationary problem

This section concerns only the stationary problem. In particular, throughout the section, all functionsu, a, f, etc., will be functions ofx∈Ω (and independent oft).

We assume Ω⊂R^N to be an open and bounded domain and a: Ω→R⁺ be a measurable and bounded non-negative function. We suppose that

Ka:={x∈Ω;a(x) = 0} ⊂Ω is a closed set in R^N. (2.1) Moreover, we assume that

Ωa:= Int(Ka)6=∅. (2.2)

Under hypothesis (2.1) we know that

H₀¹(K_a) :=H¹(R^N)∩ {u= 0 q.e. inR^N \K_a}=H¹(R^N)∩ {u= 0 a.e. inR^N\K_a}, and hypothesis (2.2) implies that

H₀¹(Ka)6={0}.

Notice that we are working with a functional space of the form H₀¹(A), where A is a closed subset of R^N. Therefore, we do not claim thatH₀¹(A) =H₀¹(Int(A)), which is true only under more regularity assumptions on the setA.

Furthermore, we define the functionalsE_λ andE onL²(Ω) as follows.

E_λ(u) = R

Ω|∇u|²+λau²dx ifu∈H₀¹(Ω)

+∞ otherwise. (2.3)

E(u) = R

Ω|∇u|²dx ifu∈H₀¹(K_a)

+∞ otherwise.

The following result was already stated and used in [1]. For the sake of completeness, we reproduce the proof here and refer the reader to [1] for the connection of this result with Γ-convergence and several examples.

Proposition 2.1. Letfλ∈L²(Ω)be a family of functions indexed by some real parameterλ >0and uniformly bounded inL²(Ω). Moreover, assume thatfλ converges to a functionf ∈L²(Ω) in the weak topology ofL²(Ω), when λtends to+∞. Then the unique solutionuλ of the problem

(P_λ^s)

−∆uλ+λau_λ=f_λ uλ∈H₀¹(Ω),

converges strongly inH¹(Ω), whenλ→+∞, to the unique solution of the problem

(P_∞^s)

−∆u=f u∈H₀¹(K_a).

Proof. This is a standard consequence of the Γ-convergence of energies Eλ, which relies on the fact that uλ is the unique minimizer inH₀¹(Ω) for

v7→E_λ(v)−2 Z

Ω

f_λv,

whereasuis the unique minimizer inH₀¹(K_a) for

v7→E(v)−2 Z

Ω

f v.

Let us write the full details of the proof. For anyλ >0, letuλ be the solution of (P_λ^s). We first prove that {uλ}λ>0 is compact inL²(Ω). This comes from the energy equality

Z

Ω

(|∇u_λ|²+λau²_λ) dx= Z

Ω

f_λu_λdx, (2.4)

which implies Z

Ω

|∇u_λ|²≤ kf_λk_L2(Ω)ku_λk_L2(Ω)≤Cku_λk_L2(Ω) withC a positive constant.

Thanks to Poincar´e’s inequality we also have that ku_λk²_L2(Ω)≤C(Ω)

Z

Ω

|∇u_λ|²dx,

which finally proves thatu_λis uniformly bounded inH₀¹(Ω).

Now letwbe any point in theL²-adherence of the family{uλ}λ>0. In other words, there exists a subsequence, still denoted by uλ, converging strongly in L² to w. Since uλ is bounded in H¹(Ω), we can assume, up to extracting a further subsequence, thatu_λconverges weakly inH¹(Ω) to a function that must necessarily bew.

Now let ube the solution of the limit problem (P_∞^s). By definition of (P_∞^s),u∈H₀¹(K_a) and in particular au= 0 almost everywhere in Ω andEλ(u) =E(u) for allλ >0. Now sinceuλis a minimizer of

u7→E_λ(u)−2 Z

Ω

f_λudx, (2.5)

anduis admissible, we have E_λ(u_λ)−2

Z

Ω

f_λu_λdx≤E_λ(u)−2 Z

Ω

f_λudx=E(u)−2 Z

Ω

f_λudx.

Hence, lettingλgo to infinity in the previous inequality, using the trivial inequalityE(uλ)≤Eλ(uλ) and then the lower-semicontinuity of the Dirichlet integral with respect to the weak convergence, it follows that

E(w)−2 Z

Ω

f wdx≤lim inf

λ

Eλ(uλ)−2 Z

Ω

fλuλdx

≤lim sup

λ

Eλ(uλ)−2 Z

Ω

fλuλdx

≤E(u)−2 Z

Ω

f udx, (2.6)

which shows that w is a minimizer, and thusw=u. By uniqueness of the adherence point, we infer that the whole sequence uλ converges strongly inL² tou(and weakly inH¹).

It remains to prove the strong convergence inH¹. To do so, it is enough to prove k∇u_λk_L2(Ω)→ k∇uk_L2(Ω).

Due to the weak convergence inH¹(Ω) (up to subsequences) we already have k∇uk_L2(Ω)≤lim inf

λ k∇uλk_L2(Ω), and going back to (2.6) we get the reverse inequality, with a limsup.

The proof of convergence of the whole sequence follows by uniqueness of the adherent point inL²(Ω).

Remark 2.2. Notice that whenuis a solution of (P_∞^s), then −∆u=f only in Int(Ka) and−∆u= 0 inK_a^c. However, in general−∆uhas a singular part on∂K_a. Typically, ifK_a is for instance a set of finite perimeter, then in the distributional sense in Ω,

−∆u=f1_Ka+∂u

∂νH^N−1|_∂Ka,

where ν is the outer normal on∂Ka andH^N−1is theN−1 dimensional Hausdorff measure.

As a consequence of Proposition 2.1, we easily obtain the following result.

Proposition 2.3. Assume that f_λ converges weakly to a function f in L²(Ω). For any λ >0, let u_λ be the solution of problem(P_λ^s). Then, when λ→ ∞,

λ Z

Ω

au²_λdx→0, (2.7)

λauλ→f1_Ω\Ka+ (∆u)|∂K_a inD⁰(Ω), (2.8)

where uis the solution of(P_∞^s). Moreover, the convergence in (2.8)holds in the weak-∗ topology of H⁻¹. Proof. Due to Proposition2.1we know thatuλconverges strongly inH¹(Ω) tou, the solution of problem (P_∞^s).

In particular, from the fact that Z

Ω

|∇uλ|²dx→ Z

Ω

|∇u|²dx= Z

Ω

uf dx,

and

Z

Ω

uλfλdx→ Z

Ω

uf dx, passing to the limit in the following energy equality

Z

Ω

|∇u_λ|²dx+λ Z

Ω

au²_λdx= Z

Ω

u_λf_λdx, (2.9)

we obtain (2.7). Next, let us now prove (2.8). Thus, sinceuλ is a solution of (P_λ^s) then, for every test function ψ∈C_c^∞(Ω), after integrating by parts in Ω we arrive at

Z

Ω

uλ(−∆ψ) dx+λ Z

Ω

auλψdx= Z

Ω

fλψdx. (2.10)

Passing to the limit we obtain that λauλ→f+ ∆uinD⁰(Ω). Now returning to (2.10), we can write, for every ψ satisfyingkψkH¹(Ω)≤1,

λ

Z

Ω

au_λψdx

≤ kfλk2+k∇uλk2≤C.

Taking the supremum inψwe get

kλau_λk_H−1 ≤C.

Therefore,λau_λ is weakly-∗ sequentially compact inH⁻¹ and we obtain the convergence by uniqueness of the limit in the distributional sense.

3. Existence and regularity of solutions for (P

)

In order to define properly a solution for (Pλ), we first recall the definition of the spaces L^p(0, T;X), with X a Banach space, which consist of all (strongly) measurable functions (see [10], Appendix E.5)u: [0, T]→X such that

kukL^p(0,T;X)= Z T

0

ku(t)k^p_Xdt

!^1/p

<+∞,

for 1≤p <+∞, and

kukL^∞(0,T;X)= ess sup

t∈(0,T)

ku(t)kX <+∞.

For simplicity we will sometimes use the following notation forp= 2 andX =L²(Ω):

k · k₂≡ k · k_L2(0,T;L²(Ω)). We will also use the notation u(x, t) =u(t)(x) for (x, t)∈Ω×(0, T).

Next, we will denote byu⁰ the derivative ofuin thetvariable, intended in the following weak sense: we say that u⁰ =v, withu, v ∈L²(0, T;X) and

Z T 0

ϕ⁰(t)u(t)dt=− Z T

0

ϕ(t)v(t)dt

for all scalar test functionsϕ∈C₀^∞(0, T). The spaceH¹(0, T;X) consists of all functionsu∈L²(0, T;X) such that u⁰ ∈L²(0, T;X).

We will often use the following remark.

Remark 3.1. By ([10], Thm. 3, p. 303), if u ∈ L²(0, T;H₀¹(Ω)) and u⁰ ∈ L²(0, T;H⁻¹(Ω)), then u ∈ C([0, T], L²(Ω)) (after possibly being redefined on a set of measure zero). Moreover, the mappingt7→ ku(t)k²_L2(Ω)

is absolutely continuous and

d

dtku(t)k²_L2(Ω)= 2hu⁰(t), u(t)iL²(Ω).

In this section, we collect some useful information about the solution uλ of problem (Pλ) coming from the classical theory of parabolic problems that can be directly found in the literature.

Firstly, existence and uniqueness of a weak solution uλ for the problem (Pλ) follows from the standard Galerkin method; see ([10], Thms. 3 and 4, Sect. 7.1) for further details. According to this theory, a weak solution means that:

(P_λ)











u∈L²(0, T;H₀¹(Ω)), u⁰ ∈L²(0, T;H⁻¹(Ω)) RT

0 hu⁰(t), v(t)i_(H−1,H₀¹)+R

QT(∇u· ∇v+λa u v) =R

QTf_λv for allv∈L²(0, T;H₀¹(Ω)),

u(0) =gλ(x) in L²(Ω).

Remember that by Remark3.1above, such weak solutionuis continuous in time with values inL²(Ω) so that the initial condition makes sense. In the rest of the paper, (P_λ) will always refer to the above precise formulation of the problem that was first stated in Section1.

Next, thanks to ([10], Thm. 5, Sect. 7.1), by consideringλauas a right hand side term (inL²(Ω×(0, T))), we have the following.

Lemma 3.2. Letλ >0,g_λ∈H₀¹(Ω),f_λ∈L²(0, T;L²(Ω)), and letu_λ be the weak solution to(P_λ). Then, u_λ∈L²(0, T;H²(Ω))∩L^∞(0, T;H₀¹(Ω)), u⁰_λ∈L²(0, T;L²(Ω)),

andu_λ satisfies the following estimate:

sup

0≤t≤T

kuλ(t)k_H1

0(Ω)+kuλk_L2(0,T;H²(Ω))+ku⁰_λk2

≤C

λkauλk2+kfλk2+kgλk_H1 0(Ω)

,

(3.1)

where the constant C depends only on ΩandT.

Remark 3.3. Notice that the bound (3.1) is not very useful whenλ→+∞since what we usually control is

√λkauk2 (shown below in Lem.5.1) but notλkauk2. Thus, the right hand side blows-upa priori.

4. Uniqueness of solution for (P

)

In this section, we focus on the following problem that will arise as the limit of (Pλ). Our notion of solution for the problem∂_tu−∆u=f inO_a will precisely be the following:

(P_∞)















u∈L²(0, T;H₀¹(Ω)), u⁰∈L²(0, T;L²(Ω)) u= 0 a.e. inQT\Oa

R

Q_T(u⁰v+∇u· ∇v) =R

Q_T f v,

for allv∈L²(0, T;H₀¹(Ω)) s.t. v= 0 a.e. in QT \Oa

u(0) =g(x) in L²(Ω).

As a byproduct of Section 5 we will prove the existence of a solution for the problem (P∞), as a limit of solutions for (P_λ). In this section, we prove the uniqueness which follows from a simple energy bound. Notice that a solution uto (P_∞) is continuous in time (see Rem.3.1) thus the initial conditionu(x,0) =g(x) makes sense inL²(Ω).

Proposition 4.1. Any solutionuof(P_∞)satisfies the following energy bound 1

4 sup

t∈(0,T)

ku(t)k²_L2(Ω)+k∇uk²₂≤1

2kgk²_L2(Ω)+Tkfk2. (4.1) Consequently, there exists at most one solution to problem (P_∞).

Proof. Let u be a solution to (P_∞), and s∈ (0, T). Choosing v =u1_(0,s) (where 1_(0,s) is the characteristic function of (0, s)) in the weak formulation of the problem, we deduce that

Z s 0

Z

Ω

u⁰udxdt+ Z s

0

Z

Ω

|∇u|²dxdt= Z s

0

Z

Ω

f udxdt. (4.2)

Now applying Remark3.1and using the fact thatu∈L²(0, T;H₀¹(Ω)) andu⁰ ∈L²(0, T;L²(Ω)) we obtain that t7→ ku(t)k²₂is absolutely continuous, and for a.e. t, there holds

d

dtku(t)k²_L2(Ω)= 2hu⁰(t), u(t)iL²(Ω). Returning to (4.2) we get

1

2ku(s)k²_L2(Ω)−1

2ku(0)k²_L2(Ω)+ Z s

0

Z

Ω

|∇u|²dxdt= Z s

0

Z

Ω

f udxdt. (4.3)

By Young’s inequality we have

Z s 0

Z

Ω

f udxdt

≤ α

2kfk²_L2(Ω×(0,s))+ 1

2αkuk²_L2(Ω×(0,s))

≤α

2kfk²_L2(Ω×(0,s))+ T 2α sup

t∈(0,T)

ku(t)k²_L2(Ω). (4.4)

Setting α= 2T, estimating (4.3) by (4.4) and passing to the supremum ins∈(0, T) finally gives 1

4 sup

t∈(0,T)

ku(t)k²_L2(Ω)+k∇uk²₂≤1

2kgk²_L2(Ω)+Tkfk²₂, as desired.

Now assume thatu₁ andu₂ are two solutions of (P_∞), and set w:=u₁−u₂. Then wis a solution of (P_∞) with f = 0 and g= 0. Therefore, applying (4.1) to w automatically givesw= 0, which proves the uniqueness of the solution of (P_∞).

5. Convergence of u

We now analyze the convergence of uλ, which will follow from energy bounds for uλ and u⁰_λ. As already mentioned before, the standard energy bound for the solutions of (Pλ) that is stated in Lemma 3.2 blows up a priori whenλgoes to +∞. Our goal in the sequel is to get better estimates, uniform in λ. The price to pay is the condition ∂ta≤0 which implies thatOa is nondecreasing in time (for the set inclusion).

5.1. First energy bound

Lemma 5.1. Assume that gλ∈L²(Ω) and fλ∈L²(0, T;L²(Ω)), and let uλ be the weak solution of problem (Pλ). Then,

1 4 sup

t∈(0,T)

kuλ(t)k²_L2(Ω)+k∇uλk²₂+λ Z T

0

Z

Ω

au²_λdxdt≤ kgλk²_L2(Ω)+Tkfλk²₂. (5.1)

Proof. Letu_λbe a solution of (P_λ) ands∈(0, T). Testing withv=u1_[0,s] in the weak formulation of (P_λ) 1

2kuλ(s)k²_L2(Ω)−1

2kuλ(0)k²_L2(Ω)+ Z s

0

Z

Ω

|∇uλ|²dxdt+λ Z s

0

Z

Ω

au²_λdxdt

= Z s

0

Z

Ω

f_λu_λdxdt.

and arguing as in the proof of Proposition 4.1, we obtain (5.1), so that we omit the details.

5.2. Second energy bound

We now derive a uniform bound on ku⁰_λk2. To this end, we will assume a time-monotonicity condition ona.

Definition 5.2 (Assumption (A)). We say that Assumption (A) hold ifa:QT →R⁺ is Lipschitz and

∂ta(x, t)≤0 for a.e. (x, t)∈QT. (5.2) Lemma 5.3. We suppose that Assumption (A) holds. Then, the solutionuλ of (Pλ)satisfies the estimate:

Z T 0

Z

Ω

(u⁰_λ)²dxdt+ sup

s∈(0,T)

Z

Ω

|∇u_λ(s)|²dx

≤ Z T

0

Z

Ω

f_λ²dxdt+ Z

Ω

|∇uλ(0)|²dx+λ Z

Ω

a(0)g²_λdx. (5.3)

Proof. Thanks to Lemma 3.2, we know that u⁰_λ ∈L²(0, T;H₀¹(Ω)). Consequently, for every s ∈ (0, T), the function v :=u⁰_λ1_(0,s) is an admissible test function in the weak formulation of (P_λ). Hence, we obtain the identity

Z s 0

Z

Ω

(u⁰_λ)²dxdt+ Z s

0

Z

Ω

∇u_λ· ∇u⁰_λdxdt+λ Z s

0

Z

Ω

au_λu⁰_λdxdt= Z s

0

Z

Ω

f_λu⁰_λdxdt,

or written differently (applying Rem.3.1), Z s

0

Z

Ω

(u⁰_λ)²dxdt+ Z s

0

1 2

Z

Ω

|∇uλ|²dx ⁰

dt+λ Z s

0

"

1 2 Z

Ω

au²_λdx ⁰

−1 2

Z

Ω

a⁰u²_λdx

# dt

= Z s

0

Z

Ω

fλu⁰_λdxdt.

This yields

Z s 0

Z

Ω

(u⁰_λ)²dxdt+1 2

Z

Ω

|∇uλ(s)|²dx+λ 2 Z

Ω

a(s)uλ(s)²dx−λ 2

Z s 0

Z

Ω

a⁰u²_λdxdt

= Z s

0

Z

Ω

f_λu⁰_λdxdt+1 2

Z

Ω

|∇uλ(0)|²dx+λ 2 Z

Ω

a(0)u_λ(0)²dx.

By Young’s inequality, Z s

0

Z

Ω

f_λu⁰_λdxdt≤1 2

Z s 0

Z

Ω

f_λ²dxdt+1 2

Z s 0

Z

Ω

(u⁰_λ)²dxdt, so that we obtain

Z s 0

Z

Ω

(u⁰_λ)²dxdt+ Z

Ω

|∇uλ(s)|²dx+λ Z

Ω

a(s)u_λ(s)²dx−λ Z s

0

Z

Ω

a⁰u²_λdxdt

≤ Z s

0

Z

Ω

|fλ|²dxdt+ Z

Ω

|∇uλ(0)|²dx+λ Z

Ω

a(0)uλ(0)²dx.

Finally, using Assumption (A), the initial condition on uλ(0) and passing to the supremum in s, we conclude that estimate (5.3) holds.

5.3. Weak convergence of solutions

Using the previous energy estimates, we first establish the weak convergence ofuλto the solutionuof problem (P_∞), under Assumption (A), and supposing certain bounds on the right hand side fλ and on the initial datag_λ.

Proposition 5.4. Assume that a satisfies Assumption (A). Let {fλ} be a bounded sequence in L²(QT) and {gλ} be a bounded sequence inH₀¹(Ω), satisfying

sup

λ

λ

Z

Ω

a(0)g_λ²dx

<∞. (5.4)

Up to extracting subsequences, we can assume that fλ converges weakly to a function f in L²(QT), and gλ

converges weakly to a function g∈H₀¹(Ω).

Let u_λ be the solution of (P_λ). Then u_λ converges weakly in L²(Q_T) to the unique solution u of problem (P_∞).

Proof. We know by Lemma5.1thatu_λis uniformly bounded inL²(0, T;H₀¹(Ω)), thus converges weakly (up to extracting a subsequence) inL²(0, T;H₀¹(Ω)) to some functionu∈L²(0, T;H₀¹(Ω)). UnderAssumption (A), we also know, thanks to Lemma 5.3, that

ku⁰_λkL²(Q_T)≤C,

so that,u⁰_λalso converges weakly inL²(QT) (up to extracting a further subsequence) to some limitw∈L²(QT), which must be equal tou⁰ by uniqueness of the limit inD⁰(QT). This shows thatu⁰∈L²(QT).

Next, due to (5.1) we know that

sup

λ

λ Z T

0

Z

Ω

au²_λdxdt

!

≤C,

which implies that, at the limit, umust be equal to zero a.e. on any set of the form {a > ε}, with ε >0. By considering the union for n∈N^∗ of those sets withε= 1/n, we obtain thatu= 0 a.e. on Q_T\O_a.

Now let us check thatusatisfies the equation in the weak sense. Letvbe any test function inL²(0, T;H₀¹(Ω)) such thatv= 0 a.e. inQT\Oa. Thenauλv= 0 a.e. inQT, and using the fact thatuλ is a solution of (Pλ), we can write

hu⁰_λ, vi_L2(Q_T)+h∇u_λ,∇vi_L2(Q_T)=hf_λ, vi_L2(Q_T). Thus, passing to the (weak) limit inuλ, u⁰_λ andfλ we get

hu⁰, vi_L2(Q_T)+h∇u,∇vi_L2(Q_T)=hf, vi_L2(Q_T).

To conclude that u is a solution of (P∞) it remains to prove that u(x,0) = g(x) for a.e. x∈ Ω. For this purpose, we let v ∈C¹([0, T], H₀¹(Ω)) be any function satisfying v(T) = 0. Testing the equation with this v, using that u_λ(0) =g_λ and integrating by parts with respect totwe obtain

−hgλ, v(0)i_L2(Ω)− Z T

0

huλ, v⁰i_L2(Ω)+ Z T

0

h∇uλ,∇vi_L2(Ω)= Z T

0

hfλ, vi.

Passing to the limit inλand using the weak convergence ofg_λ tog, we get

−hg, v(0)i_L2(Ω)− Z T

0

hu, v⁰i_L2(Ω)+ Z T

0

h∇u,∇vi_L2(Ω)= Z T

0

hf, vi_L2(Ω).

Integrating back again by parts onuyields

hg, v(0)i_L2(Ω)=hu(0), v(0)i_L2(Ω), and sincev(0) is arbitrary, we deduce thatu(0) =g inL²(Ω).

Finally, the convergence ofuλ touholdsa prioriup to a subsequence, but by uniqueness of the solution for the problem (P∞) (see Prop.4.1), the convergence holds for the whole sequence.

Corollary 5.5. Let O_a⊂Q_T be open and increasing in time (in the sense of Rem. 1.2), and letf ∈L²(Ω× (0, T))andg∈H₀¹(Ω). Then there exists a (unique) solution for (P_∞).

Proof. It suffices to apply Proposition5.4with, for instancea(x, t) := dist((x, t), Oa),fλ=f andgλ=g.

Remark 5.6. (Convergence inD⁰(Ω×(0, T))). UnderAssumption (A), lettingubeing the weak limit ofu_λ in L²(QT), we already know that

u= 0 a.e. in Q_T \O_a. Then

fλ+ ∆uλ−u⁰_λ−→f+ ∆u−u⁰ in D⁰(Ω×(0, T)), which implies that

λau_λ−→hin D⁰(Ω×(0, T)), (5.5)

for some distributionh=f+ ∆u−u⁰∈ D⁰(Ω×(0, T)), supported inO_a^c. Actually, sinceu= 0 inO_a^c, we have

∆u= 0 and u⁰ = 0 inD⁰(Oa c).

This means that

h= 0 inD⁰(Oa) and h=f inD⁰(Oa c).

Notice that, a priori,hcould have a singular part supported on∂O_a. We finally deduce that λau_λ −→

λ→+∞f1_Oc

a+ ∆u|∂Oa in D⁰(Ω×(0, T)). (5.6)

5.4. Strong convergence in L

(0, T ; H

(Ω))

We now go further using the same argument as for the stationary problem, and prove a stronger convergence which is one of our main results.

Theorem 5.7. Under the same hypotheses as in Proposition5.4, denote by u_λ the solution of(P_λ). Then, u_λ converges strongly inL²(0, T;H₀¹(Ω))to the solution uof problem(P_∞).

Proof. We already have the bound

kuλk_L2(0,T;H¹(Ω))≤C,

and we already know (by Prop. 5.4) thatuλ converges weakly inL²(0, T;H¹(Ω)) to u, the unique solution of problem (P_∞).

Moreover, by the lower semicontinuity of the norm with respect to the weak convergence, there holds kuk_L2(0,T;H¹(Ω))≤lim inf

λ kuλk_L2(0,T;H¹(Ω)).

Hence, to prove the strong convergence we only need to prove the reverse inequality, with a limsup. For this purpose we use the fact thatu(t) is a competitor foruλ(t) in the minimization problem solved byuλ attfixed.

Indeed, for a.e.t fixed,u_λsolves

−∆u_λ+λau_λ=f_λ−u⁰_λ, thus,uλ is a minimizer inH₀¹(Ω) for the energy

v7→Eλ(v)−2 Z

Ω

v(fλ−u⁰_λ),

whereE_λ is defined by (2.3). Furthermore, due to the bound (5.3) obtained in Lemma5.3, sincef_λis bounded in L²(QT) andgλ is bounded inL²(Ω) and satisfies (5.4), we know that u⁰_λ is bounded in L²(QT) , and

u⁰_λ→u⁰ weakly in L²(QT).

We also know that, up to a subsequence, uλ→ustrongly inL²(QT) (because it is bounded inH¹(QT)).

Now, using thatuis a competitor foruλ (for a.e.t fixed), we obtain Z

Ω

|∇uλ|²dx−2 Z

Ω

u_λ(f_λ−u⁰_λ)≤E_λ(u_λ)−2 Z

Ω

u_λ(f_λ−u⁰_λ)≤E_λ(u)−2 Z

Ω

u(f_λ−u⁰_λ)

= Z

Ω

|∇u|²dx−2 Z

Ω

u(f_λ−u⁰_λ).

Integrating int∈[0, T], passing to the limsup inλand since we have the convergence Z

Q_T

uλ(fλ−u⁰_λ)→ Z

Q_T

u(f−u⁰),

we get the desired inequality, which achieves the proof.

6. Simon’s exponential estimate 6.1. The stationary case

Following a similar argument to ([14], Thm. 4.1) we ascertain some strong convergence far from the set Ωa := Int(Ka), whereKa is defined by Ka :={x∈Ω;a(x) = 0}).

Lemma 6.1. Let a: Ω→R⁺ be a continuous non-negative potential and uλ be the unique weak solution in H₀¹(Ω)of−∆u_λ+λau_λ=f_λinΩ. Assume thatΩ_a := Int{a(x) = 0}= Int{K_a}is nonempty (hypothesis (2.2)).

Let ε >0 be fixed, and define

Ωε:={x∈Ω;dist(x,Ωa)> ε} and δ:= min

x∈Ωε

a(x)>0.

Then, there exists a constant C >0such that for allλ >0 and for everyW^2,∞functionη : Ω→Rthat is equal to1 in Ω2ε and to0 outsideΩε, we have

Z

Ωε

e²

√

λ^δ₂dist(x,Ω^c_2ε)η²uλ

λδ

2 uλ−fλ

dx≤C, (6.1)

with C=C(k∇ηk_∞, k∆ηk_∞, ε, sup_λkfλk2).

Proof. Letε >0 be fixed. For any functionψ∈H₀¹(Ωε) and for any functionρLipschitz satisfying|∇ρ|²≤δ/2, we start by computing the integral

Z

Ω

∇ e

√λρψ

· ∇ e⁻

√λρψ dx

= Z

Ω

√ λe

√λρ

ψ∇ρ+e

√λρ

∇ψ

·

−√ λe⁻

√λρ

ψ∇ρ+e⁻

√λρ

∇ψ dx

= Z

Ω

−λψ²|∇ρ|²+|∇ψ|² dx.

As a result, there holds the estimate Z

Ω

∇(e

√

λρψ)· ∇(e⁻

√

λρψ) +λaψ² dx≥

Z

Ω

λ(a− |∇ρ|²)ψ²dx

≥ λδ 2

Z

Ω

ψ²dx, (6.2)

by definition of δ.

Next, we apply (6.2) with the choice ψ=e

√λρηu_λ, where η∈W^2,∞(Ω,R) is equal to 1 in Ω_2εand equal to 0 outside Ωε. Thus, using the following computation:

∇ e²

√ λρη²uλ

· ∇uλ=h

∇ e²

√ λρηuλ

+e²

√

λρuλ∇ηi

·η∇uλ

=∇ e²

√ λρηuλ

·

∇(ηuλ)−uλ∇η +e²

√

λρuλη∇η· ∇uλ,

and the fact that ∇η = 0 in Ω2ε, we arrive at the following expression of the left-hand side of the previous inequality (6.2):

Z

Ω

∇(e

√

λρψ)· ∇(e⁻

√

λρψ) +λaψ² dx

= Z

Ωε

∇(e²

√

λρηuλ)· ∇(ηuλ) +λae²

√

λρη²u²_λ dx

= Z

Ω_ε

∇(e²

√λρη²u_λ)· ∇uλ+λae²

√λρη²u²_λ dx

− Z

Ωε\Ω2ε

e²

√

λρuλη∇η· ∇uλdx+ Z

Ωε\Ω2ε

∇(e²

√

λρηuλ)uλ∇ηdx.

Since the function e²

√

λρη²uλ ∈H₀¹(Ωε), it is an admissible test function for the equation satisfied by uλ, it follows that

Z

Ωε

∇(e²

√

λρη²uλ)· ∇uλ+λae²

√

λρη²u²_λ dx=

Z

Ωε

e²

√

λρη²uλfλdx.

Moreover,uλ satisfieskuλk_L2(Ω)+k∇uλk_L2(Ω)≤Ckfλk_L2(Ω) for a constantC, so that by Young inequality,

Z

Ωε\Ω2ε

e²

√

λρuλη∇η· ∇uλdx

≤Ck∇ηk_∞kfλk²_L2(Ω)e²

√ λM,

where M is defined by

M := sup

x∈Ω\Ω2ε

ρ(x).

Finally, since uλ∇η∈H₀¹(Ωε\Ω2ε), we can apply an integration by parts to obtain Z

Ω_ε\Ω2ε

∇(e²

√λρ

ηuλ)uλ∇ηdx=− Z

Ω_ε\Ω2ε

e²

√λρ

ηuλ(uλ∆η+∇uλ· ∇η) dx.

By a similar argument, we deduce the estimate

Z

Ωε\Ω2ε

∇ e²

√λρηu_λ

u_λ∇ηdx

≤C

k∇ηk_∞+k∆ηk_∞

kf_λk²_L2(Ω)e²

√λM.

Gathering the previous estimates, we conclude that λδ

2 Z

Ω_ε

e²

√

λρη²u²_λdx≤ Z

Ω_ε

e²

√

λρη²uλfλdx+Ckfλk²_L2(Ω)e²

√

λM, (6.3)

where C=C(k∇ηk∞,k∆ηk∞, ε).

Now, we specify the functionρby setting

ρ(x) :=

rδ

2dist(x,Ω^c_2ε),

which satisfies all our needed assumptions (i.e.ρis Lipschitz with|∇ρ|²≤δ/2 andρ= 0 outside Ωε). In this case,M = 0 thus (6.3) simply implies

λδ 2

Z

Ωε

e²

√λρη²u²_λdx≤ Z

Ωε

e²

√λρη²uλfλdx+Ckfλk²_L2(Ω),

or differently,

Z

Ωε

e²

√λρη²u_λ λδ

2 u_λ−f_λ

dx≤Ckf_λk²_L2(Ω), which ends the proof.

Remark 6.2. The previous lemma can be used for instance in the following two particular cases: first in the particular case when f = 0 in Ω\Ωa. Thus, we get the useful rate of convergence of uλ → 0 as λ→0 far

from Ωa:

Z

Ω_2ε

λe²

√

λ^δ₂dist(x,Ω^c_2ε)u²_λdx≤C.

This is much better compared to the usual and simple energy bound:

λ Z

Ω

au²_λ≤C.

Another application is when u_λ is an eigenfunction (this is actually the original framework of Simon [14]), i.e.whenf_λ=σ(λ)u_λand withσ(λ) standing for the first eigenvalue associated withu_λ. In this case, since we are assuming that the potential amight vanish in a subdomain (it could vanish at a single point, as performed by Simon [14]), we have that σ(λ) is bounded (cf. [2] for further details). Consequently, thanks to this bound forλlarge enough ^λδ₂ −σ(λ)≥1 which implies

Z

Ω_2ε

e²

√

λ^δ₂dist(x,Ω^c_2ε)u²_λdx≤C.

6.2. The parabolic case

We now extend the previous decay estimate to the parabolic problem.

Lemma 6.3. Leta:Q_T →R⁺be a continuous non-negative potential such thatO_a is nonempty,f_λ∈L²(Q_T), g_λ∈H₀¹(O_a∩ {t= 0})and letu_λ be the solution of(P_λ).

For every ε >0, we define Aε:=

(x, t)∈Q_T;dist (x, t), Oa

> ε and δ:= min

(x,t)∈Aε

a(x, t)>0. (6.4)

Then, for any λ≥4, and for anyW^2,∞ function η :Q_T →Requal to 1 inA_2ε and0 outsideA_ε, there exists a constant C >0such that

Z

A_ε

e

√λc_δdist((x,t),A^c_2ε)η²u_λ(x) λδ

2 u_λ−f_λ

dxdt≤C,

with cδ := 2 min(

qδ

2,^δ₂)and C=C(ε,k∇ηk_∞,k∆ηk_∞,k∂tηk_∞,sup_λkfλk2,sup_λkgλk2).

Proof. Letε >0 be fixed.

We consider any function ψ ∈ L²(0, T;H₀¹(Ω)) such that ψ = 0 outside Aε, and any Lipschitz function ρ:QT →Rsatisfying

max(|∂tρ|,|∇ρ|²)≤δ/2. (6.5)

Integrating in time estimate (6.2), and using the definition ofδ, we obtain Z

QT

∇(e

√

λρψ)· ∇(e⁻

√

λρψ) +λaψ²

dxdt≥λδ 2

Z

QT

ψ²dxdt.