ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC ... - UV

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA FRANCESCO PETITTA Abstract. Let Ω ⊆ RN a bounded open set, asymptotic behavior with respect to the time nonlinear parabolic problems whose model is ( ut (x, t) − ∆p u(x, t) = µ u(x, 0) = u0 (x)

N ≥ 2, and let p > 1; we study the variable t of the entropy solution of in Ω × (0, T ), in Ω,

where T > 0 is any positive constant, u0 ∈ L1 (Ω) a nonnegative function, and µ ∈ M0 (Q) is a nonnegative measure with bounded variation over Q = Ω × (0, T ) which does not charge the sets of zero p-capacity; moreover we consider µ that does not depend on time. In particular, we prove that solutions of such problems converge to stationary solutions.

Introduction A large number of papers was devoted to the study of asymptotic behavior for solution of parabolic problems under various assumptions and in different contexts: for a review on classical results see [12],[1], [22], and references therein. More recently in [13] the same problem was studied for bounded data and a class of operators rather different to the one we will discuss. Moreover, in [16] and [19] it was used an approach similar to our one, to face, respectively, the quasilinear case with natural growth terms of the type g(u)|∇u|2 and the linear case with general measure data. Let Ω ⊆ RN be a bounded open set, N ≥ 2, and let p > 1; we are interested in the asymptotic behavior with respect to the time variable t of the entropy solution of parabolic problems whose model is   ut (x, t) − ∆p u(x, t) = µ in Ω × (0, T ), (0.1) u(x, 0) = u0 (x) in Ω,  u(x, t) = 0 on ∂Ω × (0, T ), where T > 0 is any positive constant, u0 ∈ L1 (Ω) a nonnegative function, and µ ∈ M0 (Q) is a nonnegative measure with bounded variation over Q = Ω × (0, T ) which does not charge the sets of zero p-capacity in accordance with its definition given in [10]; 2000 Mathematics Subject Classification. 35B40, 35K55. Key words and phrases. Asymptotic behavior, nonlinear parabolic equations, measure data. 1

2

FRANCESCO PETITTA

moreover we suppose that µ does not depend on the time variable t (i.e. there exists a bounded Radon measure ν on Ω such that, for any Borel set B ⊆ Ω, and 0 < t0 , t1 < T , we have µ(B × (t0 , t1 )) = (t1 − t0 )ν(B)). Actually we shall investigate the limit as T tends to infinity of the solution u(x, T ) of the problem (0.1). Observe that by virtue of uniqueness results concerning entropy solutions of (0.1) we have uT (x, t) = uT 0 (x, t) a.e. in Ω, for all T > T 0 and t ∈ (0, T 0 ), where the index T and T 0 indicate that we deal with the solution of problem (0.1) respectively on QT and QT 0 ; so the function u(x, t) = uT (x, t), T > t is well defined for all t > 0; we are interested in the asymptotic behavior of u(x, t) as t tends to infinity. Actually, for a larger class of problem than (0.1), we shall prove that, as t tends to infinity, u(x, t) converges in L1 (Ω) to v(x), the entropy solution of the corresponding elliptic problem.  −∆p v(x) = µ in Ω, (0.2) v(x) = 0 on ∂Ω. 1. General assumptions and main result Let a : Ω × RN → RN be a Carathéodory function (i.e. a(·, ξ) is measurable on Ω, ∀ξ ∈ RN , and a(x, ·) is continuous on RN for a.e. x ∈ Ω) such that the following holds: (1.1)

a(x, ξ) · ξ ≥ α|ξ|p ,

(1.2)

|a(x, ξ)| ≤ β[b(x) + |ξ|p−1 ],

(1.3)

(a(x, ξ) − a(x, η)) · (ξ − η) > 0,

for almost every x ∈ Ω, for all ξ, η ∈ RN with ξ 6= η, where p > 1 and α, β are positive 0 constants and b is a nonnegative function in Lp (Ω). For every u ∈ W01,p (Ω), let us define the differential operator A(u) = −div(a(x, ∇u)), that, thanks to the assumption on a, turns out to be a coercive monotone operator 0 acting from the space W01,p (Ω) into its dual W −1,p (Ω). We shall deal with the solutions of the initial boundary value problem   ut + A(u) = µ in Ω × (0, T ), (1.4) u(x, 0) = u0 (x) in Ω,  u(x, t) = 0 on ∂Ω × (0, T ), where µ is a nonnegative measure with bounded variation over Q that does not depend on time, and u0 ∈ L1 (Ω).

ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 0

3

0

Remark 1.1. If µ ∈ Lp (0, T ; W −1,p (Ω)), it is well known that problem (1.4) has a unique variational solution u ∈ Lp (0, T ; W01,p (Ω)) ∩ C([0, T ]; L2 (Ω)) such that ut ∈ 0 0 Lp (0, T ; W −1,p (Ω)) and u(x, 0) = 0 , that is Z Z T Z T hf, ϕiW −1,p0 (Ω),W 1,p (Ω) dt, a(x, ∇u) · ∇ϕ dxdt = hut , ϕiW −1,p0 (Ω),W 1,p (Ω) dt + 0

0

0

Q

0

for all ϕ ∈ Lp (0, T ; W01,p (Ω)) (see [17] for the case p ≥ 2 and [15] for 1 < p < 2). Let us define the concept of elliptic p-capacity associated to our problem (for further details about elliptic p-capacity see [14]). Definition 1.2. Let K ⊆ Ω be a compact set, and W (K, Ω) = {ϕ ∈ C0∞ (Ω) : ϕ ≥ χK }, where χK represents the characteristic function of K. We define the elliptic p-capacity of the compact set K with respect to Ω as the quantity Z p e |∇ϕ| dx : ϕ ∈ W (K, Ω) , capp (K) = inf Ω

where we set inf ∅ = +∞ as usual. If U ⊆ Ω is an open set, we define the elliptic p-capacity of U as follows: capep (U ) = sup capep (K) : K ⊂ U, Kcompact . Finally, if B ⊆ Ω is a Borel set we define capep (B) = inf capep (U ) : B ⊆ U ⊆ Ω . We also recall the notion of parabolic p-capacity associated to our problem (for further details see for instance [10]). Definition 1.3. Let Q = QT = (0, T ) × Ω for any fixed T > 0, and let us call V = W01,p (Ω) ∩ L2 (Ω) endowed with its natural norm k · kW 1,p (Ω) + k · kL2 (Ω) and 0 n o p p0 0 W = u ∈ L (0, T ; V ), ut ∈ L (0, T ; V ) , endowed with its natural norm kukW = kukLp (0,T ;V ) + kut kLp0 (0,T ;V 0 ) . So, if U ⊆ Q is an open set, we define the parabolic p-capacity of U as capp (U ) = inf{kukW : u ∈ W, u ≥ χU a.e. in Q}, where again we set inf ∅ = +∞; then for any Borel set B ⊆ Q we define capp (B) = inf{capp (U ), U open set of Q, B ⊆ U }.

4

FRANCESCO PETITTA

Let us denote with M0 (Ω) the set of all measures with bounded variation over Ω that do not charge the sets of zero elliptic p-capacity, that is if µ ∈ M0 (Ω), then µ(E) = 0, for all E ∈ Ω such that capep (E) = 0; analogously we define M0 (Q) the set of all measures with bounded variation over Q that does not charge the sets of zero parabolic p-capacity, that is if µ ∈ M0 (Q) then µ(E) = 0, for all E ∈ Q such that capp (E) = 0. In [3] (for more details see also [6]) the concept of entropy solution of the elliptic boundary value problem associated to (1.4) was introduced: let µ ∈ M0 (Ω) be a measure with bounded variation over Ω which does not charge the sets of zero elliptic p-capacity; we call v an entropy solution for the boundary value problem ( A(v) = µ in Ω, (1.5) v=0 on ∂Ω, if v is finite a.e. and its truncated function Tk (v) ∈ W01,p (Ω) (recall that Tk (s) = max(−k, min(k, s)), see for instance [9] for more details), for all k > 0, and it holds Z Z (1.6) a(x, ∇v) · ∇Tk (v − ϕ) dx ≤ Tk (v − ϕ) dµ, Ω

Ω

for all ϕ ∈ W01,p (Ω) ∩ L∞ (Ω), for all k > 0; notice that the integral in the right hand side of (1.6) is well defined by virtue of the decomposition result of [6] (see below) and it turns out to be independent on the different decompositions of µ. Moreover observe that the gradient of such a solution v is not in general well defined in the sense of distributions, anyway it is possible to give a sense to (1.6), using the notion of approximated gradient of v, defined as the a.e. unique measurable function that coincides a.e. with ∇Tk (v), over the set where |v| ≤ k, for every k > 0 (see for instance [3]). Such a solution exists and is unique for all measures in M0 (Ω) (see [6]), and turns out to be a distributional solution of problem (1.5); moreover such a solution satisfies (1.6) with the equality sign (see [9]). An analogous definition will be given in the parabolic case following [21]. Definition 1.4. Let k > 0 and define Z Θk (z) =

z

Tk (s) ds, 0

as the primitive function of the truncation then we say that u(x, t) ∈ C(0, T ; L1 (Ω)) is   ut + A(u) = µ (1.7) u(x, 0) = u0 (x)  u(x, t) = 0,

function; let µ ∈ M0 (Q) and u0 ∈ L1 (Ω), an entropy solution of the problem in Ω × (0, T ), in Ω, on ∂Ω × (0, T ),

ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES

5

if, for all k > 0, we have that Tk (u) ∈ Lp (0, T ; W01,p (Ω)), and it holds Z Z Θk (u − ϕ)(T ) dx − Θk (u0 − ϕ(0)) dx Ω

Ω

Z (1.8)

T

hϕt , Tk (u − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt

+

0

0

Z

Z a(x, ∇u) · ∇Tk (u − ϕ) dxdt ≤

+ Q

Tk (u − ϕ) dµ, Q 0

0

for any ϕ ∈ Lp (0, T ; W01,p (Ω)) ∩ L∞ (Q) ∩ C([0, T ]; L1 (Ω)) with ϕt ∈ Lp (0, T ; W −1,p (Ω)). Remark 1.5. The entropy solution u of the problem (1.7) exists and is unique as shown in [21] for L1 (Q) data; this result was improved in many papers for more general measure data, in [20] it was proved via the notion of renormalized solution that turns out to be equivalent to the one of entropy solution with this kind of data (see [11]). Moreover, 1 the solution u is such that |a(x, ∇u)| ∈ Lq (Q) for all q < 1 + , even if its (N + 1)(p − 1) approximated gradient may not belong to any Lebesgue space. In [10] the authors prove the following useful result: Theorem 1.6. Let B be a Borel set in Ω, and 0 ≤ t0 < t1 ≤ T . Then we have capp (B × (t0 , t1 )) = 0

if and only if

capep (B) = 0.

Proof. See [10], Theorem 2.16.

Thanks to this result we derive that measures of M0 (Q) which do not depend on time can actually be identified with a measure in M0 (Ω); recall that, in [6] is proved that, if µ ∈ M0 (Ω), then it may be decomposed as µ = f − div(g), where f ∈ L1 (Ω) and 0 g ∈ (Lp (Ω))N . So, if µ is a measure of M0 (Q) which does not depend on time and B is a Borel set in Ω of zero elliptic p-capacity, then thanks to Theorem 1.6 we deduce that capp (B × (0, T )) = 0 and so µ(B × (0, T )) = 0; since µ is supposed to be independent on time, we have 0 = µ(B × (0, T )) = T ν(B), with ν ∈ M(Ω), and so ν(B) = 0, thus ν ∈ M0 (Ω). Hence, from now on, we shall always identify µ and ν. Moreover, notice that, if µ ≥ 0 is a measure in M0 (Ω), then f can be chosen to be nonnegative in the decomposition of [6]. Before passing to the statement and the proof of our main result let us state some interesting results about the entropy solution v of the elliptic problem (1.5). N (p−1)

According to [3] (see also [6]) we have that v is in the Marcinkiewicz space M N −p (Ω) that implies v ∈ Lq (Ω) fon any q < NN(p−1) ; hence, if p > N2N , we have v ∈ C(0, T ; L1 (Ω)). −p +1

6

FRANCESCO PETITTA

and let us observe that such a solution actually turns out to So let us suppose p > N2N +1 be an entropy solution of the initial boundary value problem (1.7) with initial datum u0 (x) = v(x), for all T > 0, since we have Z

Z Θk (v − ϕ)(T ) dx −

Ω

Θk (v − ϕ)(0) dx Ω

Z = Q

d Θk (v − ϕ) dxdt = dt

T

Z

h(v − ϕ)t , Tk (v − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

T

Z =−

hϕt , Tk (v − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

that can be cancelled out with the analogous term in (1.8) getting the right formulation (1.6) for v. For technical reasons we shall use the stronger assumption (1.9)

p>

2N + 1 N +1

throughout this paper; notice that, according to [5] (see also [10]), in this case a solution u of problem (1.4) belongs to Lr (0, T ; W01,r ) for any r < p− NN+1 . Observe that p− NN+1 > 1 if and only if (1.9); hence, in this case, the gradient of the entropy solution u (that coincides with the distributional one) actually belong L1 (Q). Moreover, this is the same assumption used in [21] since it allows, for instance, to get continuity of the solution with values in L1 (Ω) directly by using the trace result of [20]. Moreover, observe that, if µ ∈ M0 (Ω), and µ ≥ 0 then the entropy solution v of the elliptic problem is nonnegative; indeed, choosing in (1.6) as test function ϕ = Th (v + ) (where v = v + − v − is the usual decomposition of a function through its positive and negative parts), we get Z

Z

+

Tk (v − Th (v + )) dµ;

a(x, ∇v) · ∇Tk (v − Th (v )) dx ≤

(1.10) Ω

Ω 0

now, as we can write µ = f − div(g) ∈ L1 (Ω) + W −1,p (Ω), and Tk (v − Th (v + )) converges, as h tends to infinity, to Tk (−v − ) almost everywhere, ∗-weakly in L∞ (Ω), and weakly in W01,p (Ω) (see for instance [6]), we have Z

Z

Tk (v − Th (v )) dµ =

lim

h→+∞

+

Ω

Ω

Tk (−v − ) dµ ≤ 0;


7

On the other hand, observing that, for 0 ≤ v ≤ h we have Tk (v−Th (v + )) = Tk (−v − ) = 0, we can split the left hand side of (1.10) into three terms: Z Z + a(x, ∇v) · ∇Tk (v − Th (v )) dx = a(x, ∇v) · ∇Tk (v) dx {v≤0}

Ω

Z

Z a(x, ∇v) · ∇(v − h) dx +

+ {hh+k}

and the last term is obviously zero, while, using hypothesis (1.3), Z a(x, ∇v) · ∇(v − h) dx, {h 2N , and let N +1 µ ≥ 0; let u(x, t) be the entropy solution of problem (1.4) with u0 = 0, and v(x) the entropy solution of the corresponding elliptic problem (1.5). Then

lim u(x, t) = v(x),

t→+∞

in L1 (Ω). 2. Proof of Theorem 1.7 First of all, let us state and prove a comparison result that plays a key role in the proof of our main result. In the proof of this result and in what follows we will use several facts proved in [21] for L1 data and whose generalization to data in L1 (Q) + Lp (0, T ; W01,p (Ω)) is quite simple; in particular, we will use the fact that the approximating sequences of variational solutions are strongly compacts in C(0, T ; L1 (Ω)).

8

FRANCESCO PETITTA

Lemma 2.1. Let u0 , v0 ∈ L1 (Ω), such that 0 ≤ u0 ≤ v0 , and let µ ∈ M0 (Ω); if u and v are, respectively, the entropy solutions of the problems   ut + A(u) = µ in Ω × (0, T ), (2.1) u(x, 0) = u0 in Ω,  u(x, t) = 0 on ∂Ω × (0, T ), and   vt + A(v) = µ v(x, 0) = v0  v(x, t) = 0

(2.2)

in Ω × (0, T ), in Ω, on ∂Ω × (0, T ),

then u ≤ v a.e. in Ω, for all t ∈ (0, T ). Proof. First of all, let us suppose u0 , v0 ∈ L2 (Ω); we will use an approximation argument. 0 Consider the entropy solutions of the same problems with datum F ∈ W −1,p (Ω) instead of µ; let us call also these solutions u and v. These solutions coincide with the variational ones with this kind of data. Therefore we can use the variational formulation of problems (2.1) and (2.2), integrating between 0 and t, for any t ≤ T (we will use the notation Qt = Ω × (0, t)). Using ϕ = (u − v)+ as test function, and then subtracting, we get Z T 0= h(u − v)t , (u − v)+ iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

Z

(a(x, ∇u) − a(x, ∇v)) · ∇(u − v)+ dxdt

+ Qt

1 = 2

Z Z Ω

t

0

Z +

d [(u − v)+ ]2 dxdt dt

(a(x, ∇u) − a(x, ∇v)) · ∇(u − v)+ dxdt

Qt

1 = 2

Z

Z 1 [(u − v) ] (t) dx − [(u − v)+ ]2 (0) dx 2 Ω Ω Z + (a(x, ∇u) − a(x, ∇v)) · ∇(u − v)+ dxdt. + 2

Qt

Since the last term is positive we can drop it, while the second one is zero as u0 ≤ v0 . Therefore we have, for all t ∈ (0, T ) (u − v)+ = 0 a.e. in Ω, so u ≤ v a.e. in Ω, for all t ∈ (0, T ).


9

Now, let us consider u and v as the entropy solutions of problems (2.1) and (2.2) with nonnegative data in M0 (Ω), and u0 , v0 in L1 (Ω) as initial data; as we can write µ = 0 f − div(g) ∈ L1 (Ω) + W −1,p (Ω), we can approximate the L1 term f with a sequence fn of nonnegative regular functions that converges to f in L1 (Ω) and such that kfn kL1 (Ω) ≤ kf kL1 (Ω) ; moreover, let us consider two sequences of smooth functions u0,n and v0,n such that u0,n converges to u0 , and v0,n converges to v0 in L1 (Ω), with 0 ≤ u0,n ≤ v0,n . So, we can apply the result proved above finding two sequence of solutions un and vn of problems (2.1) and (2.2) with data fn − div(g) and u0,n , v0,n as initial data, for which the comparison result holds true; so un ≤ vn for all t ∈ (0, T ), a.e. in Ω. Now, it is well known (see for instance [20]) that the solutions of (2.1) and (2.2) obtained as limits of un and vn are unique and coincide with the unique entropy solution of the limit problem with datum µ, and initial datum, respectively, u0 and v0 ; then we have that the sequences un and vn converge, respectively, to u and v a.e. in Ω, for all fixed t ∈ (0, T ). So u ≤ v a.e. in Ω, for all fixed t ∈ (0, T ). Now, we are able to prove Theorem 1.7. For the sake of simplicity, in what follows, the convergences are all understood to be taken up to a suitable subsequence extraction. Proof of Theorem 1.7. To simplify the notation, during this proof, we will indicate by Q the parabolic cylinder of height one Ω × (0, 1), instead of QT as usual; let n ∈ N ∪ {0}, and define un (x, t) as the entropy solution of the initial boundary value problem  n n  in Ω × (0, 1), ut + A(u ) = µ (2.3) un (x, 0) = u(x, n) in Ω,  un (x, t) = 0 on ∂Ω × (0, 1), Recall that, since u ∈ C(0, T ; L1 (Ω)), then u(x, n) ∈ L1 (Ω) is well defined; moreover, observe that, by virtue of uniqueness of entropy solution and the definition of un , recalling that u(x, 0) = 0, we have that u(x, n) = un−1 (x, 1) a.e. in Ω, for n ≥ 1. Now, applying Lemma 2.1, and recalling that v, the entropy solution of problem (1.5), is also an entropy solution of problem (1.7) with v ≥ 0 itself as initial datum, we get immediately that (2.4)

u(x, t) ≤ v(x), for all t ∈ [0, T ], a.e. in Ω,

being u(x, t) solution of the same problem with u(x, 0) = 0 as initial datum. Moreover, applying again Lemma 2.1, we get that, for every n ≥ 0 (2.5)

un (x, t) ≤ v(x), for all t ∈ [0, 1], a.e. in Ω,

Finally, if we consider a parameter s > 0 we have that both u(x, t) and us (x, t) ≡ u(x, t + s) are solutions of problem (1.7) with, respectively, 0 and u(x, s) ≥ 0 as initial datum; so, again from Lemma 2.1 we deduce that u(x, t + s) ≥ u(x, t) for t, s ≥ 0, and so u is a monotone nondecreasing function in t. In particular u(x, n) ≤ u(x, m) for all

10

FRANCESCO PETITTA

n, m ∈ N with n < m, and so we have un (x, t) ≤ un+1 (x, t), for all n ≥ 0 and t ≥ 0. Therefore, from the monotonicity of un , it follows that exists a function u˜ such that, un (x, t) converges to u˜(x, t) a.e. on Q as n tends to infinity. Now, let us look for some a priori estimates concerning the sequence un . The symbol C will indicate any positive constant whose value may change from line to line. Let us fix n and take ϕ = 0 in the entropy formulation for un ; we get Z Z n Θk (u )(1) + α |∇Tk (un )|p dxdt Ω

(2.6)

Q

≤ k(|µ|M0 (Q) + ku(x, n)kL1 (Ω) ) ≤ k(|µ|M0 (Q) + kvkL1 (Ω) ) = Ck.

Therefore, for every fixed k > 0, from the first term on the left hand side of (2.6), recalling that un (x, t) is monotone nondecreasing in t, we get, reasoning as in [5], that un is uniformly bounded in L∞ (0, 1; L1 (Ω)) (and the bound does not depend on k), while from the second one we have that Tk (un ) is uniformly bounded in Lp (0, 1; W01,p (Ω)) for fixed k > 0. We can improve this kind of estimate by using a suitable Gagliardo-Nirenberg type inequality (see [2], Proposition 3.1) which asserts that, if w ∈ Lq (0, T ; W01,q (Ω)) ∩ L∞ (0, T ; Lρ (Ω)), with q ≥ 1, ρ ≥ 1. Then w ∈ Lσ (Q) with σ = q NN+ρ and Z Z ρq σ N |w| dxdt ≤ CkwkL∞ (0,T ;Lρ (Ω)) |∇w|q dxdt . Q

Q

Indeed, in this way we get Z (2.7)

p

|Tk (un )|p+ N dxdt ≤ Ck

Q

and so, we can write k

p p+ N

n

Z

n

meas{|u | ≥ k} ≤

p p+ N

|Tk (u )|

Z dxdt ≤

{|un |≥k}

p

|Tk (un )|p+ N dxdt ≤ Ck;

Q

then, meas{|un | ≥ k} ≤

C

p . k p−1+ N p Therefore, the sequence un is uniformly bounded in the Marcinkiewicz space M p−1+ N (Q); that implies, since in particular p > N2N , that un is uniformly bounded in Lm (Q) for +1 p all 1 ≤ m < p − 1 + N (for further properties of Marcinkiewicz spaces see for instance [23]).

(2.8)


11

We are interested about a similar estimate on the gradients of functions un ; let us emphasize that these estimate hold true for all functions satisfying (2.6), so we will not write the index n. First of all, observe that (2.9)

meas{|∇u| ≥ λ} ≤ meas{|∇u| ≥ λ; |u| ≤ k} + meas{|∇u| ≥ λ; |u| > k}.

With regard to the first term to the right hand side of (2.9) we have Z 1 meas{|∇u| ≥ λ; |u| ≤ k} ≤ p |∇u|p dx λ {|∇u|≥λ;|u|≤k} (2.10) Z Z 1 Ck 1 p |∇u| dx = p |∇Tk (u)|p dx ≤ p ; = p λ {|u|≤k} λ Q λ while for the last term in (2.9), thanks to (2.8), we can write meas{|∇u| ≥ λ; |u| > k} ≤ meas{|u| ≥ k} ≤ with σ = p − 1 +

p . N

C , kσ

So, finally, we get

Ck C + p, σ k λ and we can have a better estimate by taking the minimum over k of the right hand side; the minimum is achieved for the value 1 p σC σ+1 σ+1 k0 = λ , C meas{|∇u| ≥ λ} ≤

and so we get the desired estimate (2.11)

meas{|∇u| ≥ λ} ≤ Cλ−γ

σ with γ = p( σ+1 ) = N p+p−N = p − NN+1 ; this estimate is the same obtained in [4]. N +1 Then, coming back to our case, we have found that, for every n ≥ 0, |∇un | is equi+1 bounded in M γ (Q), with γ = p − NN+1 , and so, since p > 2N , |∇un | is uniformly N +1 bounded in Ls (Q) with 1 ≤ s < p − NN+1 . Now, we shall use the above estimates to prove some compactness results that will be useful to pass to the limit in the entropy formulation for un . Indeed, thanks to these estimates, we can say that there exists a function u ∈ Lq (0, 1; W01,q (Ω)), for all q < p − NN+1 , such that un converges to u weakly in Lq (0, 1; W01,q (Ω)). Observe that, obviously, we have u = u˜ a.e. in Q. On the other hand from the equation we deduce 0 0 q that unt ∈ L1 (Q) + Ls (0, 1; W −1,s (Ω)) uniformly with respect to n, where s0 = p−1 , for N all q < p − N +1 , and so, thanks to the Aubin-Simon type result proved in [8] we have that un actually converges to u in L1 (Q). Moreover, using the estimate (2.6) on the

12

FRANCESCO PETITTA

truncations of un , we deduce, from the boundedness and continuity of Tk (s), that, for every k > 0 Tk (un ) * Tk (u), weakly in Lp (0, 1; W01,p (Ω)), Tk (un ) → Tk (u), strongly in Lp (Q). Finally, the sequence un satisfies the hypotheses of Theorem 3.3 in [5], and so we get ∇un → ∇u a.e. in Ω. All these results allow us to pass to the limit in the entropy formulation of un ; indeed, for all k > 0, un satisfies Z (2.12) Θk (un − ϕ)(1) dx Ω

Z

Θk (un (x, 0) − ϕ(0)) dx

−

(2.13)

Ω

Z (2.14)

1

+

hϕt , Tk (un − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

Z (2.15)

+

a(x, ∇un ) · ∇Tk (un − ϕ) dxdt

Q

Z ≤

(2.16)

Tk (un − ϕ) dµ,

Q 0

0

for all ϕ ∈ Lp (0, 1; W01,p (Ω)) ∩ L∞ (Q) ∩ C([0, 1]; L1 (Ω)) with ϕt ∈ Lp (0, 1; W −1,p (Ω)). Let us analyze this inequality term by term: recalling that µ can be decomposed as 0 µ = f − div(g), where f ∈ L1 (Ω) and g ∈ (Lp (Ω))N , then, since Tk (un − ϕ) converges to Tk (u − ϕ) ∗-weakly in L∞ (Q), and Tk (un − ϕ) converges to Tk (u − ϕ) also weakly in Lp (0, 1; W01,p (Ω)), we have Z Z n n Tk (u − ϕ) dµ −→ Tk (u − ϕ) dµ; Q

Q

moreover, we can write Z (2.15) =

(a(x, ∇un ) − a(x, ∇ϕ)) · ∇Tk (un − ϕ) dxdt

Q

(2.17) Z + Q

a(x, ∇ϕ) · ∇Tk (un − ϕ) dxdt,


13

and the second term to the right hand side of (2.17) converges, as n tends to infinity, to Z a(x, ∇ϕ) · ∇Tk (u − ϕ) dxdt, Q

while to deal with the nonnegative first term of the right hand side of (2.17), we must use the a.e. convergence of the gradients; then, applying Fatou’s lemma, we get Z (a(x, ∇u) − a(x, ∇ϕ)) · ∇Tk (u − ϕ) dxdt Q

Z ≤ lim inf n

(a(x, ∇un ) − a(x, ∇ϕ)) · ∇Tk (un − ϕ) dxdt.

Q

Our goal is to prove that u = v almost everywhere in Ω; to do that, it is enough to prove that u does not depend on time, and that (2.12)+(2.13)+(2.14) converges to zero as n tends to infinity; indeed, if that holds true, we obtain that u satisfies the entropy formulation for the elliptic problem (1.5), and so, since the entropy solution is unique, we get that u = v a.e. in Ω. Let us prove first that u does not depend on time; let us denote by w(x) the almost everywhere limit of the monotone nondecreasing sequence un (x, 0) = u(x, n), hence, using the comparison Lemma 2.1, we have that, for fixed t ∈ (0, 1) un (x, 0) ≤ un (x, t) = u(x, n + t) ≤ u(x, n + 1) = un+1 (x, 0), and, since both un and un+1 converge to w(x) that does not depend on time, such happens also for the a.e. limit of un (x, t) that is u = w. Now, using the monotone convergence theorem, we get Z Z Θk (w(x) − ϕ(1)) dx − Θk (w(x) − ϕ(0)) dx lim[(2.12) + (2.13)] = n

Ω

Ω

Z Z = Ω

0

1

d Θ(w(x) − ϕ) dtdx = dt

Z

1

h(w(x) − ϕ)t , Tk (w(x) − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt, 0

0

while, since Tk (un − ϕ) converges to Tk (w − ϕ) weakly in Lp (0, 1; W01,p (Ω)), we have Z 1 n (2.14) −→ hϕt , Tk (w − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt. 0

0

Finally we can sum all these terms and, since w does not depend on time, we find Z 1 lim[(2.12) + (2.13) + (2.14)] = hwt , Tk (w − ϕ)iW −1,p0 (Ω),W 1,p (Ω) dt = 0; n

0

0

and, as we mentioned above, this is enough to prove that w(x) = v(x).

14

FRANCESCO PETITTA

3. Nonhomogeneous case Now we deal with the general case of problem (1.4) with a nonhomogeneous initial datum u0 , that is   ut + A(u) = µ in Ω × (0, T ), u(x, 0) = u0 (x) in Ω,  u(x, t) = 0 on ∂Ω × (0, T ),

(3.1)

with µ ∈ M0 (Q) satisfying the usual hypotheses used throughout this paper, and u0 a nonnegative function in L1 (Ω). We shall prove the following result: +1 Theorem 3.1. Let µ ∈ M0 (Q) be independent of the variable t, p > 2N , u0 ∈ L1 (Ω) N +1 be a nonnegative function, and let µ ≥ 0; let u(x, t) be the entropy solution of problem (3.1), and v the entropy solution of the corresponding elliptic problem (1.5). Then

lim u(x, t) = v(x),

t→+∞

in L1 (Ω). Most part of our work will be concerned with comparison between suitable entropy subsolutions and supersolutions of problem (1.4). The notion of entropy subsolution and supersolution for the parabolic problem will be given as a natural extension of the one for the elliptic case (see for instance [18]). Definition 3.2. A function u(x, t) ∈ C([0, T ]; L1 (Ω)) is an entropy subsolution of problem (3.1) if, for all k > 0, we have that Tk (u) ∈ Lp (0, T ; W01,p (Ω)) , and holds Z

+

Z

Θk (u − ϕ) (T ) dx − Ω

Ω

Z (3.2)

Θk (u0 − ϕ(0))+ dx

T

+

hϕt , Tk (u − ϕ)+ iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

Z

+

Z

a(x, ∇u) · ∇Tk (u − ϕ) dxdt ≤

+ Q

Tk (u − ϕ)+ dµ,

Q 0

0

for all ϕ ∈ Lp (0, T ; W01,p (Ω)) ∩ L∞ (Q) ∩ C([0, T ]; L1 (Ω)) with ϕt ∈ Lp (0, T ; W −1,p (Ω)) and u(x, 0) ≡ u0 (x) ≤ u0 (x) almost everywhere on Ω with u0 ∈ L1 (Ω).


15

On the other hand, u(x, t) ∈ C([0, T ]; L1 (Ω)) is an entropy supersolution of problem (3.1) if, for all k > 0, we have that Tk (u) ∈ Lp (0, T ; W01,p (Ω)) , and holds Z

−

Z

Θk (u − ϕ) (T ) dx − Ω

Ω

Z (3.3)

Θk (u0 − ϕ(0))− dx

T

+

hϕt , Tk (u − ϕ)− iW −1,p0 (Ω),W 1,p (Ω) dt 0

0

Z

−

Z

a(x, ∇u) · ∇Tk (u − ϕ) dxdt ≥

+ Q

Tk (u − ϕ)− dµ,

Q 0

0

for all ϕ ∈ Lp (0, T ; W01,p (Ω)) ∩ L∞ (Q) ∩ C([0, T ]; L1 (Ω)) with ϕt ∈ Lp (0, T ; W −1,p (Ω)) and u(x, 0) ≡ u0 (x) ≥ u0 (x) almost everywhere on Ω with u0 ∈ L1 (Ω). Thanks to Definition 3.2 it is possible to improve straightforwardly the comparison Lemma 2.1, by comparing both subsolution and supersolution with the unique entropy solution of problem (3.1); actually, we shall state this result in a simpler case that will be enough to prove Theorem 3.1. Moreover, notice that the same approximation argument we will use in the proof of next result allow us to deduce that a solution of problem (3.1) turns out to be both an entropy subsolution and an entropy supersolution of the same problem. Lemma 3.3. Let µ ∈ M0 (Ω), and let u and u be, respectively, an entropy subsolution and an entropy supersolution of problem (3.1), and let u be the unique entropy solution of the same problem. Then u ≤ u ≤ u. Proof. Let us prove only that u ≤ u being the other case analogous. We know that the entropy solution u is found as limit of regular functions un solutions of approximating 0 problems with smooth data. So, if µ = f − div(g) with f ∈ L1 (Ω) and g ∈ (Lp (Ω))N we can choose a sequence of smooth functions fn that converges to f in L1 (Ω); let us call µn = fn − div(g). Moreover, let u0,n be a sequence of smooth functions that converges to u0 in L1 (Ω); so, we call un the variational solution of problem

(3.4)

 n  (un )t + A(u ) = µn un (x, 0) = u˜0,n  u (x, t) = 0 n

in Ω × (0, T ), in Ω, on ∂Ω × (0, T ),

where u˜0,n = min(u0,n , u0 ); notice that, thanks to dominated convergence theorem, u˜0,n converges to u0 in L1 (Ω).

16

FRANCESCO PETITTA

Now, we can choose Tk (u − un )+ as test function obtaining Z T Z + h(un )t , Tk (u − un ) i dt + a(x, ∇un ) · ∇Tk (u − un )+ dxdt 0

Q

Z =

µn Tk (u − un )+ dxdt;

Q

On the other hand being u a subsolution we have Z Z + Θk (u − un ) (T ) dx − Θk (u − un )+ (0) dx Ω

Ω

Z

T

+

h(un )t , Tk (u − un )+ i dt

0

Z

Z

+

a(x, ∇u) · ∇Tk (u − un ) dxdt ≤

+ Q

Tk (u − un )+ dµdt.

Q

So, we can subtract this relation from the previous one, recalling that (u−un )+ (0) = 0, to obtain Z Z + Θk (u − un ) (T ) dx + (a(x, ∇u) − a(x, ∇un )) · ∇Tk (u − un )+ dxdt Q

Ω

Z ≤

+

Z

Tk (u − un ) dµdt − Q

+

Z

µn Tk (u − un ) dxdt + Q

Θk (u0 − uñ,0 )+ dx.

Ω

Hence, by stability results (see [21], [11], and references therein), using that u˜0,n converges to u0 in L1 (Ω), and the decomposition of µ and µn , we deduce that the right hand side of the above inequality tends to zero as n goes to infinity. Thus, the monotonicity assumption on a, and Fatou’s lemma, yield that u ≤ u˜, where u˜ is an entropy solution of problem (3.1) with u˜(0) = u0 . Hence, thanks to Lemma 2.1 we have u˜ ≤ u, and so, finally, we get u ≤ u. Observe that, we actually proved that, for every fixed T > 0, a.e. on Ω, we have u(x, T ) ≤ u(x, T ). Proof of Theorem 3.1. We will prove the result in few steps. Step 1. 0 ≤ u0 ≤ v. Let us first suppose that u solves problem (3.1) with initial datum satisfying 0 ≤ u0 ≤ v a. e. in Ω, where v is the entropy solution of problem (1.5); we proved that v is also an entropy solution of the initial boundary value problem (1.4) with v itself as initial datum, and so it turns out to be a supersolution of the same problem with initial datum u0 . Therefore, by Lemma 3.3, we get u(x, t) ≤ v(x) a.e. in Ω, for any t > 0. On the other hand, according to Lemma 2.1, the solution uˇ of problem (1.4) starting from 0 as initial datum satisfies uˇ(x, t) ≤ u(x, t) a.e. in Ω, for any t > 0.


17

Theorem 1.7 ensures that uˇ(x, t) converges toward v in L1 (Ω) as t goes to infinity. So, by comparison, we have that the solution u(x, t) of (1.4) converges to v in L1 (Ω) as t diverges. Step 2. 0 ≤ u0 ≤ v τ . Let us take uˆ(x, t) the solution of problem (3.1) with u0 = v τ as initial datum for some τ > 1, where v τ is the entropy solution of the elliptic problem (1.5) with µτ as 0 datum instead of µ = f − div(g) (f in L1 (Ω), g ∈ (Lp (Ω))N ), where ( τµ if f = 0, µτ = τ f − div(g) if f 6= 0. In [6] it was proved that, in the decomposition of µ, f can be chosen to be nonnegative and so, since τ > 1, we have that µτ ≥ µ; in fact, if f 6= 0, we can write µτ = (τ − 1)f + µ ≥ µ. Thus, we can use elliptic comparison results between entropy sub and supersolutions (see [18]) to obtain v ≤ v τ . Hence, since v τ does not depend on time, we have that it is a supersolution of the parabolic problem (3.1) with u0 = v τ , and, recalling that v is a subsolution of the same problem (being v ≤ v τ as initial data), we can apply the Lemma 3.3 again finding that v(x) ≤ uˆ(x, t) ≤ v τ (x) a.e. in Ω, for all positive t. Now, thanks to the fact that the datum µ does not depend on time, we can apply the comparison result also between uˆ(x, t + s) solution with u0 = uˆ(x, s), with s a positive parameter, and uˆ(x, t), the solution with u0 = v τ as initial datum; so we obtain uˆ(x, t + s) ≤ uˆ(x, t) for all t, s ≥ 0, a.e. in Ω. So, by virtue of this monotonicity result, and recalling that uˆ(x, t) ≥ v(x), we have that there exists a function v ≥ v such that uˆ(x, t) converges to v a.e. in Ω as t tends to infinity. Clearly v does not depend on t and we can develop the same argument used for the homogeneous case, by defining uˆn as the solution of  n  un ) = µ in Ω × (0, 1), uˆt + A(ˆ n uˆ (x, 0) = uˆ(x, n) in Ω,  uˆn (x, t) = 0 on ∂Ω × (0, 1), and starting from the analogous estimate Z Z n Θk (ˆ u )(t) + α |∇Tk (ˆ un )|p dxdt ≤ k(|µ|M0 (Q1 ) + ku0 kL1 (Ω) ) = Ck, Ω

Q1

to prove that we can pass to the limit in the entropy formulation, and so, by uniqueness result, we can obtain that v = v. So, we have proved that the result holds for the solution starting from u0 = v τ as initial datum, with τ > 1. Finally, if 0 ≤ u0 ≤ v τ for τ > 1, thanks to Lemma 3.3 we have (3.5)

uˇ(x, t) ≤ u(x, t) ≤ uˆ(x, t),

where uˇ, u and uˆ solves problem (3.1) with, respectively, 0, u0 and v τ as initial data.

18

FRANCESCO PETITTA

So, as we proved above, uˆ(x, t) tends to v as t goes to infinity, while, according to Theorem 1.7, the same happens for uˇ(x, t) ; therefore, using (3.5), we can conclude that the result holds true for u. Step 3. 0 ≤ u0 ∈ L1 (Ω) and µ 6= 0. Let us consider the general case of a solution u(x, t) with nonnegative initial datum u0 ∈ L1 (Ω) and let suppose first that µ 6= 0; let us define the monotone nondecreasing (in τ ) family of functions u0,τ = min(u0 , v τ ). As we have shown above, for every fixed τ > 1, uτ (x, t), the entropy solution of problem (3.1) with u0,τ as initial datum, converges to v a.e. in Ω, as t tends to infinity. Moreover, we have also that Tk (uτ (x, t)) converges to Tk (v) weakly in W01,p (Ω) as t diverges, for every fixed k > 0. Now, let us state the following lemma, that will be useful in the sequel: we shall prove it below, when the proof of Theorem 3.1 will be completed. Lemma 3.4. Let µ 6= 0 be a nonnegative measure in M0 (Ω), and τ > 0. Moreover, let vτ be the entropy solution of problem ( A(vτ ) = µτ in Ω, (3.6) vτ = 0 on ∂Ω, then, we have lim vτ (x) = +∞

τ →∞

almost everywhere on Ω. So, thanks to Lebesgue theorem, and Lemma 3.4, we can easily check that u0,τ converges to u0 in L1 (Ω) as τ tends to infinity. Therefore, using a stability result of entropy solution (see for instance [20]) we obtain that Tk (uτ (x, t)) converges to Tk (u(x, t)) strongly in Lp (0, T ; W01,p (Ω)) as τ tends to infinity. Now, making the same calculations used in [21] to prove the uniqueness of entropy solutions applied to u and uτ , where uτ is considered as the solution obtained as limit of approximating solutions with smooth data, we can easily find, for any fixed τ > 1, the following estimate Z Z Θk (u − uτ )(t) dx ≤ Θk (u0 − u0,τ ) dx, Ω

Ω

for every k, t > 0. Then, let us divide the above inequality by k, and let us pass to the limit as k tends to 0; we obtain (3.7)

ku(x, t) − uτ (x, t)kL1 (Ω) ≤ ku0 (x) − u0,τ (x)kL1 (Ω) ,

for every t > 0. Hence, we have ku(x, t) − v(x)kL1 (Ω) ≤ ku(x, t) − uτ (x, t)kL1 (Ω) + kuτ (x, t) − v(x)kL1 (Ω) ;


19

then, thanks to the fact that the estimate in (3.7) is uniform in t, for every fixed ε, we can choose τ¯ large enough such that ε ku(x, t) − uτ¯ (x, t)kL1 (Ω) ≤ , 2 for every t > 0; on the other hand, according to Step 2, there exists t¯ such that ε kuτ¯ (x, t) − v(x)kL1 (Ω) ≤ , 2 ¯ for every t > t, and this proves our result if µ 6= 0. Step 4. 0 ≤ u0 ∈ L1 (Ω) and µ = 0. If µ = 0 we can consider, for every ε > 0, the solution uε of problem  ε ε  ut + A(u ) = ε in Ω × (0, T ), (3.8) uε (x, 0) = u0 in Ω,  uε (x, t) = 0 on ∂Ω × (0, T ); thanks to Lemma 3.3 we get that u(x, t) ≤ uε (x, t),

(3.9)

for every fixed t ∈ (0, T ), and almost every x ∈ Ω, by the previous result we obtain lim uε (x, T ) = v ε (x),

T →∞

where v ε is the entropy solution of the elliptic problem associated to (3.8), and, moreover, by virtue of the stability result for this problem, we know that v ε (x) converges to 0 as ε tends to 0; so, almost everywhere on Ω, using (3.9), we have 0 ≤ lim sup u(x, T ) ≤ v ε (x), T →∞

and, since ε is arbitrary, we can conclude by using Vitali’s Theorem since v ε is strongly compact in L1 (Ω). 0

Proof of Lemma 3.4. Let us first suppose that µ ∈ W −1,p (Ω), so that we have µ = 0 −div(g) with g ∈ (Lp (Ω))N ; vτ then solves Z Z (3.10) a(x, ∇vτ ) · ∇ϕ dx = τ g · ∇ϕ dx, Ω

for every ϕ ∈ (1.1), we get

W01,p (Ω);

Ω

let us take ϕ = vτ as test function in (3.10); so, using assumption Z α Ω

|∇vτ |p dx ≤ τ kgk(Lp0 (Ω))N kvτ kW 1,p (Ω) ; 0

now, since vτ 6= 0, we can divide the above expression by τ kvτ kW 1,p (Ω) , getting 0 p−1 Z p−1 Z p p p 1 ∇ vτ1 dx |∇vτ |p dx ≤ kgk(Lp0 (Ω))N . = τ Ω Ω τ p−1

20

FRANCESCO PETITTA

vτ

is bounded in W01,p (Ω), and so there exists a function τ vτ v ∈ W01,p (Ω) and a subsequence, such that 1 weakly converges in W01,p (Ω) (and then τ p−1 a.e.) to v as τ tends to infinity. So, it is enough to prove that v > 0 almost everywhere on Ω to conclude our proof. To this aim, for every τ > 0, let us define 1 1 aτ (x, ξ) = a(x, τ p−1 ξ); τ we can easily check that such an operator satisfies assumptions (1.1), (1.2) and (1.3), with the same constants α and β. Notice that in the model case of the p-laplacian, vτ thanks to its homogeneity property, we have aτ ≡ a. Now, satisfies the elliptic 1 τ p−1 problem  vτ  −div(a (x, ∇ in Ω, τ  1 )) = µ   τ p−1 (3.11)  vτ    1 =0 on ∂Ω, τ p−1 Therefore, we have that

1 p−1

in a variational sense; indeed, for every ϕ ∈ W01,p (Ω), we have Z Z Z vτ 1 aτ (x, ∇ g · ∇ϕ dx. (3.12) ) · ∇ϕ dx = a(x, ∇vτ ) · ∇ϕ dx = 1 τ Ω Ω Ω τ p−1 Moreover, thanks to Theorem 4.1 in [7], we have that the family of operators {aτ } has a G-limit in the class of Leray-Lions type operators; that is, there exists a Carathéodory function a satisfying assumptions (1.1), (1.2) and (1.3), and a sequence of indices τ (k) (called τ again), such that G

aτ −→ a; vτ so, because of that, being v the weak limit of 1 in W01,p (Ω), we get that τ p−1 vτ τ →∞ a(x, ∇ ) −→ a(x, ∇v), 1 τ p−1 p0 N weakly in (L (Ω)) . Therefore, using this result in (3.12) we have Z Z a(x, ∇v) · ϕ dx = g · ∇ϕ dx, Ω

for every ϕ ∈

W01,p (Ω);

Ω

and so, v is a variational solution of problem ( −div(a(x, ∇v)) = µ in Ω, v=0 on ∂Ω.


21

Then, recalling that µ 6= 0 and using a suitable Harnack type inequality (see for instance [24]), we deduce that v > 0 almost everywhere on Ω. Now, if µ ∈ M0 (Ω), we have µτ = τ f − div(g), with f 6= 0 a nonnegative function in L1 (Ω); we can suppose, without loss of generality, that µτ = τ χE − div(g) for a suitable set E ⊆ Ω of positive measure; indeed, f , being nonidentically zero, it turns out to be strictly bounded away from zero on a suitable E ⊆ Ω, and so there exists a constant c such that f ≥ cχE , and then, once we proved our result for such a µτ , we can easily prove the statement by applying again a comparison argument. Now, reasoning vτ analogously as above we deduce that 1 solves the elliptic problem τ p−1  vτ 1  aτ (x, ( 1 )) = χB − div(g) in Ω,    τ τ p−1 (3.13)   v   τ1 = 0 on ∂Ω. τ p−1 Moreover, 1 χB − div(g) −→ χB , τ −1,p0 strongly in W (Ω) as τ tends to infinity. Therefore, since G-convergence is stable vτ under such type a of convergence of data, we have, that the weak limit v of in 1 τ p−1 W01,p (Ω), solves ( −div(a(x, ∇v)) = χB in Ω, v=0 on ∂Ω, and so we may conclude, as above, that v > 0 almost everywhere on Ω, that implies that vτ goes to infinity as τ tends to infinity. Remark 3.5. As we said before, in many cases, the convergences in norm to the stationary solution can be improved depending on the regularity of the limit solution (or equivalently to the regularity of the datum); indeed, consider (2.4) in the proof of Theorem 1.7, that is u(x, t) ≤ v(x), for all t ∈ (0, T ), a.e. in Ω; so, if, for instance, µ ∈ Lq (Ω) with q > Np , then Stampacchia’s type estimates ensure that the solution v of the stationary problem ( A(v) = µ in Ω, v=0 on ∂Ω, is in L∞ (Ω) and so the convergence of u(x, t) to v of Theorem 1.7 is at least ∗-weak in L∞ (Ω) and almost everywhere. Reasoning similarly one can refine, depending on the data, the asymptotic results of both Theorem 1.7 and Theorem 3.1.

22

FRANCESCO PETITTA

Acknowledgments I would like to thank Luigi Orsina for fruitful discussions on the subject of this paper, and the referee for useful comments. References [1] A. Arosio, Asymptotic behavior as t → +∞ of solutions of linear parabolic equations with discontinuous coefficients in a bounded domain, Comm. Partial Differential Equations 4 (1979), no. 7, 769–794. [2] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [3] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez, An L1 −theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241–273. [4] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149–169. [5] L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. , 147 (1997) no.1 , 237–258. [6] L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539– 551. [7] V. Chiad´ o-Piat, G. Dal Maso, A. Defranceschi, G-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 123–160. [8] A. Dall’Aglio, L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), 1335–1354. [9] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741–808. [10] J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (2003), no. 2, 99–161. [11] J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA (to appear). [12] A. Friedman, Partial differential equations of parabolic type, Englewood Cliffs, N.J., Prentice-Hall 1964. [13] N. Grenon, Asymptotic behaviour for some quasilinear parabolic equations, Nonlinear Anal. 20 (1993), no. 7, 755–766. [14] J. Heinonen, T. Kilpel¨ ainen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, Oxford, 1993. [15] R. Landes, V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Arkiv for Matematik, 25 (1987), no. 1, 29–40. [16] T. Leonori, F. Petitta, Asymptotic behavior of solutions for parabolic equations with natural growth term and irregular data, Asymptotic Analysis 48(3) (2006), 219–233. [17] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaire, Dunod et Gautier-Villars, (1969). [18] M. C. Palmeri, Entropy subsolution and supersolution for nonlinear elliptic equations in L1 , Ricerche Mat. 53 (2004), no. 2, 183–212. [19] F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data, C. R. Acad. Sci. Paris, Ser. I, 344 (2007) 571–576. [20] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura ed Appl. (IV), 177 (1999), 143–172.


23

[21] A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L1 data, Nonlin. Anal. TMA 28 (1997), 1943–1954. [22] S. Spagnolo, Convergence de solutions d’équations d’évolution, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), 311–327, Pitagora, Bologna, 1979. [23] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du seconde ordre à coefficientes discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258. [24] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl Math, 20 (1967), 721–747. ` La Sapienza, Piazzale A. Moro 2, 00185, Dipartimento di Matematica, Universita Roma, Italy E-mail address: [email protected]