Existence results for a Cauchy-Dirichlet parabolic problem with a

Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term

arXiv:1703.00834v2 [math.AP] 5 May 2017

Martina Magliocca A BSTRACT. We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following:  p −2 q   u t − div ( A ( t, x )∇ u |∇ u | ) = γ |∇ u | + f ( t, x ) in Q T ,  

u=0

on (0, T ) × ∂Ω,

u (0, x ) = u0 ( x )

in Ω,

where Q T = (0, T ) × Ω, Ω is a bounded domain of R N , N ≥ 2, 1 < p < N, the matrix A ( t, x ) is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition p p ( N + 1) − N max , 0, q < 2 and u0 ∈ Lσ (Ω) for σ < 2−q . In addition, the authors take into account initial data in Sobolev’s spaces, as well as the cases of attractive gradient (γ < 0) with positive initial data and of supernatural growth q ≥ 2 with σ ≥ 1 (in which existence fails). Even if this reference is concerned with the Cauchy problem, several arguments are actually local in space. In a similar spirit, we refer to [BD] for the study of the Cauchy-Dirichlet problem in the case of Laplace operator, f = 0 and q > 0, namely  q   ut − ∆u = γ|∇u| in (0, T ) × Ω, u=0 on (0, T ) × ∂Ω,   u(0, x ) = u0 ( x ) in Ω.

The authors provide existence, uniqueness and regularity results when the nature of the gradient is both repulsive and attractive and the initial datum is a function belonging to a Lebesgue space or a bounded Radon measure as well. The same problem is dealt with in [BDL] for regular initial data u0 ∈ C0 (Ω) for what concerns the long time behaviour of the solution. We underline that, because of the semigroup theory approach and the heat kernel regularity, the results proved in the works just mentioned cannot be extended to problems with more general operators like those considered in this paper, which include nonlinear operators with measurable coefficients. As for the case of p-Laplace operator with p > 2, problem (1.2) was treated in [At] for a gradient growth rate satisfying q > p − 1, f = 0 and u0 ∈ W 1,∞ (Ω). As far as general operators in divergence form are concerned, previous works have considered either the case that q = p or the case that q is sufficiently small. We refer to [BMP1, OP, DAGP, DAGSL] for what concerns the case in which the r.h.s. has natural growth (i.e. q = p). In particular, the problem (1.1) with q = p is studied in [DAGP] while [DAGSL] generalizes the r.h.s. to β(u)|∇u| p + f , β(·) bounded, polynomial or exponential as well. Furthermore, in [Po, DNFG] the authors take into account the case where the growth of the gradient is determined by the value p ( N + 1) − N (1.5) q= N+2

EXISTENCE RESULTS FOR PARABOLIC PROBLEM WITH A REPULSIVE GRADIENT TERM

3

which corresponds to a ”linear” growth, in a sense specified later. We point out that [Po] analyses the existence of weak solutions belonging at least to L1 (0, T; W01,1(Ω)) and thus a lower bound for p is required. However, the critical growth in (1.5) seems not to be always sharp for the problem to exhibit a ”linear” behaviour. We will show later some arguments aimed to justify our claim. Our purpose is filling the gap between the cases with ”sublinear” and natural growth for what concerns general operators in divergence form. This means that we deal with problems which have ”superlinear” growth in the gradient term (we will explain and comment later what we mean for ”superlinearity”). To some extent, this work is the extension to parabolic equations of similar results obtained in [GMP] for stationary problems. 1.1. Comments on the q growth and comparison with the stationary problem. We here present the stationary case of the problem (1.1) which is deeply analysed in [GMP, FM] (see also [AFM]). Such a problem reads ( α0 u − div a( x, u, ∇u) = H ( x, ∇u) in Ω, u=0 on ∂Ω and the hypotheses assumed are the following: α0 ≥ 0, the operator −div ( a( x, u, ∇u)) satisfies conditions of Leray-Lions type, the r.h.s. growth is determined by | H ( x, ξ )| ≤ γ|ξ |q + f ( x ) for p − 1 < q < p and f ∈ Lm (Ω) where m =

N (q − ( p − 1)) q

provided that m ≥ 1. We point out that the particular value of m above is optimal in the sense that it represents the minimal regularity one has to require on the source term f ∈ Lm (Ω) in order to have an existence result. We also refer to [HMV] for further comments on this sense. The a priori estimates proved in [GMP] state that

k∇[(1 + |u|)ρ ]k L p (Ω) ≤ M

(1.6)

( N − p )( q−( p −1))

where ρ = and with the constant M depends on the parameters of the problem and, above all, p ( p −q) on the forcing term. The dependence on f varies if α0 = 0 or α0 > 0. More precisely, if α0 = 0 a size condition on the data is required. On the contrary, the case α0 > 0 (which is the closest to the parabolic problem) does not need such a condition. In particular, in this last case, M remains bounded when f varies in sets which are bounded and equi-integrable in Lm (Ω). We underline that such a kind of dependence on the datum is due to the fact that we are in the superlinear growth setting. Roughly speaking, the l.h.s. grows like a ( p − 1)-power of |∇u| and thus we are saying that the r.h.s. grows faster (indeed q > p − 1). We conclude by pointing out that, on account of (1.6), depending on ρ ≥ 1 or ρ < 1 we have different ranges of q which lead to either solutions of finite energy or solutions with infinite energy. We will prove later that the parabolic problem (1.1) verifies an estimate which is similar to the one in (1.6). To be more precise, such an estimate has the form

kuk L∞ (0,T;Lσ (Ω)) + k∇[(1 + |u|) β ]k L p ( Q T ) ≤ M

(1.7)

where the value of σ, depending on p and q, will be discussed later and β = β( p, σ ). Concerning the similarity of the estimate, we want to underline that, again, the constant M remains bounded when u0 and f vary in sets which are bounded and equi-integrable in Lσ (Ω) and Lr (0, T; Lm (Ω)), for suitable values of m and r which will be largely commented in the following. We come back to emphasize that such a dependence is due to the ”superlinearity” growth rate of the r.h.s.. On the other hand, as far as the ”superlinearity” threshold of the q growth is concerned, we point out that the parabolic setting carries out noteworthy differences compared to the elliptic one. Indeed, the presence of the time derivative ut in (1.1) influences the relation between q and p and this fact clearly does not occur if we deal with stationary equations. We refer to [DNFG, Remark 3] for additional comments on this fact. We will explain soon that the threshold between linear/superlinear growth depends on the values of p we are taking into account. 2. On the superlinear setting In what follows, we are going to motivate the superlinear thresholds we will take into account during the paper. Moreover, we will highlight the link between the q growth of the gradient term and the Lebesgue spaces where the data u0 and f have to be taken in order to have an existence result. In order to explain the

4

M. MAGLIOCCA

assumptions we will require later, let us consider the Cauchy-Dirichlet problem for the standard p-Laplace operator    ut − ∆ p u = f in Q T , (2.8) u=0 on (0, T ) × ∂Ω,   u(0, x ) = 0 in Ω, where f ≥ 0 satisfies

N+2 ′ . N The function f will later play the role of the gradient term |∇u|q . The bound assumed on a allows us to give here a simple explanation through energy estimates. Indeed, for more regular forcing terms a similar explanation should make use of potential theory and Calderon-Zygmund estimates. Since the whole work will turn around energy estimates, it seems better to present a consistent argument below. In any case, the full range of a will be dealt with later in the article. We also restrict the present discussion to the case u0 = 0 (thus u ≥ 0) for simplicity. Basically, we look for a L1 ( Q T ) estimate of a suitable power of the gradient in term of the forcing terms f of the form ρ k|∇u|b k L1 ( Q T ) ≤ ck f k L a ( Q ) . f ∈ La (QT )

with

a≤

p

T

Then, when f =

|∇u|q ,

we wonder if ρ

k|∇u|b k L1 ( Q T ) ≤ ck|∇u|q k L a ( Q

T)

provides a useful estimate and in this case whether the estimate has a ”sublinear” or ”superlinear” character. The first question leads us to the condition aq ≤ b (2.9) in order to close the estimate. Then, the ”superlinear” homogeneity of the estimate holds if ρq > 1. b

(2.10) In order to find the exponents b and ρ involved, we formally multiply (2.8) by ϕ(u) = (1 + u)ν−1 − 1 with ν = ν( a) ∈ (1, 2) to be fixed. Thus, we have ¨ ¨ ˆ |∇u| p f (1 + u)ν−1 dx ds (2.11) dx ds ≤ c Φ(u(t)) dx + 2− ν Qt Q t (1 + u) Ω ú with Φ(·) defined as Φ(u) = 0 (1 + z)ν−1 − 1 dz. An application of Holder ¨ inequality with indices ( a, a′ ) ν and the inequality c(u − 1) ≤ Φ(u) provide us with the following estimate: ¨ ˆ |∇u| p 1 (u(t))ν dx + dx ds ≤ ck f k L a ( Qt ) k1 + ukν− + c. (2.12) ′ 2− ν L a ( ν −1 ) ( Q t ) Q t (1 + u) Ω ν + p −2

We now define v = (1 + u) p and rewrite (2.12) in terms of v: ¨ ˆ p ( ν −1 ) |∇v| p dx ds ≤ ck f k L a ( Qt ) kvk ν+′ pp−(ν2−1) (v(t))ν˜ dx + Qt

Ω

where ν˜ =

pν ν + p −2 .

L

a

ν + p −2 ( Q

+c

(2.13)

t)

˜ η = p and w = y allows us to deduce that v ∈ Invoking Theorem A.1 with h = ν,

+ ν˜ p NN

( Qt ). We point out that, in order to apply Gagliardo-Nirenberg inequality, we need ν˜ < p∗ : thus, 2N p . We go further imposing a′ (ν − 1) ν+ p−2 = p NN+ν˜ , algebraic computations force us to require p > N+ν otherwise a ( p − 1) − ( p − 2) ν = ν( a) = N . (2.14) N − p ( a − 1) Combining (a.4) with (2.13), we deduce that L

kv k which, being

N+p a′ N

p N + ν˜ L N

N+ p

N+ p

+ ν˜ p NN

( Qt )

≤ ck f k L aN( Q ) kvk t

+ ν˜ p NN a′ N

p N + ν˜ L N

( Qt )

+c

< 1, leads us to the following inequality: kv k

a( N+ p)

+ ν˜ p NN

L

p N + ν˜ N

( Qt )

p ( a −1 ) + c. ≤ ck f k LNa−( Q ) t

(2.15)


5

We now focus on gradient estimates and consider the gradient power |∇u|b with b ≤ p. ′ b (2− ν ) b (2− ν ) − p , so that Holder’s ¨ inequality If a < p NN+2 , we let b < p and we multiply |∇u|b by (1 + u) p (1 + u) p p with b , p−b provides us with ¨

b ¨ p−b b (2− ν ) p p |∇u| p p − b dx ds dx ds ( 1 + u ) 2 − ν Qt Q t (1 + u) b ¨ ¨ p−b p b (2− ν ) p p p ν + p −2 p − b |∇v| dx ds ≤c v dx ds .

|∇u|b dx ds ≤ Qt

¨

Qt

We thus impose

p b(2− ν ) p − b ν + p −2

Qt

= p NN+ν˜ which, in turn, implies that the exponent b has the following form:

N (ν + p − 2) + νp . N+ν Such a value of b, combined with the one previously found for ν (see (2.14)), becomes b = b( ν ) =

b = b( a ) = a

(2.16)

p ( N + 1) − N . N−a+2

(2.17)

(2.18)

′

If a = p NN+2 , then we set b = p and we have ν = ν˜ = 2 thanks to the definition of ν˜ and to (2.14). Then, (2.12) implies that ˆ t ˆ p + c, (2.19) (u(t))2 dx + k∇u(s)k L p (Ω) ds ≤ ck f k L a ( Qt ) k1 + uk p N+2 L

0

Ω

N

( Qt )

whether (2.15) becomes

kuk

p ( N +2)( N + p )

p NN+2

p N +2 L N

( Qt )

≤ ck f k LNa((pQ( N)+1)− N) + c. t

(2.20)

We point out that, since ν = 2, then (2.17) provides us with b = p. ′ Note that the assumption a ≤ p NN+2 ensures that b ≤ p. The bounds in (2.13), (2.15) and the inequality in (2.16) give the desired estimate of the gradient we were looking for, namely b ( N +2 )

a ( N +2 )

a +2 k|∇u|b k L1 ( Qt ) ≤ ck f k LNa−( Q + c = ck f k Lp(aN( Q+1))− N + c ) t

t

where the equality is due to the value of b in (2.18). Now we let f = |∇u|q , thus our last estimate becomes b ( N +2 )

k|∇u|b k L1 ( Qt ) ≤ ck|∇u|q k Lp(aN( Q+1))− N + c. t

We are ready to check the conditions in (2.9) and (2.10). We first require that aq ≤ b which implies that the estimate be closed giving q ( N +2 )

k|∇u|b k L1 ( Qt ) ≤ ck|∇u|b k Lp1( N( Q+1))− N + c. t

The condition aq ≤ b, combined with (2.14) and (2.17), implies that ν≥

N (q − ( p − 1)) . p−q

(2.21)

Moreover, (2.14) and (2.21) lead us to a≥

N (q − ( p − 1)) + 2q − p . q

We also notice that the same computation would remain unchanged if we had an initial datum u0 belonging N (q − ( p − 1)) to Lσ (Ω) with σ = . In particular, we have found the relations between the growth rate q and p−q the summability of u0 and f which are needed in order to have an existence result for (1.2). As far as the homogeneity of the estimate is concerned, we notice that q

N+2 >1 p ( N + 1) − N

6

M. MAGLIOCCA

leads us to the following superlinearity threshold for the growth q: q> If 1 < p ≤

N ( p − 1) + p . N+2

2N N +ν ,

we cannot apply Gagliardo-Nirenberg regularity result but we know that, at least, we b(2 − ν ) = ν in (2.16) (i.e. b = νp can impose a′ (ν − 1) = ν in (2.12) (i.e. a = ν) and 2 ). We underline that the p−b above conditions can be asked (and thus the estimate below holds) for every value of p. Then, another gradient estimate is given by k|∇u|b k L1 ( Qt ) ≤ ck f kνLν ( Qt ) . Letting f = |∇u|q in our last inequality provides that p

k|∇u| 2 ν k L1 ( Qt ) ≤ ck|∇u|q kνLν ( Qt ) + c.

(2.22)

Now the estimate is closed whenever we have that p 2 and, in this case, we always fall within a sublinear type of estimate. In particular, this means that the r.h.s. of p p (1.2) shows a sublinear (linear) growth for q < 2 (q = 2 ). q≤

We are going to clarify the meaning of the thresholds discovered above. N ( q−( p −1)) We first point out that, once we take ν = σ = , which gives the least (and therefore optimal) p−q integrability condition on the data, then the following double implication holds: p 2N ⇐⇒ q > . N+ν 2 o n p N ( p −1)+ p in order to have a superlinear character in the This means that we have to require q > max 2 , N +2 growth of the r.h.s.. We conclude that the superlinearity thresholds are p if 1 2 N ( p − 1) + p if p ≥ 2. q> N+2 In this range, the superlinear character of the estimate does not allow us to deduce an a priori estimate from the above arguments. The reader will see additional arguments, based on equi-integrability and continuity, in the proof of our result. p>

In the above explanation of the natural thresholds of the problem, we have supposed that the forcing term in (2.8) fulfils the same regularity in space and time. We now consider the case in which time and spatial summability may be different, i.e. we take f ∈ Lr (0, T; Lm (Ω)), and we wonder which is the curve where the exponents (m, r ) can live on in order to have an existence result. ¨ inequality with indices (m, m′ ) To this aim, we come back to (2.11) and, thanks to twice applications of Holder’s ′ and (r, r ) in the r.h.s., we get ˆ ¨ p ( ν −1 ) −2 ν˜ (2.23) (v(t)) dx + |∇v| p dx ds ≤ ck f k Lr (0,T;Lm(Ω)) kvk Lν+r¯ (p0,T;L ¯ ( Ω )) + c m Qt

Ω

where r¯ = r ′ (ν − 1)

p ν+ p−2

and

¯ = m ′ ( ν − 1) m

p . ν+ p−2

If N2N +ν < p < N, we apply again Theorem A.1 with h = ν˜ and η = p but we now focus on the case w 6 = y. Then, we have ˆ t ˆ t y y− p p kv(s)k Lw (Ω) ds ≤ ckvk L∞ (0,T;Lν˜ (Ω)) k∇v(s)k L p (Ω) ds (2.24) 0

0

where (w, y) satisfy the relation N ν˜ N ( p − ν˜ ) + pν˜ + = N. w y Observe that, if r 6= m, then y 6= w and vice versa. ¯ and y ≥ r¯: in this way, (2.25) leads us to the condition We go further requiring w ≥ m Nν N ( p − 2) + pν + ≤ N ( p − 1) + pν m r

(2.25)


7

i.e. the admissible range of the values (m, r ) for initial datum u0 fixed in Lν (Ω). In particular, we have that such a value of ν fulfils ν = Nm

r ( p − 1) − ( p − 2) . Nr − pm(r − 1)

(2.26)

¯ and y = r¯, i.e., when we assume the lowest regularity on (m, r ). if w = m ∞ ν 1 ν Finally, if 1 < p ≤ N2N +ν , the regularity u ∈ L (0, T; L ( Ω )) allows us to take f ∈ L (0, T; L ( Ω )) as the best choice. Observe that, letting ν = m in (2.26), gives

[ p( N + ν) − 2N ](r − 1) = 0 and thus there is continuity between the case 1 < p ≤ (m, r ).

2N N +ν

and

2N N +ν

< p < N for what concerns the values

3. Assumptions and statements 3.1. Assumptions. Let us consider the following nonlinear parabolic Cauchy-Dirichlet problem:    ut − div a(t, x, u, ∇u) = H (t, x, ∇u) in Q T , u=0 on (0, T ) × ∂Ω,   u(0, x ) = u0 ( x ) in Ω,

(P)

where Ω is an open bounded subset of R N , N ≥ 2, Q T = (0, T ) × Ω, a(t, x, u, ξ ) : (0, T ) × Ω × R × R N → R N and H (t, x, ξ ) : (0, T ) × Ω × R N → R are Caratheodory functions (i.e. measurable with respect to (t, x ) and continuous in (u, ξ )). We assume that the functions a(t, x, u, ξ ) and H (t, x, ξ ) are such that • the classical Leray-Lions structure conditions hold:

∃α > 0 : ∃λ > 0 :

α|ξ | p ≤ a(t, x, u, ξ ) · ξ,

(A1) p′

| a(t, x, u, ξ )| ≤ λ[|u| p−1 + |ξ | p−1 + h(t, x )] where h ∈ L ( Q T ),

( a(t, x, u, ξ ) − a(t, x, u, η )) · (ξ − η ) > 0 with 1 < p < N, for almost every (t, x ) ∈ Q T , for every u ∈ R and for every ξ, η in R N , ξ 6= η; • the r.h.s. satisfies the growth condition: ∃ γ s.t. | H (t, x, ξ )| ≤ γ|ξ |q + f (t, x ) p p ( N + 1) − N k} k Lr (0,T;Lm( Ω)) < 0 ∀k ≥ k0 . 2 Moreover, for k ≥ k0 , we set

(4.4)

(4.5)

(4.6)

T ∗ := sup{s ∈ [0, T ] : k Gk (wn (t))kσLσ ( Ω) ≤ δ0 ∀t ≤ s}. Since, thanks to [G, Theorem 1.1], {wn } ⊆ C ([0, T ]; Lν (Ω)) for every 1 ≤ ν < ∞, we have that T ∗ > 0 due to (4.5). If we suppose that t ≤ T ∗ in (4.4), then the definition of δ0 implies that ˆ ˆ ˆ 1 σ−1 t σ | G (wn (t))| dx + α˜ |∇[| Gk (wn )| β ]| p dx ds σ Ω k 2β p 0 Ω (4.7) ˆ yβ 1 yβ −( σ −1) σ | G (u0 )| dx + c2 k| f |χ{| f |>k} k Lr (0,T;Lm(Ω)) ≤ σ Ω k for every k ≥ k0 . We can extend the inequality (4.7) to the whole interval [0, T ] observing that, if t = T ∗ < T, then (4.5) and (4.6) lead to ˆ Ω

| Gk (wn ( T ∗ ))|σ dx < δ0


13

which is in contrast with the definition of T ∗ because of the continuity regularity C ([0, T ]; Lσ (Ω)). Therefore, we have that T ∗ = T and (4.7) holds for all t ≤ T, that is p

sup k Gk (un (t))kσLσ ( Ω) + k∇[| Gk (un )| β ]k L p ( Q

t ∈(0,T )

T)

≤ M1

∀k ≥ k0 .

(4.8)

The proof of the Theorem will be concluded once we show that |∇[| Tk0 (wn )| β ]| satisfies a bound like the one proved in (4.8). With this purpose, we multiply the equation in (4.2) for | Tk0 (wn )|σ−2 Tk0 (wn ) and integrate over Qt , so that we have: ¨ ¨ ˆ ( σ − 1) σ −1 dx ds + α˜ |wn || Tk0 (wn )| |∇(| Tk0 (wn )| β )| p dx ds Θk0 (wn (t)) dx + βp Q Q Ω t t ¨ ¨ ˆ ≤ γ˜ |∇wn |q | Tk0 (wn )|σ−1 dx ds + | f || Tk0 (wn )|σ−1 dx ds + Θk0 (u0 ) dx Qt

Qt

where Θk0 ( s) =

ˆ

s

0

Ω

| Tk0 (z)|σ−2 Tk¯ (z) dz.

The last integral in the r.h.s. is uniformly bounded in n thanks to the assumption on the initial datum u0 . As far as the first integral is concerned, we use the decomposition wn = Gk0 (wn ) + Tk0 (wn ) and estimate as below ¨ ¨ p−q q γ˜ ( σ −1 ) p + p γ˜ |∇wn |q | Tk0 (wn )|σ−1 dx ds ≤ q |∇[|wn | β ]|q | Tk0 (wn )| dx ds β Qt Qt ¨ ¨ ≤c |∇[| Tk0 (wn )| β ]|q dx ds |∇[| Gk0 (wn )| β ]|q dx ds + Qt

Qt

and the bound obtained in where c = c(σ, p, q, k0). Twice applications of Young’s inequality with (4.8) give us ¨ ¨ β q |∇[| Gk0 (wn )| β ]| p dx ds + c ≤ c0 [ M1 + 1] |∇[| Gk0 (wn )| ]| dx ds ≤ c

Qt

and

p p q , p−q

QT

¨

Qt

|∇[| Tk0 (wn )| β ]|q dx ds ≤ α˜

σ−1 2β p

¨

Qt

|∇[| Tk0 (wn )| β ]| p dx ds + c˜0

where both c0 and c˜0 depend on |Ω|, T and k0 . Finally, since ¨ Qt

| f || Tk0 (wn )|σ−1 dx ds ≤ c¯0 k f k Lr (0,T;Lm (Ω))

with c¯0 = c¯0 ( T, |Ω|, k0 ), we collect all the previous estimates in the following inequality: ˆ ¨ σ−1 Θk0 (wn (t)) dx + α˜ |∇(| Tk0 (wn )| β )| p dx ds p 2β Ω Qt i ˆ h |u0 |σ dx. ≤ c M1 + 1 + k f k Lr (0,T;Lm(Ω)) + Ω

In the end, we have found that:

p

k∇[| Tk0 (wn )| β ]k L p ( Q

T)

≤ M2

where M2 = M2 (δ0 , k0 , T, |Ω|, f , u0 ) besides the parameters given by the problem. Then, the inequality (4.1) follows with M depending on α, p, q, γ, N, |Ω|, T and k0 . In particular, since k0 = k0 (δ0 ), M remains bounded when u0 and f vary in sets which are bounded and equi-integrable, respectively, in Lσ (Ω) and Lr (0, T; Lm (Ω)). Part 2. 1,p We observe that the boundedness of {|un | β }n in L p (0, T; W0 (Ω)) does not provide that {un }n is uniformly 1,p

p bounded in L (0, T; W0 (Ω)) as well. However, choosing wn as test function and thanks to Young’s inequality p p with q , p−q , we get ˆ ¨ ¨ 1 |wn (t)|2 dx + |wn |2 dx dt + α˜ |∇wn | p dx dt 2 Ω Qt Qt ˆ ¨ ¨ 1 q |u0 | dx ≤ γ˜ |∇wn | |wn | dx dt + | f ||wn | dx dt + 2 Ω Qt Qt ¨ ¨ ¨ ˆ p 1 α˜ | f ||wn | dx dt + ≤ |∇wn | p dx dt + c |wn | p−q dx dt + |u0 |2 dx 2 Qt 2 Ω Qt Qt

14

M. MAGLIOCCA

from which ¨ α˜ |wn (t)| dx + |wn | dx dt + |∇wn | p dx dt 2 Qt Ω Q t ¨ ¨ ˆ p 1 | f ||wn | dx dt + ≤c |wn | p−q dx dt + |u0 |2 dx. 2 Ω Qt Qt 1 2

ˆ

2

¨

2

p

Then, since {wn }n is bounded in L p−q ( Q T ) (indeed, (since

m′ β

≤

m′ β ( σ − 1)

p β( p −q)

≤ p

N + βσ N

′

′

being σ ≥ 2) and in Lr (0, T; Lm (Ω))

and similar for r ′ ), the assertion follows.

p

C OROLLARY 4.3. Assume (A1), (A2), (H) with p − NN+2 ≤ q < p if N2N +2 < p < N and 2 < q < p if 2N 1 < p ≤ N +2 , (F1) and (ID1). Moreover, let {un }n be a sequence of solutions of (Pn ). Then, up to subsequences, un converges strongly to some function u in L p ( Q T ). P ROOF. Standard compactness results (see [S, Corollary 4]) guarantee that, up to subsequences, un converges strongly to u in L p ( Q T ). We here recall the hypotheses of [S, Corollary 4]: Let X, B and Y be Banach spaces such that X ֒→ B ֒→ Y where the embedding X ֒→ B is compact. Then, if {un }n ⊆ L p (0, T; X ), 1 ≤ p < ∞, and {(un )t }n ⊆ L1 (0, T; Y ), we have that {un }n is relatively compact in L p (0, T; B). 1,p

′

We thus apply the result above for p > 1, X = W0 (Ω), B = L p (Ω) and Y = W −1,s (Ω) and s greater than N. 4.2. The a.e. convergence of the gradient. We prove here that the a.e. convergence ∇un → ∇u holds. This last step is essential in order to prove the desired existence result: indeed, even if we would deal with linear operator in the l.h.s., the nature of the r.h.s. requires this step. P ROPOSITION 4.4. Assume 1 < p < N, (A1), (A2), (A3), (H) with p − NN+2 ≤ q < p if N2N +2 < p < N and p 2N 2 < q < p if 1 < p ≤ N +2 , (F1) and (ID1). Then there exists a subsequence (still denoted by un ) and a function u such that

∇un → ∇u a.e. Q T . Moreover Hn (t, x, ∇un ) converges strongly to H (t, x, ∇u) in L1 ( Q T ). P ROOF. Theorem 4.2 ensures that {|∇un |}n is bounded in L p ( Q T ). In particular, this means that the r.h.s. is bounded in L1 ( Q T ). Then, we can reason as in [BDAGO, Theorem 3.3] and deduce the a.e. convergence of the gradient. Now, we want to apply the Vitali Theorem in order to get the strong convergence of Hn (t, x, ∇un ). The a.e. convergence of Hn (t, x, ∇un ) to H (t, x, ∇u) holds by the a.e. convergence of the gradient seen above. It remains only to show that ¨ | Hn (t, x, ∇un )| dx dt → 0, lim sup | E |→0 n

E

E ⊂ Q T . The assumption (H) ensures that ¨ ¨ ¨ | Hn (t, x, ∇un )| dx dt ≤ |∇un |q dx dt + | f | dx dt. E

E

E

and thus, having {|∇un | p }n uniformly bounded in L1 ( Q T ), q < p and (F1), the assertion follows.

4.3. The existence result. We are now able to prove the following existence result. T HEOREM 4.5. Assume 1 < p < N, (A1), (A2), (A3), (H) with p − p 2

N N +2

≤ q < p if

< q < p if 1 < p ≤ N2N +2 , (F1) and (ID1). Then, there exists at least one finite energy solution u of (P) in the sense of Definition 4.1 which satisfies u ∈ C ([0, T ]; Lσ (Ω))

2N N +2 < p < N and 1,p ∈ L p (0, T; W0 (Ω))

(4.9)

and 1,p

|u| β ∈ L p (0, T; W0 (Ω)) with β =

σ−2+ p . p

(4.10)


15

1,p

P ROOF. Let {un }n ⊆ L p (0, T; W0 (Ω)) be the sequence of solutions of (Pn ).

1,p

Theorem 4.2 implies that {un }n and {|un | β }n are uniformly bounded in L∞ (0, T; Lσ (Ω)) ∩ L p (0, T; W0 (Ω)) 1,p

and in L p (0, T; W0 (Ω)) respectively and the inequality (4.1) holds. Moreover, Corollary 4.3 ensures that un → u in L p ( Q T ) (up to subsequences) and, in particular, un → u a.e. (again, up to subsequences). Then, since ∇un → ∇u a.e., we let n → ∞ in (4.1) and conclude that ˆ T p sup ku(t)kσLσ ( Ω) + k∇[(1 + |u|) β ]k L p (Ω) ≤ M 0

t ∈(0,T )

and (4.10) is proved. The continuity regularity follows by the Vitali Theorem. Indeed, let us consider the limit on n → ∞ in the inequality in (4.7), so that we have yβ ! ˆ ˆ ˆ Ω

| Gk (w(t))|σ dx ≤

Ω

| Gk (u0 )|σ dx + c

T

0

k| f (s)|χ{| f |>k} krLm (Ω) ds

r [ yβ −( σ −1)]

(w(t))|σ dx

´

converges to zero if k → ∞. This fact provides that for every k ≥ k0 . Thus, we deduce that Ω | Gk ˆ ˆ |w(t)|σ dx ≤ | Gk (w(t))|σ dx + kσ | E| E

Ω

converges to 0 if | E| → 0 and k → ∞. Now, let {t j } j be a sequence such that t j → t, t ∈ [0, T ], as j → ∞. The continuity regularity C ([0, T ]; L1 (Ω)) proved in [P] allows us to say that w(t j ) → w(t) in L1 (Ω) and conclude the proof of (4.9). The a.e. convergence of the gradient and (A2) imply that ′

a(t, x, un , ∇un ) ⇀ a(t, x, u, ∇u) weakly in ( L p ( Q T )) N and, from Proposition 4.4, we have Hn (t, x, ∇un ) → H (t, x, ∇u)

strongly in L1 ( Q T ).

Moreover, we observe that, by definition of {u0,n }n , the convergence u0,n → u0 in Lσ (Ω) holds. Thus, we take the limit on n in the weak formulation of (Pn ), so we get 1,p

u ∈ L p (0, T; W0 (Ω)) and

−

ˆ

Ω

u0 ( x ) ϕ(0, x ) dx +

¨

QT

−uϕt + a(t, x, u, ∇u) · ∇ ϕ dx dt =

¨

H (t, x, ∇u) ϕ dx dt QT

for every test function ϕ such that ′

1,p

ϕ ∈ L p (0, T; W0 (Ω)) ∩ L∞ ( Q T ), ϕt ∈ L p ( Q T ) and ϕ( T, x ) = 0, otherwise, we have recovered Definition 4.1.

5. Solutions of infinite energy p

Let us suppose that < p < N and the gradient growth rate satisfying max{ 2 , p − NN+1 } < q < p − NN+2 . In this range of q, the optimal conditions on the data (ID1) and (F1) do not allow us to have finite energy solutions as in Section 4: in particular, (ID1) implies that 1 < σ < 2, then u0 ∈ Lσ (Ω) does not necessarily belong to L2 (Ω). This is why we are going to consider a different notion of solution. 2N N +2

1,p

We define the set of functions T0 ( Q T ) as the set of all measurable functions u : Q T → R almost every1,p

where finite and such that the truncated functions Tk (u) belong to L p (0, T; W0 (Ω)) for all k > 0. Moreover, 1,p

in the spirit of [BBGGPV], we define the generalized gradient of a function u in T0 ( Q T ) as follows:

∇ Tk (u) = ∇uχ{|u| 0 as test function in (Pn ) and integrate over Qt for 0 ≤ t ≤ T. Thus, thanks to the assumptions (A1) and (H), we have: ˆ

Θε ( Gk (wn (t))) dx + α¯ Ω |

≤

ˆ

Ω

¨

Qt

|∇ Gk (wn )| p [ε + | Gk (wn )|]σ−3 | Gk (wn (s))| dx ds {z } A

B

z

Θε ( Gk (u0 )) dx + γ¯

¨

Qt

|∇ Gk (wn )| +

where we have set Θε (v) = define the function Φε (v) =

´v ´z

σ −3 | s| ds 0 0 ( ε + | s|) σ −3 1 ´v p | z | p dz so 0 ( ε + | z |)

A = α¯

¨

Qt

q

ˆ

0

¨ |

}|

Gk ( w n )

(ε + |z|) |f|

{| f |>k }∩ A k}

σ −3

ˆ

{

|z| dz dx ds (ε + |z|)σ−3 |z| dz dx ds {z }

Gk ( w n )

0

C

dz, Ak = Ak,n := {(s, x ) ∈ Qt : |wn (s, x )| > k}. We also we can rewrite the A term as

|∇Φε ( Gk (wn ))| p dx ds.

Now we are going to deal with the r.h.s.. Let us start with the B term. The definition of Φε (·) allows us to estimate as follows ˆ G ( wn ) ¨ p−q p−q k ( σ −3 ) p q p |z| dz dx ds B ≤ γ¯ |∇Φε ( Gk (wn ))| (ε + |z|) Q 0 ¨ t |∇Φε ( Gk (wn ))|q |Φε ( Gk (wn ))| p−q | Gk (wn )|q− p+1 dx ds ≤ γ¯ Qt

1 1 , where the last step is due to Holder’s ¨ inequality with indices p− q q−( p −1) (recall that q > p − 1). An p p∗ N application of the Holder ¨ inequality with indices q , p−q , p−q , Sobolev’s embedding and the definition of σ

(we just recall here that σ =

N ( q−( p −1)) ) p−q

B ≤ c1

ˆ

0

t

give us

σ

p−q

p

k Gk (wn (s))k Lσ (NΩ) k∇Φε ( Gk (wn (s)))k L p (Ω) ds

¯ N, q, T ). where c1 = c1 (γ, As far as the C term is concerned, we first observe that since σ < 2 and the equality σ − 1 = holds, then we have σp(+σp−−12) ˆ x ˆ x 1 σ −3 σ −3 (ε + |y|) |y| dy ≤ c (ε + |y|) p |y| p dy 0

0

σ −2 p

+1

p ( σ −1 ) σ + p −2

(5.3)


17

for some c > 0. Then, this estimate and twice applications of Holder’s ¨ inequality with (m, m′ ) and (r, r ′ ) imply that we can deal with C as below: C ≤ c2

¨

|f| {| f |>k }∩ A k

ˆ

Gk ( w n )

(ε + |z|)

σ −3 p

1 p

|z| dz

0

≤ c2 k| f |χ{| f |>k} k Lr (0,T;Lm(Ω)) kΦε ( Gk (wn ))k

σp(+σp−−12)

dx ds

σ −1 β m ′ σ −1 r ′ σ −1 β ( Ω )) L β (0,t;L

Then, recalling the Gagliardo-Nirenberg inequality in Theorem A.1, the definition of Φε (·) and the assumption (F1), we proceed as in Theorem 4.2 getting σ −1 β

C ≤ c2 k| f |χ{| f |>k} k Lr (0,T;Lm( Ω)) kΦε ( Gk (wn ))k Ly (0,t;Lw ( Ω)) yβ

β( y − p )

σ −1 ) ≤ c3 k| f |χ{| f |>k} k Lyβr (−(0,T;L m ( Ω )) + c4 k Gk ( w n )k L ∞ (0,t;L σ ( Ω ))

ˆ

t 0

p

k∇Φε ( Gk wn (s))k L p (Ω) ds.

We collect the estimates above saying that it holds ˆ t ˆ p Θε ( Gk (wn (t))) dx + α¯ k∇Φε ( Gk wn (s))k L p (Ω) ds Ω 0 ˆ t p−q σ p β( y − p ) k∇Φε ( Gk wn (s))k L p (Ω) ds ≤ c4 k Gk (wn )k L∞ (0,t;Lσ (Ω)) + c5 k Gk (wn )k L∞N(0,t;Lσ (Ω)) 0

+c3 k| f |χ{| f |>k} k

yβ yβ −( σ −1) L r (0,T;L m ( Ω))

+

ˆ

Ω

Θε ( Gk (u0 )) dx.

Next, we continue reasoning as in Theorem 4.2, i.e., we fix a value δ¯ such that satisfies the equality n o p−q β(y− p) 2 max c4 δ¯ σ , c5 δ¯ N = α¯ . Furthermore, we let k¯ large enough so that 2

k Gk (u0 )kσLσ (Ω) < σ (σ − 1)c3 k| f |χ{| f |>k} k

δ¯ 2

¯ ∀k ≥ k,

yβ yβ −( σ −1) L r (0,T;L m ( Ω))

0 : || Gk (wn (s))||σLσ ( Ω) ≤ δ, ¯ we We notice again that T ∗ > 0 due to (5.4) and the continuity of wn (t) in Lσ (Ω). Then, for t ≤ T ∗ and k ≥ k, have ˆ ˆ α¯ t p Θε ( Gk (wn (t))) dx + k∇Φε ( Gk (wn (s)))k L p (Ω) ds 2 Ω 0 (5.6) ˆ yβ yβ −( σ −1) ≤ Θε ( Gk (u0 )) dx + c3 k| f |χ{| f |>k} k Lr (0,T;Lm ( Ω)). Ω

We claim that

T∗

= T. Indeed, taking t = T ∗ < T leads to ˆ ˆ yβ yβ −( σ −1) ∗ Θε ( Gk (u0 )) dx + c3 k| f |χ{| f |>k} k Lr (0,T;Lm ( Ω)) ∀k ≥ k¯ Θε ( Gk (wn ( T ))) dx ≤ Ω

Ω

| Gk ( w n ( s ))|σ , σ ( σ −1 ε →0

which, by the convergence Θε ( Gk (wn (s))) −→ )

implies that

k Gk (wn ( T ∗ ))kσLσ (Ω) ≤ k Gk (u0 )kσLσ (Ω) + σ (σ − 1)c3 k| f |χ{| f |>k} krLr (0,T;Lm(Ω)) .

(5.7)

Thus, the conditions (5.4) and (5.5) and the inequality in (5.7) would give us ˆ | Gk (wn ( T ∗ ))|σ dx < δ¯ Ω

T∗

which is in contrast with the definition of and the continuity regularity. Recalling that wn = Gk (wn ) + Tk (wn ), we have just proved that {wn }n is bounded (uniformly in n) in L∞ (0, T; Lσ (Ω)).

18

M. MAGLIOCCA

As far as the proof of (5.2) is concerned, we note that (5.6) guarantees that |∇[Φε ( Gk (wn ))]| is uniformly bounded, in n and in ε, in L p ( Q T ). Then, being ˆ T ˆ T p p k∇Φ1 ( Gk (wn (s)))k L p (Ω) ds k∇Φε ( Gk (wn (s)))k L p (Ω) ds ≥ 0 ¨0 |∇ Gk (wn )| p (1 + | Gk (wn )|)σ−3 | Gk (wn )| dx ds = QT ¨ |∇ Gk+1 (wn )| p (1 + | Gk+1 (wn )|)σ−2 dx ds ≥c QT

we get an estimate on k(1 + | Gk (wn

(s))|) β−1 G

k ( w n ( s))k L p ( Q T )

for k ≥ k¯ + 1.

Finally, since ¨

≤c

¨

QT

|∇((1 + |wn |) β−1 wn )| p dx dt ≤ c

Q T ∩{|w n |> k }

¨

|∇wn | p dx dt p (1− β) Q T (1 + | w n |) ¨

|∇ Gk (wn )| p dx dt + c (1 + | Gk (wn )|) p(1− β)

Q T ∩{|w n |≤ k }

|∇ Tk (wn )| p dx dt

the inequality (5.2) follows taking Tk (wn ) as test function and reasoning as in the second part of Theorem 4.2. 5.2. Some convergence results. p

N P ROPOSITION 5.3. Let N2N +2 < p < N and assume (A1), (A2), (A3), (H) with max{ 2 , p − N +1 } < q < N p − N +2 , (F1), (ID1) and let {un }n be a sequence of solutions of (Pn ). Then, for some function u, we have that

un → u

a.e. in Q T ,

∇un → ∇u a.e. in Q T , L1 ( Q T )

Hn (t, x, ∇un ) → Hn (t, x, ∇u) strongly in and Tk (un ) → Tk (u)

strongly in

1,p

L p (0, T; W0 (Ω))

∀k > 0.

P ROOF. The boundedness of {|∇un |η }n . First of all, we prove that {|∇un |η }n is uniformly bounded in L1 ( Q T ) for some q < η < p. Indeed, this fact will allow us to reason as in [BDAGO, Theorem 3.3] (being the r.h.s. uniformly bounded in L1 ( Q T )) and get the a.e. convergence of the gradient. Since (1 + |un |) β−1 un < (1 + |un |) β for every n ∈ N, Theorem 5.2 implies that {(1 + |un |) β−1 un }n is σ

1,p

uniformly bounded in L∞ (0, T; L β (Ω)) ∩ L p (0, T; W0 (Ω)). Note that q

> 2p : we Nβ + σ p Nβ

in L

σ β

< p∗ if and only if p >

thus apply Gagliardo-Nirenberg regularity results and deduce that {(1 + |un

|) β−1 u

1,p L p (0, T; W0 (Ω))

( Q T ). Moreover, being {(1 + |un n }n bounded in ¨ |∇[(1 + |un |) β−1 |un |]| p dx dt c> QT ¨ |∇un | p (1 + β|un |) p dx dt = p (2− β) ( 1 + | u |) QT n ¨ |∇un | p > βp dx dt. p (1− β) Q T (1 + | un |)

|) β−1 u

2N N +σ

n }n

which is

is bounded

too, we have that

(5.8)

We now look for a bound for a suitable power of the gradient, that is, we employ (5.8) so that ¨ ¨ |∇un |η (1 + |un |)η (1− β) dx dt |∇un |η dx ds = η (1− β) ( 1 + | u |) QT QT n ¨ ηp ¨ p−p η (1− β ) |∇un | p pη p −η (1 + |un |) dx dt ≤ dx dt p (1− β) QT Q T (1 + | un |) ≤ c.

(1 − β ) Nβ + σ Nβ + σ = p . This condition leads to η = p = p−η N N+σ Nβ+ σ N (q − ( p − 1)) + 2q − p and thus it holds q 1.

Hence, we have to choose η such that pη


The a.e. convergence un → u. So far, we have that {|∇un |η }n is bounded in L1 ( Q T ). In particular, if p > 2 − 1,η W0 (Ω),

1 N +1 ,

19

then {|∇un |}n is bounded ′

B = Lη ( Q T ) and Y = W −1,s (Ω) in Lη ( Q T ). This means that we can invoke [S, Corollary 4] with X = with s > N obtaining that {un }n is compact in Lη ( Q T ). We thus deduce that, up to subsequences, un → u a.e.. 1 If, otherwise, N2N +2 < p ≤ 2 − N +1 , we reason as in [P, Theorem 2.1]. In particular, we point out that [S, Corollary 4] is applied (as in Corollary 4.3) to a regularization of the truncation function Tk (un ) instead of to un itself. The a.e. convergence ∇un → ∇u. Since the r.h.s. is bounded in L1 ( Q T ) and un → u a.e., the a.e. convergence of the gradients follows from [BDAGO]. The strong convergence Hn (t, x, ∇un ) → Hn (t, x, ∇u) in L1 ( Q T ). We conclude saying that, since η > q, we get the equi-integrability of the r.h.s. Hn (t, x, ∇un ) in L1 ( Q T ) and so an application of the Vitali Theorem gives us the desired convergence as well. 1,p

The strong convergence Tk (un ) → Tk (u) in L p (0, T; W0 (Ω)). The convergence in L1 ( Q T ) of the r.h.s. allows us to reason as in [BM, P], getting the strong convergence of the truncation functions. 5.3. The existence result. p

N T HEOREM 5.4. Let N2N +2 < p < N and assume (A1), (A2), (A3), (H) with max{ 2 , p − N +1 } < q < p − (F1) and (ID1). Then, there exists at least one solution of the problem (P) in the sense of Definition 5.1 satisfying 1,p

(1 + |u|) β−1 u ∈ L p (0, T; W0 (Ω)) with β = and

σ−2+ p p

N N +2 ,

(5.9)

u ∈ C ([0, T ]; Lσ (Ω)).

(5.10)

1,p

P ROOF. Let {un }n ⊆ L p (0, T; W0 (Ω)) be a sequence of weak solutions of (Pn ). The uniform bound (5.2) and the a.e. convergences un → u and ∇un → ∇u imply that u satisfies the following inequality: p sup ku(t)kσLσ ( Ω) + k∇((1 + |u|) β−1 u)k L p ( Q ) ≤ c. T

t ∈[0,T ]

1,p

This means that u ∈ L∞ (0, T; Lσ (Ω)) and (1 + |u|) β−1 u ∈ L p (0, T; W0 (Ω)). The continuity regularity (5.10) is a consequence of the Vitali Theorem and can be proved as in Theorem 4.5 (taking into account the inequality (5.7)). Now, we focus on (5.1). We multiply the equation in (Pn ) for S′ (un ) ϕ and integrate over Q T , getting ¨ ¨ ˆ S′ (un ) a(t, x, un , ∇un ) · ∇ ϕ dx ds −S(un ) ϕt dx ds + S(un (0)) ϕ(0, x ) dx + − Q Q Ω T T ¨ ¨ (5.11) H (t, x, ∇un )S′ (un ) ϕ dx ds S′′ (un ) a(t, x, un , ∇un ) · ∇un ϕ dx ds = + QT

QT

1,p L p (0, T; W0 (Ω)) ∩

p′

where ϕ ∈ L∞ ( Q T ), ϕt ∈ L ( Q T ) and ϕ( T, x ) = 0. The proof will be concluded once we show that the limit on n → ∞ can be taken in (5.11). Note that, being supp(S′ (un )) ⊆ [− M, M ], the equation (5.11) takes into account only TM (un ). The previous remark and Proposition 5.3 imply that S′ (un ) a(t, x, un , ∇un ) = S′ (un ) a(t, x, TM (un ), ∇ TM (un )) ′

→ S′ (u) a(t, x, TM (u), ∇ TM (u)) in ( L p ( Q T )) N . (5.12) Moreover, since a(t, x, un , ∇un ) · ∇un = a(t, x, TM (un ), ∇ TM (un ) · ∇ TM (un ). 1,p

the strong convergence of TM (un ) → TM (u) in L p (0, T; W0 (Ω)) implies that a(t, x, TM (un ), ∇ TM (un )) · ∇ TM (un ) → a(t, x, TM (u), ∇ TM (u)) · ∇ TM (u)

in L1 ( Q T ).

Having S′′ (un ) → S′′ (u) pointwisely and being S′′ (·) bounded by assumptions give us the following convergence: S′′ (un ) a(t, x, TM (un ), ∇ TM (un )) · ∇ TM (un ) → S′′ (u) a(t, x, TM (u), ∇ TM (u)) · ∇ TM (u)

in L1 ( Q T ). (5.13)

20

M. MAGLIOCCA

As far as the r.h.s. is concerned, Proposition 5.3 guarantees that S′ (un ) Hn (t, x, ∇un ) = S′ (un ) Hn (t, x, ∇ TM (un )) → S′ (u) H (t, x, ∇ TM (u))

in L1 ( Q T ).

(5.14)

Finally, since S( un ) → S( u)

(S(un ))t → (S(u))t

L p (0, T; W 1,p (Ω)),

in in

′

′

L p (0, T; W −1,p (Ω)) + L1 ( Q T )

and thus S(un ) → S(u) strongly in C ([0, T ]; L1 (Ω)) thanks to [P, Theorem 1], we can take the limit in (5.11) finding Definition 5.1. N R EMARK 5.5. We briefly present the case when N2N +1 < p < N and the growth rate of the gradient is q = p − N +1 1 + ω and the initial datum is taken in the Lebesgue space L (Ω) for ω ∈ (0, 1) and we are looking for renormalized solutions in the sense of Definition 5.1. Note that this value of q implies that σ = 1, so our running assumptions are not the sharp ones. However, having a stronger regularity on the initial datum allows us to repeat the proofs presented in Section 5. In particular, the a priori estimate reads as below: N N +1 , u0 {(1 + |un |)ν−1 un }n , ν

T HEOREM 5.6. Assume (A1), (A2), (H) with q = p − sequence of solutions of (Pn ). Then, {un }n and in

L∞ (0, T; L1+ω (Ω))

and in

1,p L p (0, T; W0 (Ω)).

∈ L1+ω (Ω), ω > 0, (F2) and let {un }n be a =

p −1 + ω , p

are uniformly bounded, respectively,

Moreover, the following estimate holds: p

ν −1 ω un )k L p ( Q sup kun (t)k1L+ 1+ ω ( Ω ) + k∇((1 + | u n |)

t ∈[0,T ]

T)

≤M

(5.15)

where the constant M depends on α, p, q, γ, N, |Ω|, T, ω, u0 , f and remains bounded when u0 and f vary in sets which are, respectively, bounded in L1+ω (Ω) and equi-integrable in L1 ( Q T ). We observe that the case q = p −

N N +1

could be dealt with taking the initial datum u0 in the Orlicz space u0 ∈ L1 ((log L)1 ). 6. Case of L1 data p p ( N +1)− N

} < q < p − NN+1 . We finally take into account the case with N2N +1 < p < N and max{ 2 , N +2 These ranges of q imply that the value σ (defined in (ID1)) is smaller than one. In this range even measure data could be considered, however, we focus on L1 (Ω) data for the sake of simplicity. We go further introducing our current notion of renormalized solution. 1,p

D EFINITION 6.1. We say that a function u ∈ T0 ( Q T ) is a renormalized solution of (P) if satisfies H (t, x, ∇u) ∈ L1 ( Q T ), ˆ ¨ −S(u) ϕt + a(t, x, u, ∇u) · ∇(S′ (u) ϕ) dx ds − S(u0 ) ϕ(0, x ) dx + Ω ¨ QT H (t, x, ∇u)S′ (u) ϕ dx ds =

(RS.1)

QT

1 n→∞ n lim

¨

a(t, x, u, ∇u) · ∇u = 0,

(RS.2)

{ n≤|u|≤2n} 1,p

for every S ∈ W 2,∞ (R ) such that S′ has compact support and for every test function ϕ ∈ L p (0, T; W0 (Ω)) ∩ ′ L∞ ( Q T ) such that ϕt ∈ L p ( Q T ) and ϕ( T, x ) = 0. R EMARK 6.2. Note that the main difference between Definitions 5.1 and 6.1 relies in the condition (RS.2). Indeed, the setting considered in Section 5 ensures that a solution in the sense of Section 5.1 enjoys 1,p

(1 + |u|) β−1 u ∈ L p (0, T; W0 (Ω)),

β=

σ+ p−2 < 1, p

which implies (RS.2). Roughly speaking, (RS.2) regards the behaviour for ”large” values of u (i.e., the case |u| = ∞ which is excluded by the truncated function) and it is a standard request in the renormalized framework. For further comments on the notion of renormalized solution we mention [LM, BGDM, DMMOP] for the stationary setting and [BM, BP] for what concerns the evolution framework.


21

We will need different spaces from the Lebesgue and the Sobolev’s ones we have used so far. In particular, we will use the Marcinkievicz space of γ order M γ ( Q T ). So, let us recall the definition and a few properties of this space. Let 0 < γ < ∞. Then M γ ( Q T ) is defined as the set of measurable functions f : Q T → R such that meas{(t, x ) ∈ Q T : | f | > k} ≤ ck−γ and it is equipped with the norm 1

k f k Mγ ( Q T ) = sup {kγ meas{(t, x ) ∈ Q T : | f | > k}} γ . k >0

Moreover, the following embeddings hold Lγ ( Q T ) ֒→ M γ ( Q T ) ֒→ Lγ−ω ( Q T ) for every ω ∈ (0, γ − 1], γ > 1. 6.1. The a priori estimate and convergence results. p ( N +1)− N

T HEOREM 6.3. Let N2N < q < p − NN+1 , (F2), (ID2) and +1 < p < N and assume (A1), (A2), (H) with N +2 ∞ let {un }n be a sequence of solutions of (Pn ). Then {un }n is uniformly bounded in L (0, T; L1 (Ω)) and {|∇un |q }n is uniformly bounded in L1 ( Q T ). P ROOF. We first prove that we are in theframework of Lemma B.1.

With this purpose, we take 1 − (1+| G 1( w )|)δ sign(un ), δ > 1 arbitrary large, as test function. Thus, dropping n k the gradient term in the l.h.s., we have ˆ ˆ 1 1 1− dx | Gk (wn (t))| dx − δ−1 Ω (1 + | Gk (wn (t))|)δ−1 Ω ¨ ¨ ˆ ˆ 1 1 q dx + γ |∇ G ( w )| dx dt + | f |χ{| f |>k} dx dt. ≤ | Gk (u0 )| dx − 1− n k δ−1 Ω (1 + | Gk (u0 )|)δ−1 QT QT Ω Letting δ → ∞, we obtain ˆ ˆ ¨ | Gk (wn (t))| dx ≤ | Gk (u0 )| dx + γ Ω

Ω

QT

|∇ Gk (wn )|q dx dt +

¨

QT

| f |χ{| f |>k} dx dt.

This means that Gk (wn ) satisfies an inequality of the type k Gk (wn )k L∞ (0,T;L1( Ω)) ≤ M where ˆ ¨ ¨ | Gk (u0 )| dx. | f |χ{| f |>k} dx dt + |∇ Gk (wn )|q dx dt + M=γ QT

QT

Ω

If, instead, we take Tj ( Gk (wn )) as test function then we can estimate as below: ¨ |∇ Tj ( Gk (wn ))| p dx dt α QT ¨ ˆ ¨ | f |χ{| f |>k} dx dt + | Gk (u0 )| dx |∇ Gk (wn )|q dx dt + ≤j γ QT

QT

and, in particular, we deduce that α

¨

QT

(6.1)

Ω

|∇ Tj ( Gk (wn ))| p dx ds ≤ jM.

Thus we can apply Lemma B.1 with v = Gk (wn ) obtaining

k|∇ Gk (wn )| "

≤ c k|∇ Gk (wn )|

p ( N +1)− N N +2

2 q p ( NN++1)− N

q p ( NN++12)− N L

Our current assumptions ensure that q L

k

p ( N +1)− N N +2

q

N +2 p ( N +1)− N ( Q

T)

k

N +2

M N +1 ( Q T )

#

+ k f |χ{| f |>k} k L1 ( Q T ) + k Gk (u0 )k L1 (Ω) .

N +2 N+2 N+2 < and so we have that the embedding M N+1 ( Q T ) ⊂ p ( N + 1) − N N+1

( Q T ) holds. We thus go further estimating from below as follows: k|∇ Gk (wn )|

≤ c k|∇ Gk (wn )|

p ( N +1)− N N +2

k

p ( N +1)− N N +2

L

q p ( NN++12)− N L

q

N +2 p ( N +1)− N ( Q

k

T)

q

N +2 p ( N +1)− N ( Q

T)

+ k f |χ{| f |>k} k L1 ( Q T ) + k Gk (u0 )k L1 (Ω) .

22

M. MAGLIOCCA

Now, set Yn,k = k|∇ Gk (wn )|

p ( N +1)− N N +2

k L

q

N +2 p ( N +1)− N ( Q

T)

and hk = k f χ{| f |>k} k L1 ( Q T ) + k Gk (u0 )k L1 ( Ω) so we rewrite q ( N +2 ) p ( N +1)− N

Yn,k − c1 Yn,k

≤ c2 h k .

We thus can reason as in [GMP], otherwise, we define the function q ( N +2 )

F (Y ) = Y − c1Y p( N+1)− N p ( N +1)− N p ( N +1)− N ( N +1)(q − p +1)+q −1 . In particular, F ( Z ∗ ) = F ∗ = F ∗ ( p, N, q, α, γ). which has a unique maximizer Z ∗ = c q( N +2) 1 Coming back to the inequality F (Yn,k ) ≤ c2 hk we observe that it is not trivial only if c2 hk < F ∗ . Hence, taking in mind the definition of hk , we define k∗ = inf{k > 0 : c2 hk < F ∗ }. Such a value of k∗ ensures that, being hk non increasing in k, we have that c2 hk < F ∗ for every k ≥ k∗ . We now consider the equation F (Yn,k ) = c2 hk and observe that it admits two roots, say Z1 and Z2 , which satisfy 0 ≤ Z1 < Z ∗ < Z2 . Thus the inequality F (Yn,k ) ≤ c2 hk implies that either Yn,k ≤ Z1 or Yn,k ≥ Z2 . Since the continuity of the function k → Yn,k and the convergence to zero of Yn,k for k → ∞ imply that Yn,k ≤ Z1 for all k ≥ k∗ we can say that ¨ QT

Finally, being ¨

q

QT

|∇un | dx dt ≤ c

∀k ≥ k∗ .

|∇ Gk (wn )|q dx dt ≤ c

¨

q

QT

|∇ Gk∗ (wn )| dx dt +

¨

q

|∇ Tk∗ (wn )| dx dt

QT q {|∇un | }n

we need an estimate on the last integral in order to prove that take Tk∗ (un ) as test function, obtaining ¨ ¨ ¨ q p ∗ |∇un | dx dt + |∇ Tk∗ (un )| dx dt ≤ k γ α QT

QT

(6.2)

is uniformly bounded in L1 ( Q T ). We

| f | dx dt + QT

ˆ

Ω

|u0 | dx .

from which ¨

¨ |∇ Tk∗ (wn )| dx dt ≤ c k∗ q

QT

¨ ≤ c k∗

q

QT

QT

|∇un | dx dt +

¨

|∇ Tk∗ (wn )|q dx dt

| f | dx dt + QT

q p

ˆ

Ω

|u0 | dx

qp

+c

thanks to (6.2). Young’s inequality allows us to conclude saying that ¨ |∇ Tk∗ (wn )|q dx dt ≤ c QT

where c depends on

k∗ ,

above the parameters of the problem.

P ROPOSITION 6.4. Let N2N +1 < p < N and assume (A1), (A2), (A3), (H) with (ID2) and let {un }n be a sequence of solutions of (Pn ). Then, for some function u, un → u

a.e. in Q T ,

Tk (un ) → Tk (u)

in L

p

< q < p−

N N +1 ,

(F2), (6.3)

∇un → ∇u a.e. in Q T , Hn (t, x, ∇un ) → H (t, x, ∇u)

p ( N +1)− N N +2

(6.4) 1

in L ( Q T ),

1,p (0, T; W0 (Ω))

∀k > 0.

(6.5) (6.6)

P ROOF. So far, we know that the r.h.s. { Hn (t, x, ∇un )}n is bounded in L1 ( Q T ). Classical estimates (see [BG, BDAGO]) allow us to get the a.e. convergence (6.3) (see also Proposition 5.3) and the boundedness of p ( N +1)− N

{|∇un |}n in M N+1 ( Q T ). The a.e. convergence in (6.3) and the boundedness of the r.h.s. in L1 ( Q T ) give us the a.e. convergence of the gradients (6.4) thanks to [BDAGO]. p ( N +1)− N

p ( N +1)− N

, the boundedness of {|∇un |}n in M N+1 ( Q T ) implies the equi-integrability Moreover, being q < N +1 of the r.h.s. and thus the strong convergence in L1 ( Q T ) of the r.h.s. through an application of the Vitali Theorem. Finally, we deduce the strong convergence of the truncation (6.6) recalling [BM, P].


23

6.2. The existence result. p ( N +1)− N

T HEOREM 6.5. Let N2N < q < p − NN+1 , (F2) and +1 < p < N and assume (A1), (A2), (A3), (H) with N +2 (ID2). Then, there exists at least one renormalized solution of the problem (P) in the sense of Definition 6.1 satisfying u ∈ C ([0, T ]; L1 (Ω)). P ROOF. The renormalized formulation (RS.1) can be proved reasoning as in Theorem 5.4; the continuity regularity C ([0, T ]; L1 (Ω)) can be deduced recalling the trace result [P] as well. Finally, the energy growth conditions (RS.2) is a consequence of the proof of [B, Theorem 2] (see also [BM, Lemma 3.2 and Remark 2.4]). 7. On the sublinear problem We are going to briefly discuss what we called the sublinear case. As we have already mentioned, this case was previously analysed in [Po, DNFG]. More precisely, the authors take into account a parabolic problem of Cauchy-Dirichlet type with lower order terms which grow as a power of the gradient |∇u|q for N ( p − 1) + p . We have already pointed out that such a threshold is not sharp for 1 < p < 2 since the q ≤ N+2 p borderline for the superlinear growth becomes q = 2 . We refer to Section 2 for further details on the argument presented to justify this assertion. So, let us our parabolic Cauchy-Dirichlet problem:    ut − div a(t, x, u, ∇u) = H (t, x, ∇u) in Q T , (Psub ) u=0 on (0, T ) × ∂Ω,   u(0, x ) = u0 ( x ) in Ω, where p is assumed to be 1 < p < 2. The hypotheses (Psub) are listed below: • the Leray-Lions structure conditions (A1), (A2), (A3) hold; • the function H (t, x, ξ ) : (0, T ) × Ω × R N → R grows at most as a power of the gradient plus a forcing term, namely

∃ γ s.t. | H (t, x, ξ )| ≤ γ|ξ |q + f (t, x ) with 0 < q ≤ 2p ,

(Hsub )

a.e. (t, x ) ∈ Q T , for all ξ ∈ R N and with f = f (t, x ) is some Lebesgue space. As far as the data are concerned, we assume that the initial datum verifies p u0 ∈ Lm (Ω) with m > 1 if 0 < q ≤ 2 and p u0 ∈ Lm (Ω) with m ≥ 1 if 0 < q < 2 and the forcing term satisfies f ∈ L1 (0, T; Lm (Ω)). We note here that, if m > 1, we could only deal with the linear case q =

|∇u|q

p 2

p 2.

≤ |∇u| + c when q < However, we will separate the growths q < features of the sublinear and linear settings.

p 2

p 2

(ID1sub ) (ID2sub ) (Fsub )

since, by Young’s inequality,

and q =

p 2

in order to stress the

7.1. The a priori estimate. T HEOREM 7.1. Assume 1 < p < 2, (A1), (A2), (Hsub ), either (ID1sub) or (ID2sub), (Fsub ) and let {un }n be a sequence of solutions of (Pn ). Then, {un }n is uniformly bounded in L∞ (0, T; Lm (Ω)). Moreover, we have that: 1,p

• if m > 1, then {(1 + |un |)µ−1 un }n is uniformly bounded in L p (0, T; W0 (Ω)) and the following estimate holds: m+ p−2 p µ −1 . (7.1) sup kun (t)km un )k L p ( Q ) ≤ M, µ= L m ( Ω) + k∇((1 + | un |) T p t ∈[0,T ] 1,p

In particular, if m ≥ 2, then {un }n is uniformly bounded in L p (0, T; W0 (Ω)). The constant M above depends on α, p, q, γ, N, |Ω|, T, k f k L1 (0,T;Lm( Ω)) and ku0 k Lm ( Ω) . p • If m = 1 and q < 2 , then {un }n satisfies the estimates of Lemmas B.1 and B.3. R EMARK 7.2. Note that the constant M depends on the initial datum u0 and on the forcing term f through their norms an not through an equi-integrability relation. We recall that this fact is due to the sublinear behaviour of the r.h.s..

24

M. MAGLIOCCA

p

P ROOF. The case m ≥ 2 and q < 2 . Having m ≥ 2 allows us to multiply the equation in (Pn ) by ϕ(un ) = |un |m−2 un so that an integration over Qt gives us ¨ ˆ α ( m − 1) 1 |∇[|un |µ ]| p dx ds |un (t)|m dx + m Ω µp Qt ¨ ˆ ¨ 1 (7.2) q m −1 m −1 |∇un | |un | dx ds + |u |m dx. ≤γ | f kun | dx ds + m Ω 0 Qt Qt | {z } | {z } B

A

Dealing with the A term turns out to be simpler than before, since the sublinear growth guarantees that we p p can proceed estimating by Young’s inequality with indices q , p−q as below ¨ p−q q γ ( m −1 ) p + p dx ds A= q |∇[|un |µ ]|q |un | µ Qt ¨ ¨ q α ( m − 1) m −1 + p − q µ p ≤ | u | |∇[| u | ]| dx ds + c dx ds. n n 1 2µ p Qt Qt p

q

We point out that having q < 2 implies that the exponent m − 1 + p−q is strictly smaller than m and this fact allows us to apply again Young’s inequality to the last term, so that ¨ ˆ α ( m − 1) 1 µ p A≤ |∇[|un | ]| dx ds + sup |un (t)|m dx + c2 2µ p 2m t∈(0,T ) Ω Qt where c2 = c2 ( T, |Ω|, m). The B term can be estimated by Holder’s ¨ and Young’s inequalities with (m, m′ ) as follows: ˆ t 1 −1 m B≤ k f k Lm ( Ω) kun km ds ≤ kun km L ∞ (0,T;L m ( Ω)) + c3 k f k L1 (0,T;L m ( Ω)) . Lm ( Ω) 4m 0

(7.3)

p

We summarize saying that, if m ≥ 2 and q < 2 , then it holds that ˆ ¨ α ( m − 1) 1 m |∇[|un |µ ]| p dx ds sup |un (t)| dx + 4m t∈(0,T ) Ω 2µ p QT ˆ 1 m ≤ c3 k f k L1 (0,T;Lm(Ω)) + |u0 |m dx + c2 . m Ω p

The case m ≥ 2 and q = 2 . We come back to (7.2) and assume that t ≤ t1 ≤ T, where t1 has to be fixed. Now, we have m − 1 + and we estimate A as follows: ¨ ¨ α ( m − 1) µ p |∇[|un | ]| dx ds + c˜1 |un |m dx ds A≤ 2µ p Qt Qt ˆ ¨ α ( m − 1) µ p ˜ |un (t)|m dx + c˜3 . |∇[| u | ]| dx ds + c t sup ≤ n 2 1 2µ p Ω Qt t ∈(0,t )

q p−q

=m

1

Then, letting c˜2 t1
0 4

(7.7)

and recalling (7.6), we deduce ¨ |∇un | p α ( m − 1) 3bm − c˜2 t1 sup kun (t)km + dx ds m ( Ω) L 2− m 4 2 Q t1 (1 + | un |) t ∈[0,t1 ] ˆ ≤ am +C |u0 |m dx + c2 k f km L1 (0,T;L m ( Ω)) Ω

where C = C ( T, m, |Ω|). We conclude partitioning the time interval [0, T ] into a finite number of subintervals [t j , t j+1 ], 0 ≤ j ≤ n − 1, where t0 = 0 and tn = T so that (7.7) is fulfilled in each subinterval. We have thus proved (7.1).

26

M. MAGLIOCCA

p

The case m = 1 and q < 2 . We claim that we are within the assumptions of Lemma B.3. Indeed, we can reason as in the first part of Theorem 6.3 and obtain the L1 data classical estimates (see [BG]): ˆ sup |un (t)| dx ≤ M t ∈[0,T ] Ω

and α where M=γ

¨

QT

¨

QT

|∇ Tk (un )| p dx ds ≤ kM

|∇un |q dx ds +

¨

| f | dx ds + QT

ˆ

Ω

|u0 | dx.

Then, Lemma B.3 implies that p

k|∇un |q k 2q

p M 2q ( Q

from which, being q
0.

P ROOF. We just deal with the case 1 ≤ m < 2, since having m ≥ 2 allows us to reason as in Corollary 4.3 and Proposition 4.4. We start proving that {|∇un |b }n is bounded in L1 ( Q T ) for some q N2N +m : we thus reason as in Proposition 5.3, getting b = p N + m . Algebraic computations p show that such a value of b satisfies the inequalities q ≤ < b, being m > 1, and b < p, since µ < 1. 2

The case 1 < m < 2 and 1 < p ≤ N2N +m . Again, we reason as in Proposition 5.3 estimating the b power of the gradient as bp ¨ p−p b ¨ ¨ (1− µ ) |∇un | p pb p −b b (1 + |un |) dx dt . dx dt |∇un | dx ds ≤ p (1− µ ) QT QT Q T (1 + | un |) However, since our current range of p implies leads to b =

m µ

> p∗ , we are forced to require pb

p p m which verifies q < 2 m < p since 1 < m < 2. 2

(1 − µ) = m. This condition p−b

Since the r.h.s. is bounded in L1 ( Q T ), we can prove the a.e. convergences of un → u and ∇un → ∇u as in Proposition 5.3. More precisely, we have to split the proof between b greater and smaller than one and follows


27

the line of the Proposition quoted above. Now, having q < b, we deduce the equi-integrability of the r.h.s. and the strong convergence in L1 ( Q T ) of the r.h.s. follows, again, by Vitali Theorem. The strong convergence of the truncation function can be deduced as in Proposition 5.3. The cases m = 1. We just recall Proposition 6.4 and say that we can proceed in the same way.

7.3. The existence result. T HEOREM 7.4. Let 1 < p < 2 and assume (A1), (A2), (A3), (Hsub ), either (ID1sub) or (ID2sub) and (Fsub ). Then, there exists at least one weak solution of the problem (Psub) such that

• if m ≥ 2, then 1,p

u ∈ L p (0, T; W0 (Ω)) and satisfies the weak formulation ¨ ˆ ¨ −uϕt + a(t, x, u, ∇u) · ∇ ϕ dx dt = − u0 ( x ) ϕ(0, x ) dx + QT

Ω

H (t, x, ∇u) ϕ dx dt QT

1,p

′

for every test function ϕ ∈ L p (0, T; W0 (Ω)) ∩ L∞ ( Q T ) such that ϕt ∈ L p ( Q T ) and ϕ( T, x ) = 0. Moreover, this solution fulfils the following regularity: 1,p

|u|µ ∈ L p (0, T; W0 (Ω)) with µ =

m−2+ p ; p

• if 1 < m < 2, then 1,p

u ∈ T0 ( Q T ) , H (t, x, ∇u) ∈ L1 ( Q T ),

−

ˆ

Ω

S(u0 ) ϕ(0) dx +

¨

=

¨Q T

−S(u) ϕt + a(t, x, u, ∇u) · ∇(S′ (u) ϕ) dx dt

H (t, x, ∇u)S′ (u) ϕ dx dt,

QT 1,p

W 2,∞ (R ) such that S′ (·) has compact support and for every test function

for every S ∈ ϕ ∈ L p (0, T; W0 (Ω)) ∩ ′ L∞ ( Q T ) such that ϕt ∈ L p ( Q T ) and ϕ( T, x ) = 0. Moreover, such a solution fulfils the following regularities: 1,p

(1 + |u|)µ−1 u ∈ L p (0, T; W0 (Ω)) with µ =

m−2+ p ; p

• if m = 1, then 1,p

u ∈ T0 ( Q T ) , H (t, x, ∇u) ∈ L1 ( Q T ),

−

ˆ

Ω

1 lim n→∞ n

¨

S(u0 ) ϕ(0) dx +

¨

=

a(t, x, u, ∇u) · ∇u = 0, { n≤|u|≤2n}

¨Q T

−S(u) ϕt + a(t, x, u, ∇u) · ∇(S′ (u) ϕ) dx dt

H (t, x, ∇u)S′ (u) ϕ dx dt,

QT 1,p

for every S ∈ W 2,∞ (R ) such that S′ (·) has compact support and for every test function ϕ ∈ L p (0, T; W0 (Ω)) ∩ ′ L∞ ( Q T ) such that ϕt ∈ L p ( Q T ) and ϕ( T, x ) = 0. P ROOF. The a priori estimate proved in Theorem 7.1 and the convergence results in Proposition 7.3 allow us to reason as in Theorem 4.5 if m ≥ 2, Theorem 5.4 whenever 1 < m < 2 and, finally, as in Theorem 6.5 as m = 1. R EMARK 7.5. In particular, |∇u|b ∈ L1 ( Q T ) for b = p < N.

p 2m

if 1 < p ≤

2N N +m

and b =

N ( m −2+ p )+ pm N +m

if

2N N +m

k} ≤

´T

= cM

0

η

k Tk (v(t))k Lη (Ω) dt kη

p+ N N

k−

p ( N +1)− N N

≤ cM η − p+1k−(η −1)

.

N+ p

Taking k = h p( N+1)− N , we obtain h meas{(t, x ) ∈ Q T : |v|

p ( N +1)− N N+ p

N

> h} N+ p ≤ cM

and so the first part is concluded. We go further defining the function ϕ(k, λ) = meas{(t, x ) ∈ Q T : |∇v| p > λ and |v| > k} and observing that, being ϕ(k, ·) non increasing in the λ variable, the following inequalities hold: ˆ ˆ ˆ 1 λ 1 λ 1 λ ϕ(0, θ ) dθ = ϕ(k, θ ) + [ ϕ(0, θ ) − ϕ(k, θ )] dθ ≤ ϕ(k, 0) + ϕ(0, θ ) − ϕ(k, θ ) dθ. ϕ(0, λ) ≤ λ 0 λ 0 λ 0 Moreover, since ϕ(0, θ ) − ϕ(k, θ ) = meas{(t, x ) ∈ Q T : |∇v| p > θ and |v| ≤ k} and thus ˆ

∞

[ ϕ(0, θ ) − ϕ(k, θ )] dθ =

ˆ

0

0

T

p

k∇ Tk (v(t))k L p (Ω) dt ≤ kM

we finally get ϕ(0, λ) ≤ ϕ(k, 0) +

Mk . λ

The definition of ϕ(k, λ) allows us to say that meas{(t, x ) ∈ Q T : |∇v| p > λ} ≤ cM

p+ N N

k−

p ( N +1)− N N

+

Mk λ

30

M. MAGLIOCCA

which, minimizing the r.h.s. in k, becomes N +2

meas{(t, x ) ∈ Q T : |∇v| p > λ} ≤ cM N+1 λ The estimate (b.2) follows taking λ = h

2 p p ( NN++1)− N

−

p ( N +1)− N p ( N +1 )

.

.

R EMARK B.2. Note that we have just proved that the assumptions given in the previous statement ensure v ∈ p ( N +1)− N N

( Q T ) and |∇v| ∈ M M [AMSLT, ST]).

p ( N +1)− N N +1

( Q T ) which are the regularities satisfied by the heat problem (we refer to

1,p

L EMMA B.3. Let 1 0. Then, we have that |∇v| 2 ∈ M1 ( Q T ). Moreover, the following estimate holds: p

k|∇v| 2 k M1 ( Q T ) ≤ cM.

(b.3)

where the constant c depends on N, p and q. ˇ P ROOF. We just proceed as before changing the Gagliardo-Nirenberg inequality into Cebyˇ se¨ v’s one, so our starting step reads M meas{(t, x ) ∈ Q T : |v| > k} ≤ c . k R EMARK B.4 (Comparison between estimates (b.2) and (b.3)). We point out that (b.2) and (b.3) respectively imply N +2

k∇vk M

p ( N +1)− N N +1 (QT )

k∇vk

p M2

(QT )

≤ cM p( N+1)− N 2

≤ cM p

which hold for every 1 N2N +1 . If instead we take into account the homogeneity exponent, we have that (b.2) exhibits a preferable bound than (b.3) only when p ≥ 2. C. On the sharpness of the assumptions. We go on with our analysis observing that the assumption (ID1) is the weakest one, within the class of Lebesgue spaces, which allows us to have an existence result. Roughly speaking, assuming (A1), (A2), (A3), (H) and u0 in a Lebesgue space Lη (Ω) with 1 ≤ η < σ does not allow (P) to necessarily admit a solution (in some sense). For the sake of simplicity, let us consider the problem (P) with the p-Laplace operator, the q power of the gradient in the r.h.s. and zero forcing term f :  q   ut − ∆ p u = γ|∇u| in Q T , (c.1) u=0 on (0, T ) × ∂Ω,   u(0, x ) = u0 ( x ) in Ω for some positive γ. Our goal is showing that there exists some initial datum u0 ∈ L η ( Ω ) ,

1 ≤ η < σ,

(c.2)

such that the problem (c.1) does not admit any solution u such that p

1,p

u ∈ Lloc (0, T; W0 (Ω)) ∩ C ([0, T ]; Lη (Ω)),

|u|

η + p −2 p

(c.3)

∈L

p

1,p (0, T; W0 (Ω)).

Note that we are no longer dealing with functions belonging to the set σ + p −2 1,p u solving (c.1) : |u| p ∈ L p (0, T; W0 ) .


since

η + p −2 p

0, Br = Br (0), c = c( N, p) and λ = p( N + 1) − 2N. The particular choice of u0 implies that Br

u0 ( x ) dx ≥ cr

−N η +ω

and so, taking r such that t ≫ cr we have r

−N η +ω

≫

p r t

1 p −2

p+

( N − ωη )( p −2) η ,

(c.5)

. Then, we are allowed to say that N

inf U (t, y) ≥ ct− λ r

pN η −1 ω η +p λ λ

y ∈ Br

and also to deduce ˆ

N

U (t, y) dy ≥ ct− λ r

pN η −1 ω η +p λ +N λ

.

(c.6)

Br

Upper bound for u. We now look for a bound from above for the integral in the space variable of the solution of (c.1). First, we observe that such a solution u should fulfil:

|u|

η + q −1 q

1,q

∈ Lq (0, T; W0 (Ω))

(c.7)

where this last regularity follows from the boundedness of ¨ |∇u|q |u|η −1 dx dt < ∞. QT

Note that the above boundedness holds thanks to (c.3) and follows reasoning as in (4.3). Then, there exists at least a sequence {t j } j satisfying t j → 0 such that, applying also Holder’s ¨ inequality with indices ′ η + q −1 η + q −1 , we have q∗ q , q∗ q ˆ

Ω

|∇u(t j )|q |u(t j )|η −1 dx = k∇(|u(t j )|

η + q −1 q

q

)k Lq (Ω) ≤

1 . tj

Then, we obtain ˆ

Nq

Br

u(t j , y) dy ≤ ku(t j )k

r η + q −1 q∗ q L ( Ω)

≤ ck∇(|u(t j )|

η + q −1 q

q

)k Lη q+(qΩ−1) r

− η +1q−1 N − N −q η + q −1

≤ ct j

r

N − q ∗ ( η + q −1 )

.

Nq

N − q ∗ ( η + q −1 )

(c.8)

32

M. MAGLIOCCA

Conclusion. We already know that U ≤ u where u and U are, respectively, solutions of (c.1) and (c.4). Then we take advantage of this information gathering (c.6) and (c.8), so we get ˆ ˆ η −1 ω − η +1q−1 N − N −q − N N + pN η +p λ ≤ λ r η + q −1 u(t j , y) dy ≤ ct j U (t j , y) dy ≤ tj λ r Br

Br

from which

1 η + q −1

tj

η pη + N ( p −2)− ωη ( p−2)

We recall (c.5) and set r = ωt j

r

pN η −1 N −q N ω λ η + p λ + η + q −1 − λ

≤ c.

(c.9)

, where 0 < ω ≪ 1, obtaining ϕ

t j ≤ c(ω )

(c.10)

for ϕ = ϕ(η ) defined by ϕ=−

η( N + ω) − N N 1 p η ( N − q) + + + . λ η + q − 1 λ η ( p − ω ( p − 2)) + N ( p − 2) (η + q − 1)(η ( p − ω ( p − 2)) + N ( p − 2))

This means that, as j → ∞, we need to have ϕ ≥ 0 in order to have (c.10) fulfilled. Algebraic computations lead us to the equivalent request h i (η + q − 1) − Nη ( p − ω ( p − 2)) − N 2 ( p − 2) + pη ( N + ω ) − N p

+λ[( p − ω ( p − 2))η + N ( p − 2) + η ( N − q)]

= (η + q − 1) [ηω ( N ( p − 2) + p) − N ( N ( p − 2) + p)] +λ[η ( N + p − q − ω ( p − 2)) + N ( p − 2)] = −λ(η + q − 1)( N − ωη ) + λ[η ( N + p − q − ω ( p − 2)) + N ( p − 2)] ≥ 0 by the definition of λ. Then, looking for ϕ ≥ 0, we erase λ getting η ( p − q) − N (q − ( p − 1)) − ωη (η + q − p + 1) ≥ 0 from which we deduce σ−η ≤

ωη ( η + q − p + 1) p−q

(c.11)

thanks to the definition of σ. Since we can choose ω sufficiently small such that (c.11) is violated, then we deduce the assertion by contradiction. Acknowledgements The author wishes to thank Andrea Dall’Aglio and Alessio Porretta for their advices, comments and support during the composition of this work. References

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