Journal of Differential Equations

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1 J. Differential Equations ) Contents lists available at ScienceDirect Journal of Differential Equations Renormalized solutions for a nonlinear parabolic equation with variable exponents and L 1 -data M. Bendahmane a,, P. Wittbold b, A. Zimmermann b a Institut Mathématiques de Bordeaux, Université Victor Segalen Bordeaux 2, Bordeaux, France b Institut für Mathematik, Technische Universität Berlin, MA 6 4, Strasse des 17. Juni 136, D-1623 Berlin, Germany article info abstract Article history: Received 15 May 29 Revised 15 April 21 Available online 31 May 21 MSC: primary 35J7 secondary 35D5 We prove the well-posedness existence and uniqueness) of a renormalized solution to nonlinear parabolic equations with variable exponents and L 1 -data. The functional setting involves Lebesgue Sobolev spaces with variable exponents. 21 Elsevier Inc. All rights reserved. Keywords: Parabolic equation Variable exponents Renormalized solution Existence Uniqueness 1. Introduction We consider a bounded open spatial domain R N N 2) with a Lipschitz boundary denoted by. Fixing a final time T >, we set Q T =, T ) and Σ T =, T ).Ouraimistoprove the existence and uniqueness of renormalized solutions u to the nonlinear parabolic equation t u div u ) px) 2 u = f in Q T, u = onσ T, 1) u, ) = u ) in, * Corresponding author. addresses: mostafa_bendahmane@yahoo.fr M. Bendahmane), wittbold@math.tu-berlin.de P. Wittbold), zmorzyns@math.tu-berlin.de A. Zimmermann) /$ see front matter 21 Elsevier Inc. All rights reserved. doi:1.116/j.jde

2 1484 M. Bendahmane et al. / J. Differential Equations ) where f L 1 Q T ), u L 1 ) and p : 1, + ) is a continuous function. The study of problems with variable exponent is a new and interesting topic which raises many mathematical difficulties see [19,31,34,3,4]). One of our motivations for studying 1) comes from applications to electrorheological fluids we refer to [31] for more details) as an important class of non-newtonian fluids sometimes referred to as smart fluids). Other important applications are related to image processing see [19]) and elasticity see [34]). Eq. 1) can be viewed as a generalization of the classical p-laplacian equation t u div u ) p 2 u = f in Q T, u = onσ T, u, ) = u ) in, with constant p 1, + ). Note that 1) has a more complicated nonlinearity than the classical p-laplacian 2) since it is nonhomogeneous. Existence and uniqueness of renormalized solutions to problem 2) is nowadays well known and was established by Blanchard and Murat in [12]. We recall that the notion of renormalized solutions was introduced in [2] by DiPerna and Lions in their study of the Boltzmann equation. This notion was adapted to the study of some nonlinear elliptic problems with Dirichlet boundary conditions by Boccardo, Diaz, Giachetti, and Murat [15] and Lions and Murat see Lions [28]). Later it was extended to more general problems of parabolic, elliptic parabolic and hyperbolic type see [28,13,14,29,11]). Let us also mention that an equivalent notion of solutions, called entropy solutions, was introduced independently by Bénilan et al. in [8] see also [2]). In two former papers see [7,33]) we have already studied the corresponding elliptic problem for the px)-laplacian and also more general elliptic equations with variable exponents involving lower order terms. In particular, we have established an existence and uniqueness result for renormalized solutions of the stationary problem with arbitrary L 1 -data. Relying on these results and using nonlinear semigroup theory, it is easy to deduce existence of a unique mild solution for the abstract Cauchy problem corresponding to 1) and arbitrary L 1 -data cf. Section 4). In this paper we use the results from abstract semigroup theory to prove existence and uniqueness of renormalized solutions to the parabolic problem 1) for arbitrary L 1 -data. The paper is organized as follows: In Section 2 we recall some basic notations and properties of Sobolev spaces with variable exponents. In addition, we prove an interpolation result that will be used later to obtain a-priori-estimates. In Section 3, the definition of renormalized solution is given as well as the main result, Theorem 3.1, on existence of renormalized solutions to problem 1). Section 4 is devoted to the study of some properties of renormalized solutions and to the proof of existence of approximate solutions to 1). Theorem 3.1 is proved in Section 5. In Section 6 we prove uniqueness of renormalized solutions. Finally we make some remarks on the equivalence between the solution concept used in this paper, the notion of renormalized solution, and the notion of entropy solution which is another suitable solution concept for elliptic and parabolic problems with L 1 -data cf. e.g. [8, 3,2,32]). 2. Mathematical preliminaries 2.1. Sobolev spaces with variable exponents We recall in some definitions and basic properties of the generalized Lebesgue Sobolev spaces L p ) ), W 1,p ) ) and W 1,p ) ), where is an open subset of R N. We refer to Fan and Zhao [23] for further properties of variable exponent Lebesgue Sobolev spaces. Let p : [1, + ) be a continuous, real-valued function the variable exponent) and let p = min x px) and p + = max x px). We define the variable exponent Lebesgue space { L p ) ) = u: R; u is measurable with ux) px) dx < }. 2)

3 M. Bendahmane et al. / J. Differential Equations ) We define a norm, the so-called Luxemburg norm, on this space by the formula u L p ) ) {μ = inf > ; ux) μ px) dx 1 }. The following inequality will be used later { min u p L p ) ), } u p+ L p ) ) ux) px) { p dx max u L p ) ), } u p+ L p ) ). 3) If p > 1, then L p ) ) is reflexive and the dual space of L p ) ) can be identified with L p ) ), where p ) p ) = 1. For any u L p ) ) and v L p ) ) the Hölder type inequality 1 uv dx p + 1 ) p u L p ) ) v L p ) ) 4) holds true. Extending a variable exponent p : [1, ) to Q T =[, T ] by setting pt, x) := px) for all t, x) Q T, we may also consider the generalized Lebesgue space { L p ) Q T ) = u: Q T R; u is measurable with Q T ut, x) px) dt, x)< }, endowed with the norm u L p ) Q T ) {μ = inf > ; Q T ut, x) μ px) } dt, x) 1, which, of course, shares the same type of properties as L p ) ). We define also the variable Sobolev space W 1,p ) ) = { u L p ) ); u L p ) ) }. On W 1,p ) ) we may consider one of the following equivalent norms u W 1,p ) ) = u L p ) ) + u L p ) ) or u W 1,p ) ) {μ = inf > ; ux) μ px) + ux) px)) } μ dx 1. We define also W 1,p ) ) := Cc ) W 1,p ) ). Assuming p > 1 the spaces W 1,p ) ) and W 1,p ) ) are separable and reflexive Banach spaces. The space W 1,p ) )) denotes the dual of W 1,p ) ).

4 1486 M. Bendahmane et al. / J. Differential Equations ) Remark 2.1. Note that generalized Lebesgue and Lebesgue Sobolev spaces can also be defined in the same way for only measurable real-valued variable exponents p ) satisfying 1 p inf p sup < where p inf = ess-inf x px), p sup = ess-sup x px). According to [23], such variable exponent Lebesgue spaces are Banach spaces, the Hölder type inequality holds, they are reflexive if and only if 1 < p inf p sup <. The inclusion between Lebesgue spaces also generalizes naturally: if < < and r 1, r 2 are variable exponents so that r 1 ) r 2 ) almost everywhere in then there exists the continuous embedding L r 2 ) ) L r 1 ) ), whose norm does not exceed +1. For u W 1,p ) ) with p C) and p 1, the Poincaré inequality holds cf. [24]) u L p ) ) C u L p ) ), 5) for some universal constant C which depends on and the function p. Forp C) with 1 < p p + < N the Sobolev embedding holds see e.g. [22]) W 1,p ) ) L r ) ), 6) for any measurable function r : [1, + ) such that ) Npx) ess inf x N px) rx) >. Remark 2.2. The variable exponent p : [1, ) is said to satisfy the log-continuity condition if x 1, x 2, x 1 x 2 < 1, px1 ) px 2 ) < w x 1 x 2 ), 7) where ω :, ) R is a nondecreasing function with lim sup α + wα) ln α 1 )<+. Logcontinuity condition 7) is used to obtain several regularity results for Sobolev spaces with variable exponents; in particular, C ) is dense in W 1,p ) ) and W 1,p ) ) = W 1,p ) ) W 1,1 ). Moreover, if p satisfies the log-continuity 7) condition and 1 < p p + < N, then the Sobolev embedding holds also see e.g. [21] for more details) for r ) = p ), i.e. W 1,p ) ) L p ) ). We do not need these regularity properties to prove our results and will most exclusively work with Lebesgue and Lebesgue Sobolev spaces with only continuous variable exponents p : [1, ) such that p > 1. Remark 2.3. Note that the following inequality u px) dx C u px) dx, in general does not hold see [23] for more details). We will also use the standard notations for Bochner spaces, i.e., if q 1 and X is a Banach space, then L q, T ; X) denotes the space of strongly measurable functions u :, T ) X for which t ut) X L q, T ). Moreover,C[, T ]; X) denotes the space of continuous functions u :[, T ] X endowed with the norm u C[,T ];X) := max t [,T ] ut) X.

5 M. Bendahmane et al. / J. Differential Equations ) Technical lemma We now prove a version of a weak Lebesgue space estimate that goes back to Boccardo and Gallouët [16] for parabolic equations with constant exponents p ) = p constant). We establish the following result. Lemma 2.1. Let p C) with 2 1 N+1 < p p + < Nandletβ>. Then there exists a constant c >, depending on β, such that, for any function g L p, T ; W 1,p ) )) L, T ; L 1 )) with g L,T ;L 1 )) = ess sup t,t ) gt, x) dx β, 8) and sup γ Bγ g px) dxdt β, 9) where, for γ, B γ ={γ u γ + 1}, it follows that for all continuous functions q ) on satisfying g L q,t ;W 1,q ) c, 1) )) Npx) 1) + px) 1 qx)< N + 1 for all x. 11) Proof. In a first step let q + be a constant satisfying 1 q + < Np 1) + p. 12) N + 1 Note that 12) in particular implies q + < p. According to the continuous embedding W 1,p ) ) into W 1,p ), for any integer γ 1, we deduce from 9) and Hölder inequality T γ 1 g q+ dxdt = g q+ dxdt + g q+ dxdt γ = γ =γ Bγ Bγ c γ + g q+ dxdt γ =γ Bγ ) q+ c γ + g p p dxdt measbγ ) ) 1 q+ p γ =γ Bγ c γ + c 1 measbγ ) ) q 1 + p, 13) γ =γ

6 1488 M. Bendahmane et al. / J. Differential Equations ) for some constants c γ, c 1 > depending only on β, and γ.letr 1. Then, for γ >, we have 1 gt, x) r dxdt measbγ ), γ r 14) Bγ so that T g q+ dxdt c γ + c 1 γ =γ 1 γ r p q + p Bγ g r dxdt ) p q + p c γ + c 1 γ =γ 1 γ r p q + q + ) q+ p γ =γ Bγ g r dxdt ) p q + p. 15) Note that q + < N and thus q + > 1 where q + = Nq + N q + ; moreover, if we choose r = q+ N+1),then N 1 r q +. Applying now the interpolation inequality for L p -norms, by 8), we get gt, ) Lr ) gt, ) 1 a L 1 ) gt, ) a L q+ ) c 2gt, ) a for a.e. t, T ), for some constant c 2 >, where a = q+ 1 r) r1 q + ).Thus L q+ ), g r L r,t ;L r )) c 2 As r = q+ N+1), the last estimate simplifies to N T q+ 1 r) gt,.) 1 q + ) L q+ ) dt. By the Sobolev inequality, we have g r L r,t ;L r )) c 2 g q+ L q+,t ;L q+. 16) )) ) 1 gt, x) q + q + dx c 3 ) 1 gt, x) q + q + dx, 17) for some constant c 3 >. Using Hölder inequality, and the estimates 15), 16), 17), the result is T g L q +,T ;L q+ )) c 4 gt, x) ) 1 q+ q + dxdt q + p q + ) q c 5 + c 6 g + p L q+,t ;L q+ )) γ =γ 1 γ p q + )q + N+1) q + N ) 1 p

7 M. Bendahmane et al. / J. Differential Equations ) for some constants c 4, c 5, c 6 >. Since p q + ) p c 5 + c 6 g L q+,t ;L q+ )) γ =γ 1 γ p q + )N+1) N ) 1 p, 18) p q + )N + 1) > 1, 19) N which ensures that the series which appears in 18) is convergent, we deduce from 18) g L q +,T ;L q+ )) c 7 ; thus from 15) and 16) we obtain g L q + Q T ) c 8, for some constant c 8 >. In particular, there exists a constant c 9 > such that g L1 Q T ) c 9. 2) Now let us consider a continuous variable exponent q on satisfying the pointwise estimate 11). By the continuity of p ) and q ) on there exists a constant δ> such that Npy) 1) + py) max qy)< min y Bx,δ) y Bx,δ) N + 1 for all x. 21) Observe that is compact and therefore we can cover it with a finite number of balls B i ) i=1,...,k. Moreover, there exists a constant α > such that measb i ) >α for all i = 1,...,k. 22) We denote by q i and q + i respectively p i and p + i ) the local minimum and the local maximum of q on B i respectively the local minimum and the local maximum of p on B i ). Using now the same arguments as before locally, we see that the inequality 13) holds on B i B γ and B i, respectively. In particular, it is easy to check that, instead of the global estimate 18), we find g L q+ i,t ;L q+ i B i )) c 5 + c 6 g p q + ) i i p i L q+ i,t ;L q+ i B i )) γ =γ 1 γ p q + )N+1) i i N ) 1 p i. 23) Denote by g i the average of g i over B i : 1 g i t) = measb i ) B i gt, x) dx for almost t, T ). In view of 2) and 22), we deduce

8 149 M. Bendahmane et al. / J. Differential Equations ) T gi t) c 9 dt α. 24) By Poincaré Wirtinger inequality, we obtain g g i L q+ c 1 g q i B i ) +, 25) L i B i ) for some constant c 1 >. Keeping in mind 21), we deduce from 24), 25) and 23) g L q+ i,t ;L q+ i B i )) c 11 + c 12 g p q + ) i i p i L q+ i,t ;L q+ i, B i )) for some constants c 11, c 12 >. Obviously, this implies that, for some constant c 13, depending on p ), q ),, g L q+ i,t ;L q+ c 13 for all i = 1,...,k. 26) i B i )) = Nq + i N q + i Finally, since q + i q x) qx) and q + i conclude from 23) and 26) that qx) for all x B i and for all i = 1,...,k, we g L q,t ;L q ) )) + g L q,t ;W 1,q ) )) c 14, for some constant c 14 depending on p ), q ) and. This concludes the proof of the lemma. Remark 2.4. Note that the result obtained in Lemma 2.1 also holds for any measurable function q : [1, + ) such that b := ess inf x Npx) 1) + px) N + 1 ) qx) >. In fact, in this case there exists a continuous function s ) such that sx) qx) for almost every x, and min x Npx) 1) + px) N + 1 ) sx) > b/2)>. From Lemma 2.1, we deduce the bound of g in L s, T ; W 1,s ) )). Finally the result follows from the continuous embedding L s, T ; W 1,s ) )) into L q, T ; W 1,q ) )).

9 M. Bendahmane et al. / J. Differential Equations ) Renormalized solutions We start by defining truncation/renormalization functions. For any given γ >, we define the truncation function T γ : R R by γ, if z γ, T γ z) = z, if z < γ, γ, if z γ. Moreover, we will need the following associated function renormalization) φ γ r) = T γ +1 r) T γ r), for any γ >. Notice that T γ and φ γ are Lipschitz continuous, piecewise C 1 functions satisfying T γ ) γ and φ γ ) 1. Pick any positive C R) function s ) such that sz) = 1 if z 1, sz) = if z 2, and sz) 1forallz R. Foranyn 2, define the function S n r) by S n r) = r s nz) dz, where { 1, if z n 1, s n z) = sz n 1) signz)), if z n 1, where signz) denotes the sign of z. Foreachintegern 2, the function S n satisfies { Sn r) = S n Tn+1 r) ), S n L R) s L R), supp S n [ n + 1),n + 1 ], supp S n [ n + 1), n ] [n,n + 1]. We shall use the following definition of renormalized solutions for the parabolic equation 1): Definition 3.1. A renormalized solution of 1) is a measurable function u : Q T R satisfying the following conditions: T γ u) L p, T ; W 1,p ) ) ) for any γ >, 27) T γ u) L p ) Q T ) ) N for any γ >, 28) lim u px) dxdt =, 29) γ {γ u γ +1} and, for any renormalization S C R) such that supp S [ M, M] for some M >, t Su) div S u) u px) 2 u ) + S u) u px) = fs u) in D Q T ). 3) Moreover, the initial condition is satisfied in the sense Su) t= = Su ) a.e. in. 31) Several remarks are in order. In particular, we have to make sure that all the terms in 3) as well as the initial condition make sense.

10 1492 M. Bendahmane et al. / J. Differential Equations ) Remark 3.1. In the preceding definition as well as in the following, we identify, as usual, an abstract function like v L p, T ; L p ) )) with the real-valued Lebesgue measurable function v defined by vt, x) = vt)x) for almost all t, T ), for almost all x. In the same way we associate to any function v L p ) Q T ) an abstract Bochner measurable function v :, T ) L p ) ) by setting vt) := vt, ) for almost every t, T ). Note that, with this identification, we have the following continuous dense embeddings L p+, T ; L p ) ) ) d L p ) Q T ) d L p, T ; L p ) ) ). 32) In fact, for v L p ) Q T ), the corresponding abstract function v :, T ) L p ) ) is strongly Bochner measurable by the Dunford Pettis theorem, since it is weakly measurable and L p ) ) is separable). Moreover, using 3) and Hölder inequality, we find the estimate T vt) p T T L p ) ) dt max { vt, x) px) dx, T vt, x) px) dxdt + T 1 p /p + max { v p L p ) Q T ), v p+ L p ) Q T ) vt, x) px) dx ) p /p +} dt ) p /p + vt, x) px) dxdt } + T 1 p /p + max { v p ) 2 /p + L p ) Q T ), v p L p ) Q T ) }. Therefore, v L p, T ; L p ) )), and the embedding of L p ) Q T ) into L p, T ; L p ) )) is continuous. The first embedding in 32) can be proved in a similar way. Note that both embeddings are dense. We consider the first embedding and fix u L p ) Q T ).SinceDQ T ) is dense L p ) Q T ),we find a sequence u n ) n DQ T ) converging to u in L p ) Q T ) as n.butdq T ) is also densely) embedded into L p+, T ; L p+ )), henceu n is in L p+, T ; L p ) )) for all n N. Toprovethesecond embedding, we fix v L p, T ; L p ) )), take a standard sequence of mollifiers ρ n ) n DR) and extend v by zero onto R. The regularized in time) function ρ n v) ) := R ρ n s)vs) ds is in L p+ R; L p ) )) for each n N, henceinl p ) Q T ) and converges to v in L p, T ; L p ) )). Remark 3.2. Note that the inclusions in 32) are, in general, strict for nonconstant exponent p )). In particular, a function that satisfies 27) does not automatically satisfy 28). As an example, consider N = 2, = 1, 1) 1, 1), px, y) = 3/2 x /4, x, y). Thenp = 5/4, p + = 3/2 and the function v = vt, x, y) = t 2/3 1 x )1 y ), t [, T ],x, y), isanelementofl p, T ; W 1,p ) )), but x v, y v / L p ) Q T ). Indeed, it is clear that v C[, T ] ) and that v = ontheboundary, T ).Moreover, x v = x vt, x, y) = t 2/3 signx)1 y ) a.e., and x v p L p,t ;L p+ )) = = T T x v 3/2 dxdy) 2/3 5/4 dt ) ) 5/6 t 1 3/2 1 y dxdy dt

11 M. Bendahmane et al. / J. Differential Equations ) T 4 t 5/6 dt <, i.e., x v L p, T ; L p+ )) L p, T ; L p ) )). By symmetry the same holds for v, and thus y v L p, T ; W 1,p ) )). On the other hand, x v / L p ) Q T ) as T t 2/3 1 y )) T 3/2 x /4 dxdy dt 1/2) 3/2 1/2 1/2 t 1 t x /6 dxdt = 3 2 T t 1/12 1 dt, t lnt) and this last improper integral diverges. Remark 3.3. Note that, if p ) = p is constant, i.e., if we consider the classical evolution problem 2) for the p-laplacian, then, of course, 27) implies 28) and thus the problem can be settled within the classical functional setting of the Bochner Lebesgue spaces L p, T ; W 1,p )). Aswehaveseenin the preceding remark, this is not true for the general case of a variable exponent p ). Condition 28) plays a crucial role in order to get a well-posed problem. In view of the definition of a renormalized solution and the preceding remarks, we are naturally led to introduce the functional space which, endowed with the norm V = { f L p, T ; W 1,p ) ) ) ; f L p ) Q T ) }, 33) f V := f L p ) Q T ), or, the equivalent norm f V := f L p,t ;W 1,p ) )) + f L p ) Q T ), is a separable and reflexive Banach space. The equivalence of the two norms is an easy consequence of the continuous embedding L p ) Q T ) L p, T ; L p ) )) and the Poincaré inequality. We state some further properties of V in the following lemma. Their proofs are straightforward and therefore omitted here: Lemma 3.1. Let V be defined as in 33) and its dual space be denoted by V.Then, i) we have the following continuous dense embeddings: L p+ 1,p ), T ; W ) ) d V d L p 1,p ), T ; W ) ). 34)

12 1494 M. Bendahmane et al. / J. Differential Equations ) In particular, since DQ T ) is dense in L p+, T ; W 1,p ) )), it is dense in V and for the corresponding dual spaces we have L p ), T ; W 1,p ) ) ) ) V L p+ ), T ; W 1,p ) ) ) ). 35) ii) One can represent the elements of V as follows: If T V, then there exists F = f 1,..., f N ) L p ) Q T )) N such that T = div x Fand T T,ξ V,V = F Dξ dxdt for any ξ V.Moreover,wehave T V = max { f i L p ) Q T ), i = 1,...,n}. Remark 3.4. Note that, if u is a renormalized solution of 1), then Su) V L Q T ) for any renormalization S. Let us also remark that V L Q T ), endowed with the norm v V L Q T ) := max { v V, v L Q T )}, v V L Q T ), is a Banach space. In fact, it is the dual space of the Banach space V + L 1 Q T ), endowed with the norm v V +L 1 Q T ) := inf{ v 1 V + v 2 L 1 Q T ) ; v = v 1 + v 2, v 1 V, v 2 L 1 Q T ) }. Note also that, if u is a renormalized solution of 1), then Su) t V + L 1 Q T ). In fact, for any ξ DQ T ), using Hölder s inequality, we find T S u) u px) 1 ξ = T S u) T γ u) px) 1 ξ 2 S L T R) γ u) px) 1 L p ) Q T ) ξ L p ) Q T ) 2 S { L R) max T γ u) ) 1 px) p, T γ u) ) 1 } px) p + ξ L p ) Q T ) Q T Q T where γ > is such that supps ) [ γ, γ ]. As DQ T ) is dense in V, we deduce that divs u) u px) 2 u) V, and then it follows from Eq. 1) that Su) t V + L 1 Q ). Remark 3.5. The initial condition 31) makes sense. In fact, if u is a renormalized solution, then, for any renormalization S, Su) V L Q T ) and Su) t V + L 1 Q T ), and therefore Su) C[, T ]; L 2 )) according to Lemma 3.2 below.

13 M. Bendahmane et al. / J. Differential Equations ) Lemma 3.2. W := { u V ; u t V + L 1 Q T ) } C [, T ]; L 1 ) ) 36) and W L Q T ) C [, T ]; L 2 ) ). 37) The proof of this lemma follows the same lines as the proof of the corresponding result in the case of a constant exponent p, Theorem 1.1 of [29], and therefore is omitted here. Remark 3.6. Using the embedding W 1,p ) ) W 1,1 ), we can associate to every measurable function u : Q T R satisfying T γ u) V for all γ >, a generalized gradient still denoted by u), defined as the unique measurable function satisfying u = T γ u) a.e. on { u < γ }, for all γ > see e.g. [8]). It follows that all the terms in 3) make sense. Moreover, using the fact that for any function ϕ V L Q T ) there exists functions ϕ n DQ T ) that converge strongly to ϕ in V and weak- in L Q T ), we see that in 3) we can not only use test functions in DQ T ), but also functions in V L Q T ). In fact, we can replace 3) by for all ϕ V L Q T ). t Su), ϕ T V +,V T + S u) u px) ϕ dxdt = S u) u px) 2 u ϕ dxdt T fs u)ϕ dxdt, 38) Remark 3.7. Observe that 29) implies lim u px) dxdt =, 39) γ {γ u γ +c} for all c >. Our main result is the following theorem: Theorem 3.1. Assume p C) with 1 < p p + < N. Then, there exists at least one renormalized solution u of the initial boundary value problem 1). Moreover, if p > 2 1 N+1,thenu Lq, T ; W 1,q ) )) for all continuous variable exponents q on satisfying 1 qx)< Npx) 1)+px) for all x. N+1 Remark 3.8. If 2 1 N+1 < p, we can apply Lemma 2.1 to obtain the additional regularity result of Theorem 3.1 see Lemma 4.2 for the proof). In this case we deduce that u L 1, T ; W 1,1 )), and then from the Stampacchia theorem [26] it follows that

14 1496 M. Bendahmane et al. / J. Differential Equations ) T γ u) = 1 { u <γ } u a.e. on Q T, where 1 { u <γ } denotes the characteristic function of the measurable set { u < γ } Q T. In particular, the generalized gradient as defined in Remark 3.6 coincides with the usual gradient in the sense of distributions. In the general case 1 < p < N, we are still able to prove existence of a renormalized solution u to 1), but the gradient of u only has a meaning in the sense of Remark Properties of renormalized solutions In order to find more estimates for renormalized solutions and also to get useful a-priori-estimates of approximate solutions to the equation, the following integration-by-parts-formula plays a crucial role: Lemma 4.1. Let S W 1, R) with supp S compact, v L 1 Q T ) with T γ v) V for all γ > and such that Sv) t V + L 1 Q T ),Sv)) = Sv ) with v L 1 ). Then Sv)t, hv w)ξ V +L 1 Q T ),V L Q T ) = Q T ξ t v v hr w) dsr) dt, x), 4) for any h W 1, R) with supp h compact, w W 1,p ) ) L ) and ξ D, T ) R N ) such that hv w)ξ V. Remark 4.1. From the assumptions of Lemma 4.1 it follows immediately that Sv) V L Q T ) and thus, according to Lemma 3.2, Sv) C[, T ]; L 2 )). Therefore, the condition Sv)) = Sv ) makes sense. Proof of Lemma 4.1. The proof follows the same lines as the proof of the corresponding result in the classical variational setting when V = L p, T ; W 1,p )) see, e.g., [1,18] and the references therein). The essential point is that the Steklov average in time) v η ) = η 1 +η vs) ds, η >, of a function v V L Q T ) appropriately prolongated outside, T )) still belongs to V L Q T ) and converges, as η, strongly to v in V and weak- in L Q T ). Now we may state the first result on additional properties of renormalized solutions: Lemma 4.2. Let u be a renormalized solution, then u L,T ;L 1 )) f L 1 Q T ) + u L1 ), 41) Tγ u) L p ) Q T ) γ max{ f L1 Q T ) + u 1/p L1 )), ) f L 1 Q T ) + u 1/p +} L 1 ), 42) for all γ >,and sup γ > {γ u γ +1} u px) dxdt f L 1 Q T ) + u L 1 ). 43) Moreover, if 2 1 N+1 < p )<Nthen

15 M. Bendahmane et al. / J. Differential Equations ) u L q, T ; W 1,q ) ) ), 44) for all continuous variable exponents q C) satisfying 1 qx)< Npx) 1)+px) N+1 for all x. Proof. Proof of 41). We take S = S n and choose ϕ = γ 1 T γ u) in 38). Employing the integration-byparts-formula for the evolution term, letting first n and second γ, it follows that t ut, ) L 1 ) u L1 ) + f τ, ) L 1 ) dτ, for all t, T ). Proof of 42). We take S = S n and ϕ = T γ u) in 38). After letting n,wefind T T γ u) px) dxdt γ f L 1 Q T ) + u L 1 )), which yields 42). Proof of 43). Take S = S n and ϕ = φ γ u) in 38) and letting n, we obtain {γ u γ +1} u px) dxdt f L1 Q T ) + u L1 ), which yields 43). Proof of 44). We take S = S n and ϕ = φ γ T k u)) in 38), k >. As n,wenowfind {γ T k u) γ +1} for all k >. By Lemma 2.1 and 41), we deduce from 45) Tk u) px) dxdt f L1 Q T ) + u L1 ), 45) Tk u) L q,t ;W 1,q ) C for all k >, 46) )) for some constant C > independent of k. Herein, q is a continuous variable exponent on satisfying 1 qx)< Npx) 1)+px) for all x. Finally, by an application of Fatou s lemma, 44) follows after N+1 letting k in 46). Note that, in view of the regularity proved in Lemma 4.2, it is clear that if 2 1 N+1 < p arenormalized solution is also a solution in the sense of distributions. Note also the partial converse result: Lemma 4.3. Suppose f V and u L 2 ). Then a weak solution u of 1) is also a renormalized solution. By a weak solution u of 1) we understand a solution in the sense of distributions that belongs to the energy space, i.e., u V, t u div u px) 2 u ) = f in D Q T ) 47)

16 1498 M. Bendahmane et al. / J. Differential Equations ) and u, ) = u. Proof of Lemma 4.3. As u V and f V, from Eq. 47), we deduce that t u V. By density of DQ T ) in V, 47) is equivalent to T t u,ϕ V,V + u px) 2 u ϕ = f,ϕ V,V 48) for all ϕ V. Takingϕ = S u)ψ in 48), where S C R), supps [ M, M] for some M >, and ψ DQ T ), we deduce easily that u satisfies the renormalized equation 38). Condition 29) holds since u V implies Du p x) L 1 Q T ) and {γ < u < γ + 1} asγ. The remaining conditions for being a renormalized solution hold trivially Approximate solutions: the semigroup approach In [33] we have studied elliptic problems of the form E) f ) { βu) div u px) 2 u F u) ) = f in, u = on, 49) where β is a maximal monotone graph in R R and F : R R N is a locally Lipschitz continuous function. For p C) with 1 < p p + < N existence and uniqueness of a renormalized solution of E) f ) has been proved for any f L 1 ) see also [32] for results in the case 1 < p p + < N for the case of a log-continuous exponent and [7] for regularity properties of renormalized solutions in the case p > 2 1 N ). In particular, it follows from the results of [33] that, for all f L1 ), λ>, there exists a unique renormalized solution u : R of the problem E) λ f ) { u λ div u px) 2 u ) = f in, u = on, 5) i.e. u is a measurable function such that T γ u) W 1,p ) ) for all γ >, S u)u div S u) u px) 2 u ) + S u) u px) = S u) f in D ), and lim u px) dx =. γ {γ < u <γ +1} Moreover, if u, v are renormalized solutions of E) λ f ), E) λ g), f, g L 1 ), respectively, the following comparison principle holds: u v) + L1 ) f g) + L1 ), 51) u v) + L ) f g) + L ). 52)

17 M. Bendahmane et al. / J. Differential Equations ) As the roles of u and v, respectively, f and g, can be exchanged, the contraction does not only hold for the positive parts, but for the absolute values of the corresponding functions. By interpolation it follows u v Ls ) f g Ls ) for all 1 s. 53) The proof of 51) follows the same lines as the proof of the comparison principle, Theorem 7.1, in [33]. The second estimate can be proved by using in the renormalized equation for u, v, respectively, the renormalization function S n and the test function γ 1 T γ u v f g) + L )) + ), passing to the limit in the difference of the two resulting equations successively with n and then γ. In terms of nonlinear operators the preceding result reads as follows: If A is the nonlinear operator defined in L 1 ) by A = { u, w) L 1 ) L 1 ); u is a renormalized solution of E)w) }, then A is m-completely accretive in L 1 ), i.e., RI + λa) = L 1 ) for all λ>, 54) and the resolvent of A, the mapping f L 1 ) I + λa) 1 f L 1 ), is an order-preserving contraction with respect to the L 1 - and L -normandthuswithrespectto any L s -norm, 1 s ) see [9] for the theory of completely accretive operators). In particular, the nonlinear operator A is m-accretive in L 1 ), i.e., the range condition 54) holds and the resolvent is a contraction in the L 1 -norm. By the general theory of nonlinear semigroups see, e.g., [1] or [5]) we conclude that the abstract evolution problem corresponding to 1), i.e., the Cauchy problem for the nonlinear operator A CP)u, f ) du + Au = f on, T ), dt u) = u admits a unique mild solution u C[, T ]; L 1 )) for any initial datum u DA) L 1 ) = L 1 ) as we shall see below) and any right-hand side f L 1, T ; L 1 )) = L 1 Q T ). Roughly speaking, a mild solution is a continuous function u C[, T ]; L 1 )) which is the uniform limit of piecewise constant functions u ε = v ε i on ]t ε i 1, tε i ], i = 1,...,m ε, u ε ) = v ε, solutions of time-discretized problems given by an implicit Euler scheme of the form v ε v ε i i 1 t ε t ε i i 1 + Av ε i = f ε i, i = 1,...,m ε, 55) T and f ε i L 1 ), i = 1,...,m ε such that where ε >, m ε N, = t ε < tε 1 < < tε mε m ε t ε i i=1 t ε f t) f ε i L 1 ) dt, max i=1,...,mε tε t ε i i 1 ), T tε mε and u v ε L 1 ) asε. i 1 Let us recall that the mild solution of CP)u, f ) depends continuously on the data; more precisely, if u, v C[, T ]; L 1 )) are the mild solutions of CP)u, f ), CP)v, g), respectively, then the following contraction principle holds: for any t T,

18 15 M. Bendahmane et al. / J. Differential Equations ) t ut) vt) L1 ) u v L 1 ) + f s) gs) L1 ds. 56) ) In the case where the operator A is m-completely accretive, a corresponding contraction principle for mild solutions also holds with respect to any L r -norm, 1 r. More precisely, in this case, for any 1 r, wehave ut) vt) ) + L r ) u v ) + t L r ) + f s) gs) ) + L ds. r 57) ) Moreover, a function u C[, T ]; L 1 )) is the unique mild solution of CP)u, f ) if and only if u is the unique integral solution of CP)u, f ) in the sense of Bénilan [1,5]: Definition 4.1. Afunctionu C[, T ]; L 1 )) is called an integral solution of CP)u, f ), ifu satisfies the following family of integral inequalities: for any v, w) A, forany s t T,wehave ut) vl1 ) t us) vl1 ) + [ ] uτ ) v, f τ ) w dτ, 58) s where, for g, h L 1 ), the bracket [g, h] denotes the right-hand side Gâteaux derivative of the L 1 - norm at g in the direction of h, i.e., g + λh L1 ) [g, h]= g lim L1 ) λ λ = sign g)hdx+ h dx, where r R sign r) is the usual sign-function which is equal to 1 on], [, = 1on], [ and = forr =. As A is m-completely accretive, by the results of [25], we know that, if u DA) and f W 1,1, T ; L 1 )) such type of data will be called smooth data for short in the following), then the mild solution u of CP)u, f ) is already a strong solution, i.e., u W 1,1, T ; L 1 )), u) = u and {g=} du t) + Aut) = f t) for almost all t, T ). dt In other words, according to the definition of A, for smooth data u, f, the mild solution u satisfies, for almost all t, T ), and, for any renormalization S, ) 1,p ) T γ ut) W ) for any γ >, lim ut) px) = γ {γ < ut) <γ +1}

19 M. Bendahmane et al. / J. Differential Equations ) dsut)) div S ut) ) ut) px) 2 ut) ) + S ut) ) ut) px) = f t)s ut) ) dt in D ). By the definition of A, using in the discretized approximate equation 55) the renormalization S = S n and the test function T γ v ε i ), integrating over tε i 1, tε ) and over, summing up the equations i over i = 1,...,m ε, using the estimate S n v ε) i v ε i v ε i 1 ) Tγ v ε i ) S n v ε) vε i i v ε i 1 T γ r) dr, which holds by the monotonicity of T γ and S n, and then passing to the limit with n yields the a-priori-estimate uεt mε ε ) uε) t ε mε T γ r) dr dx + T γ u ε ) px) dxdt mε = vε i i=1 v ε i 1 mε tε i i=1 t ε i 1 mε T γ r) dr dx + f ε i T γ u ε ) dxdt. tε i i=1 t ε i 1 T γ v ε px) i dxdt As a consequence, we find t ε mε T γ u ε ) px) dxdt uεt ε mε ) t ε mε T γ r) dr dx + T γ u ε ) px) dxdt v mε γ ε L1 ) + tε i i=1 t ε i 1 f ε i L1 ), ) dt i.e., we get a uniform a-priori-bound in L p ) Q T ) N of T γ u ε )) and thus of T γ u ε )) ε in L p, T ; W 1,p ) )). It follows that the mild =strong for smooth data) solution u of CP)u, f ) obtained as the L, T ; L 1 ))-limit of such approximate solutions satisfies and T γ u) L p, T ; W 1,p ) ) ) for all γ >,

20 152 M. Bendahmane et al. / J. Differential Equations ) T γ u) L p ) Q T ) ) N for all γ >. In the same way, using in the time-discretized equations the renormalization S = S n function φ γ v ε i ) = T γ +1v ε i ) T γ v ε i ), yields the a-priori-estimate and the test t mε ε {γ < uε <γ +1} mε = tε i u ε px) dxdt i=1 t ε {γ < v ε <γ +1} i 1 i v ε px) i dxdt vε mε φ γ r) dr dx + tε i i=1 t ε i 1 f ε i φ γ u ε ) dxdt. Passing to the limit with ε in this inequality, using 3), we get the following estimate for the mild solution u: φ γ u) { L p ) Q T ) max { u γ } { u γ } u dx + u dx + { u γ } { u γ } f dt dx ) 1/p + ) 1/p } f dt dx. 59) For smooth data u and f which is also essentially bounded on and Q T, respectively, according to the L -contraction principle for mild solutions of the Cauchy problem for an m-completely accretive operator A and as A =, for almost all t, T ), wealsohave, ut) L ) u L ) + t f s) L ds <, ) thus the mild =strong) solution is also essentially bounded on Q T, and therefore u is also a weak solution of the parabolic equation, i.e., u V L Q T ). Let now u L 1 ), f L 1 Q ) be arbitrary, and consider smooth and essentially bounded approximations of this data, functions f ε W 1,1, T ; L 1 )) L Q T ) and u,ε DA) L ) satisfying f ε f in L 1 Q T ), u,ε u in L 1 ), as ε, T T f ε dxdt f dxdt, u,ε dx u dx. As to the right-hand side f it is clear that this type of approximation exists. As to the initial data we may, of course, always approximate u L 1 ) by T 1/ɛ u ) L ) which converges to u in L 1 ) as ɛ +. Consequently, in the following, we may assume that the initial data u L ). 6)

21 M. Bendahmane et al. / J. Differential Equations ) For such L -datum consider the function u,ε = I + ε A) 1 u ). By the complete accretivity of A and as A =, we have u,ε Ls ) u Ls ) for all 1 s. 61) Moreover, by definition of A and as u,ε is essentially bounded, u,ε W 1,p ) ) and u,ε ε div u,ε px) 2 u,ε ) = u in D ). Testing this equation with u,ε yields, for all ε >, the estimate ε It follows that, for any ϕ D), by the Hölder inequality, ε u,ε px) 2 u,ε ϕdx ε u,ε px) 1 ϕ dx u,ε px) dx u 2 L 2 ). 62) 2ε u,ε px) 1 L p ) ) ϕ L p ) ) { ) 1/p ) 1/p +} 2ε max u,ε px) dx, u,ε px) dx ϕ L p ) ) ) 1/p ) 1/p +} = 2 max {ε 1 1/p ε u,ε px) dx,ε 1 1/p+ ε u,ε px) dx ϕ L p ) ), and according to 62), the right-hand side of the preceding inequality tends to as ε. As a consequence, we have ε div u,ε px) 2 u,ε ) ind ), and thus u,ε u in D ) as ε. Taking into account 61) we can conclude that u,ε u in L 2 ) as ε, and therefore also in L 1 ). By the way, the above proof shows that DA) is dense in L 1 ) which was claimed before. By the aforementioned results, for any ε >, the mild solution u ε of CP)u,ε, f ε ) with u,ε, f ε as in 6), is already a strong solution and also a weak and thus renormalized solution of the parabolic problem 1). Moreover, by the general theory of nonlinear semigroups, the mild solution u ε converges, as ε, in C[, T ]; L 1 )) totheuniquemildsolutionofcp)u, f ). Our aim is to prove that this mild solution is also a renormalized solution. The proof of this result consists of two main steps. First, we prove ε-uniform a-priori-estimates in certain Bochner spaces as well as in appropriate variable exponent Lebesgue spaces for u ε and u ε. Second, we pass to the limit in the renormalized equations as ε. 5. Proof of Theorem A-priori-estimates Lemma 5.1. The estimates in Lemma 4.2 hold with u replaced by u ε, and all the constants are independent of ε.

22 154 M. Bendahmane et al. / J. Differential Equations ) Proof. See the proof of Lemma 4.2. By Lemma 5.1 and Lemma 4.2, we obtain the following result. Lemma 5.2. If p > 2 1, there exists a constant C, not depending on ε,suchthat N+1 u ε L q,t ;W 1,q ) C, 63) )) for any continuous variable exponents q with 1 q )< Np ) 1)+p ) N+1 on Basic convergence results The a-priori-estimates in Lemmas 5.1 and 5.2, together with the C[, T ]; L 1 ))-convergence guaranteed by nonlinear semigroup theory, imply the following basic convergence results: Lemma 5.3. For a subsequence as ε u ε u a.e. in Q T 64) and, if p > 2 1 N+1,alsostronglyinLq, T ; L q ) )) for any continuous variable exponent q with 1 q )< Np ) 1)+p ) N+1 on ).Moreover, and T γ u ε ) T γ u) weakly in L p ) Q T ) ) N, 65) T γ u ε ) T γ u) weakly in L p, T ; W 1,p ) ) ), 66) for any γ >. By the preceding lemma, for the mild solution u of CP)u, f ), wehavet γ u) V for any γ >. Lemma 5.4. The sequence u ε ) ε<1 satisfies the following energy estimate lim lim sup u ε px) dxdt =. n ε {n uε n+1} Proof. According to the preceding section see 59)), the weak solution u ε satisfies the energy estimate φ n u ε ) { L p ) Q T ) max { u,ε n} { u,ε n} u,ε dx + { uε n} ) 1/p } u,ε dx +. f ε dt dx Since φ n = T n+1 T n and thus φ n u ε ) = 1 {n u ε n+1} u ε a.e. in Q T,wededuce ) 1/p +,

23 M. Bendahmane et al. / J. Differential Equations ) lim sup ε max {n uε n+1} { { u n} { u n} u ε px) dxdt u dx + { u n} u dx + ) p+ /p }. f dt dx) p /p+, Since u L 1 ) and f L 1 Q T ), u L, T ; L 1 )), passing to the limit n yields the desired result Strong convergence We start by recalling a suitable time-regularization procedure, which was first introduced by Landes [27], and employed by several authors to solve nonlinear time dependent problems with L 1 or measure data see e.g. [29,13]). We denote this time regularized function to T γ u) by T γ u)) μ,withμ >. It is defined as the unique solution T γ u)) μ V L Q T ) of the equation t Tγ u) ) μ + μ T γ u) ) μ T γ u) ) = ind Q T ), 67) with the initial condition Tγ u) ) μ t= = w μ in, 68) where w μ is a sequence of functions such that w μ W 1,p ) ) L ), w μ T γ u ) μ w L ) γ, a.e. in as μ, 1 μ w μ W 1,p ) asμ. 69) ) Following [27] we can easily prove t Tγ u) ) μ V L Q T ), T γ u) ) μ L Q T ) γ, Tγ u) ) μ T γ u) a.e. in Q T,weak- in L Q T ), and strongly in V,asμ. 7) The proof of the following lemma is very similar to that in [6,13] with constant exponents. Lemma 5.5. Fix γ >. LetS W 1, R) be a nonincreasing function such that Sr) = rfor r γ and supp S [ M, M] for some M >.Then lim inf μ lim ε T t t Su ε ) T γ u ε ) T γ u) ) μ) dxdsdt. 71)

24 156 M. Bendahmane et al. / J. Differential Equations ) To continue our proof of Theorem 3.1 we need the following proposition. Proposition 5.1. For any truncation level γ >, wehave T lim ε Tγ u ε ) px) 2 T γ u ε ) Tγ u) px) 2 T γ u) ) T γ u ε ) T γ u) ) dxdt =, 72) T γ u ε ) T γ u) strongly in L p ) Q T ) ) N, 73) and T γ u ε ) T γ u) strongly in L p, T ; W 1,p ) ) ). 74) Proof. As u ε u in L, T ; L 1 )) and L p ) Q T ) L p, T ; L p ) )), it is clear that 73) implies 74). Let us show next how 73) can be deduced from 72). Proof of 73). We recall the following well-known inequalities: a a p 2 b b p 2) a b) cp) { a b p, if p 2, a b 2 a + b ) 2 p, if 1 < p < 2, 75) for any two real vectors a, b and a real p, where cp) = 2 2 p when p 2 and cp) = p 1 when 1 < p < 2. Observe that T 2 2 p+ Tγ u ε ) T γ u) px) dxdt T {x : px) 2} {x : px) 2} T γ u ε ) T γ u) ) dxdt T T γ u ε ) px) 2 T γ u ε ) T γ u) px) 2 T γ u) ) Tγ u ε ) px) 2 T γ u ε ) Tγ u) px) 2 T γ u) ) T γ u ε ) T γ u) ) dxdt =: Eε), 76) and Eε) by 72). Next in the set where 1 < px)<2 we use 75) as follows: T T γ u ε ) T γ u) px) dxdt {x ; 1<px)<2} T {x ; 1<px)<2} T γ u ε ) T γ u) px) T γ u ε ) + T γ u) ) px)2 px)) 2

25 M. Bendahmane et al. / J. Differential Equations ) Tγ u ε ) + Tγ u) ) px)2 px)) 2 dxdt T T γ u ε ) T γ u) px) T γ u ε ) + T γ u) ) px)2 px)) 2 T γ u ε ) + T γ u) ) px)2 px)) 2 2 max T max T { T T γ u ε ) T γ u) 2 T γ u ε ) + T γ u) ) T γ u ε ) T γ u) 2 T γ u ε ) + T γ u) ) { T L 2/px) ) L 2/2 px)) ) dt 2 px) dxdt 2 px) dxdt ) p + /2} ) p /2 T γ u ε ) + T γ u) ) px) dxdt ) 2 p + )/2 T γ u ε ) + T γ u) ) px) dxdt ) 2 p )/2},, 2 max { p 1 ) p /2 Eε) ) p /2, p 1 ) p + /2 Eε) ) p + /2} { T max T γ u ε ) + T γ u) ) ) 2 p + )/2 px) dxdt, T T γ u ε ) + T γ u) ) ) 2 p )/2} px) dxdt. 77) Since T γ u ε ) is bounded in L p ) Q T ), using 72), it follows that the right-hand side of the preceding inequality tends to as ε. Combining the two preceding convergence results yields T lim ε T γ u ε ) T γ u) px) dxdt =, 78) i.e., 73) holds. Proof of 72). As T γ u ε ) T γ u) weakly in L p ) Q T ) N as ε and T γ u)) μ T γ u) strongly in L p ) Q T ) N as μ, in order to prove 72), it is actually sufficient to prove T lim μ lim ε =. T γ u ε ) px) 2 T γ u ε ) T γ u) ) μ px) 2 T γ u) ) ) μ T γ u ε ) T γ u) ) μ) dxdt

26 158 M. Bendahmane et al. / J. Differential Equations ) In order to prove this last estimate, we use ϕ = S n u ε)v ε,μ in 48), where V ε,μ = T γ u ε ) T γ u) ) μ. The result is T t t S n u ε )V ε,μ dxdsdt + J 1 ε,μ,n + J 2 ε,μ,n = J 3 ε,μ,n, 79) where T Jε,μ,n 1 = t T Jε,μ,n 2 = S n u ε) u ε px) 2 u ε V ε,μ dxdsdt, 8) t T Jε,μ,n 3 = t S n u ε) u ε px) V ε,μ dxdsdt, 81) f ε S n u ε)v ε,μ dxdsdt. 82) Our next goal is to pass to the limit in 79) 82) as, successively, ε, μ and then n. Using the definitions of S n see Section 3) and V ε,μ, and Lemma 5.5 with S = S n to deduce that for any n γ lim inf μ lim ε T t t S n u ε )V ε,μ dxdsdt. 83) By the definition of V ε,μ, 64), and Lemma 5.3, we deduce for any μ > V ε,μ T γ u) T γ u) ) μ weakly in V as ε, V ε,μ L Q T ) 2γ for any ε >, V ε,μ T γ u) T γ u) ) μ a.e. in Q T and weak- in L Q T ) as ε. 84) Next, as supp S n [ n + 1), n] [n,n + 1], wehaveforanyn N and any μ > From 84) we deduce that J 2 ε,μ,n T S n u ε) L R) V ε,μ L R) lim sup μ lim sup J 2 ε,μ,n C lim sup ε ε {n uε n+1} {n uε n+1} u ε px) dxdt. u ε px) dxdt, 85)

27 M. Bendahmane et al. / J. Differential Equations ) for any n. Herein, C > is a constant independent of n. From Lemma 5.4 and 85) we deduce lim n lim sup lim sup Jε,μ,n 2 =. μ ε An application of Lebesgue s dominated convergence theorem, we get from 84) T lim Jε,μ,n 3 = ε t fs n u) T γ u) T γ u) ) μ) dxdsdt, 86) for any μ > and any n N. We use 7) and that fs n u) L1 Q T ) when passing to the limit μ in 86). This yields for any fixed n lim n lim sup lim sup Jε,μ,n 3 =. μ ε Now we can pass to the limit sup in 79) when ε, μ, and n, respectively. The result is that for any γ T lim lim sup lim sup n μ ε t S n u ε) u ε px) 2 u ε V ε,μ dxdsdt. 87) By the definition of S n see Section 3), we have for any n γ S n u ε) u ε px) 2 u ε T γ u ε ) = T γ u ε ) px) 2 T γ u ε ) T γ u ε ). Moreover, by the definition of S n, S n u ε) u ε px) 2 u ε T γ u) ) μ = S n u ε) T n+1 u ε ) px) 2 T n+1 u ε ) T γ u) ) μ. 88) As T n+1 u ε )) ɛ is bounded in L p ) Q T ) N, passing to a subsequence if necessary, we may assume that T n+1 u ε ) px) 2 T n+1 u ε ) converges weakly in L p ) Q T ) N to some function Z n+1.inviewof Lemma 5.3 and 7) it follows lim μ lim ɛ T t T γ u) ) μ dxdsdt χ { u ε >γ } S n u ε) Tn+1 u ε ) px) 2 T n+1 u ε ) T = =. t χ { u >γ } S n u)z n+1 T γ u) dxdsdt Consequently, 87) implies

28 151 M. Bendahmane et al. / J. Differential Equations ) lim lim sup lim sup n μ ε T t T γ u ε ) T γ u) ) μ) dxdsdt. T γ u ε ) px) 2 T γ u ε ) Taking into account the weak convergence of T γ u ε ) T γ u) in L p ) Q T ) as ɛ, the strong convergence of T γ u)) μ T γ u) in V as μ and the nonnegativity of the integrand, we obtain 79). The proof of Proposition 5.1 is completed Concluding the proof of Theorem 3.1 Let S C R) be such that supp S [ M, M] for some M >. As, according to Lemma 4.3, u ε is also a renormalized solution of 1), we have t Su ε ) div S u ε ) u ε px) 2 u ε ) + S u ε ) u ε px) = f ε S u ε ) in D Q T ). 89) In the following we pass to the limit ε in the sense of distributions) in each of the terms in 89). As u ε u in C[, T ]; L 1 )) as ε, it is easy to pass to the limit in the first and the right-hand side term. Let us study the second term. We have S u ε ) u ε px) 2 u ε = S u ε ) T M u ε ) px) 2 T M u ε ), and, because of 73), S u ε ) T M u ε ) px) 2 T M u ε ) S u) T M u) px) 2 T M u) strongly in L p ) Q T ).Since S u) T M u) px) 2 T M u) = S u) u px) 2 u, this concludes passing to the limit in the second term in 89). Let us now consider the third term. Since supp S [ M, M], clearly we have S u ε ) u ε px) = S u ε ) TM u ε ) px) a.e. in Q T. Using the boundedness of S and that S u ε ) converges to S u) a.e. in to deduce from 73) that S u ε ) T M u ε ) px) S u) T M u) px) strongly in L 1 Q T ) as ε. This concludes the treatment of the third term in 89), as S u) T M u) px) = S u) u px) and the proof of the existence result is complete.

29 M. Bendahmane et al. / J. Differential Equations ) Uniqueness of renormalized solutions In this section we prove uniqueness of a renormalized solution of 1). In fact, the uniqueness result as well as the comparison principle for renormalized solutions is an immediate consequence of the following result. Theorem 6.1. Let u be a renormalized solution of problem 1) for data u, f ) L 1 ) L 1, T ; L 1 )). Then u is an integral solution and thus the unique mild solution of the abstract Cauchy problem CP)u, f ). In particular, a renormalized solution of 1) is unique. As mild solutions of the Cauchy problem for an m-completely accretive operator A satisfy the comparison principle 57), an immediate consequence of the preceding theorem is Corollary 6.1. Let u, v be renormalized solutions of 1) corresponding to data u, f ), v, g) L 1 ) L 1, T ; L 1 )), respectively. Then, ut) vt) ) + L 1 ) u v ) + t L 1 ) + f t) gt) ) + L 1 ) for any t T. Proof of Theorem 6.1. Let v, w) A L ) L )), i.e., v W 1,p ) ) L ), w L ) and div v px) 2 v ) = w in D ). 9) Let < s < t < T, and k N with 1/k < min{s, T t}. Letσ k W 1,, T ) be the piecewise affine interpolation of the function which is constant equal to 1 on s, t) and equal to on [, s 1/k] [t + 1/k, T ]. Now, taking S = S n as a renormalization in 3) and choosing 1 γ T γ u v)σ k V L Q T ), γ >, as a test function, we obtain t S n u), 1 γ T γ u v)σ k + 1 γ T = T + T V +L 1 Q T ),V L Q T ) σ k S n u) u px) 2 u T γ u v) dτ, x) σ k S n u) u px) 1 γ T γ u v) dτ, x) σ k fs n u) 1 γ T γ u v) dτ, x). According to the integration-by-parts-formula, the first term reads

30 1512 M. Bendahmane et al. / J. Differential Equations ) t S n u), 1 γ T γ u v)σ k = T T V +L 1 Q T ),V L Q T ) uτ ) σ k ) τ Sn r) 1 γ T γ r v) dr dτ, x) u uτ ) σ k ) τ u 1 γ T γ r v) dr dτ, x) as n, by Lebesgue s dominated convergence theorem, since S n u) 1 a.e. on Q T as n and S n u) L Q T ) 1foralln N. There is no difficulty in passing to the limit with n inthesecondtermontheleft-andthe term on the right-hand sides. As to the third term on the left-hand side, according to the energy estimate 29) satisfied by a renormalized solution, we find T σ k S 1 n u) u px) γ T γ u v) dτ, x) {n u n+1} u px) dτ, x) Consequently, passing to the limit with n in 91) yields T + 1 γ T = uτ ) σ k ) τ T u 1 γ T γ r v) dr dτ, x) asn. σ k u px) 2 u T γ u v) dτ, x) σ k f 1 γ T γ u v) dτ, x). Using now 1 γ T γ ut) v)σ k as a test function in 9), integrating the resulting equation over, T ) and then subtracting it from the preceding equation, we get T uτ ) 1 σ k ) τ γ T γ r v) dr dτ, x) u + 1 σ γ k u px) 2 u v px) 2 ) v u v) dτ, x) { u v <γ } T = σ k f w) 1 γ T γ u v) dτ, x).

31 M. Bendahmane et al. / J. Differential Equations ) Obviously, the second term in the equation is nonnegative. As to the first term, we have T uτ ) σ k ) τ u 1 γ T γ r v) dr dτ, x) T σ k ) τ uτ ) v L 1 ) u v L1 )) dτ as γ ut) v L1 ) us) v L1 ) as k. For the term on the right-hand side, we find T t σ k f w) 1 γ T γ u v) dτ, x) s t s t Combining all estimates we get s f w) 1 γ T γ u v) dτ, x) as γ f w) sign u v) dτ, x) as k [ uτ ) v, f τ ) w ] dτ. ut) vl 1 ) t us) vl 1 ) [ ] uτ ) v, f τ ) w dτ, s for all s t T. As A is the closure of A L ) L )) in L 1 ) L 1 ), using the upper semicontinuity of the bracket and Fatou s lemma, it follows that the preceding inequality still holds for all v, w) A, and thus u is the unique integral solution and thus the unique mild solution of CP)u, f ). 7. Remarks 7.1. Entropy solutions In the case of a constant exponent, a notion of entropy solution for 2) has been introduced in [3]. The next definition will be a straightforward generalization for the case of a variable exponent: Definition 7.1. For k >, let us define θ k : R R by θ k r) := r T kσ ) dσ, r R and E := { φ V L Q T ); φ t V + L 1 Q T ) }.