NONLINEAR PARABOLIC PROBLEMS WITH VARIABLE EXPONENT AND L 1 -DATA

Electronic Journal of Differential Equations, Vol. 217 217), No. 32, pp. 1 32. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR PARABOLIC PROBLEMS WITH VARIABLE EXPONENT AND L 1 -DATA STANISLAS OUARO, AROUNA OUÉDRAOGO Communicated by Jesus Ildefonso Diaz Abstract. In this article, we prove the existence and uniqueness of entropy solutions to nonlinear parabolic equation with variable exponent and L 1 -data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. 1. Introduction The purpose of this article is to study the existence and uniqueness of entropy solutions to the nonlinear parabolic problem involving the px)-laplacian type operator u t div ax, u) f u in, T ) on Σ T, T ) u, ) u ) in, 1.1) where R N N 3) is a bounded open domain with smooth boundary and T is a positive fixed final time. The study of various mathematical problems with variable exponent has received considerable attention in recent years. These problems concern applications see [2, 1, 11, 21, 22]) and raise many difficult mathematical problems. The operator div ax, u) is called px)-laplacian type operator and is a generalization of the px)-laplace operator px) u) : div u px) 2 u) and the generalized mean curvature operator div 1 u 2) px) 2)/2 u ). Therefore, the problem 1.1) can be viewed as a generalization of the px)-laplace problem u t div u px) 2 u) f in, T ) u on Σ T, T ) u, ) u ) in. 1.2) 21 Mathematics Subject Classification. 35K55, 35D5. Key words and phrases. Parabolic equation; variable exponent; entropy solution; L 1 -data. c 217 Texas State University. Submitted October 9, 213. Published January 29, 217. 1

2 S. OUARO, A. OUÉDRAOGO EJDE-217/32 and the generalized mean curvature problem 1 u t div u 2 ) ) px) 2)/2 u f in, T ) u on Σ T, T ) u, ) u ) in. 1.3) The existence and uniqueness of renormalized solutions to problems 1.2) and 1.3) are nowadays well-known see [3, 24]). We recall that the notion of renormalized solutions was introduced for the first time by Diperna and Lions [13] in their study of the Boltzmann equation. An equivalent notion of solutions, called entropy solutions, was introduced independently by Bénilan and al. in [4]. Following [4] and using the same notion of solution, Ouaro and Traoré see [19]) studied the problem u div ax, u) f u on, in 1.4) where they proved the existence and uniqueness of entropy solution for a data f L 1 ). Relying on these results and applying nonlinear semigroup theory, it is easy to deduce the existence of a unique mild solution for the abstract Cauchy problem corresponding to 1.1) and arbitrary L 1 data cf. section 4). In this paper, we use the abstract semigroup theory to prove the existence and uniqueness of entropy solution to 1.1) for arbitrary L 1 data. We recall that Wittbold and Zimmermann in [23] studied and proved the existence and uniqueness of a renormalized solution to the stationary problem βu) div ax, Du) div F u) f in, u on, 1.5) where f L 1 ), a bounded domain of R N N 1) with Lipschitz boundary if N 2), F : R R N locally Lipschitz continuous, β : R 2 R a set valued, maximal monotone mapping such that β), a : R N R N a Carathéodory function and p ) : 1, ) a continuous variable exponent such that 1 < min x px) < N. Relying on these above results and applying nonlinear semigroup theory see [6]), Bendahmane, Wittbold and Zimmermann proved see [3]) the existence and uniqueness of a renormalized solution to the problem 1.2). Apart from the work by Bendahmane and al [3], Zhang and Zhou [24] studied the problem 1.2) by using other methods, where they proved the existence and uniqueness of entropy solutions. They also proved the equivalence between entropy and renormalized solutions of 1.2). The method used in [24] was the following: They employed first the difference and variation methods to prove the existence and uniqueness of a weak solution for the approximate problem of 1.2) under appropriate assumptions. Then they constructed an approximate solution sequence and established some a priori estimates. Next, they drew a subsequence to obtain a limit function and proved that this function is a renormalized solution. Based on the strong convergence on the truncations of approximate solutions, they obtained that the renormalized solution to problem 1.2) is also an entropy solution, which leads to an equality in the entropy formulation. Finally, by choosing suitable test functions, they proved the uniqueness of renormalized solutions and entropy solutions and thus, the equivalence of renormalized solutions and entropy solutions. The main

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 3 operator in problem 1.1) is more general than the p )-Laplace operator of 1.2) as we will see later. The aim of our paper is to extend the results in [19], to the case of parabolic equations. Inspired by [3] and [24], we first define two notions of solutions of problem 1.1): The notion of entropy solution and the notion of renormalized solution. Next, we show that the two notions are equivalent which will permit us to use both notion when convenient. After that, according to the results in [19], we prove some properties of the entropy solutions of problem 1.1), by using nonlinear semigroup theory. Next, we prove the existence and uniqueness of entropy solutions to problem 1.1). This article is organized as follows: In section 2 we recall some results of [19], the assumptions of problem 1.1) and some basic notations and properties of Lebesgue and Sobolev spaces with variable exponents. In section 3, we give the definition of entropy and renormalized solutions to problem 1.1) and prove that the two notions are equivalent. In section 4, using the results of [19], we prove some properties of entropy solutions to problem 1.1). Finally, in section 5 we prove the existence and uniqueness of entropy solutions of 1.1). 2. Preliminaries In this article, we study problem 1.1) with the following assumptions on the data: p ) : R is a measurable function such that 1 < p p <, 2.1) where p : ess inf x px) and p : ess sup x px). For the vector field a, ), we assume that ax, ξ) : R N R N is Carathéodory and is the continuous derivative with respect to ξ of the mapping A : R N R, i.e. ax, ξ) ξ Ax, ξ) such that: Ax, ) for almost every x. 2.2) There exists a positive constant C 1 such that ax, ξ) C 1 jx) ξ px) 1 ) 2.3) for almost every x and for every ξ R N where j is a nonnegative function in L p ) ), with 1 px) 1 p x) 1. The following inequalities hold ax, ξ) ax, η)) ξ η) >, 2.4) for almost every x and for every ξ, η R N, with ξ η and 1 C ξ px) ax, ξ).ξ Cpx)Ax, ξ) 2.5) for almost every x, C > and for every ξ R N. Assumption 2.4) is imposed to obtain uniqueness of the solution to problem 1.1). Remark 2.1. 1) Strict monotonicity see assumption 2.4)) of the vector field is certainly not needed to prove uniqueness of the entropy solution. It was assumed it here only just for simplicity. 2) ax, ) for a.e. x. Indeed for a.e. x fixed, denote z ax, ) R N. By the continuity of ax, ), we have lim ξ ax, ξ) z. Suppose now that

4 S. OUARO, A. OUÉDRAOGO EJDE-217/32 z if z, there is no need to make a proof; this is the case for example when ax, ξ) ξ p 2 ξ) and choose ξ sz with s > used to tend toward ; then ax, ξ ) ξ sz s)) z s z 2 szs) s z 2 s z s), where lim s s). Therefore, for s sufficiently small, s z 2 s z s) <, which is a contradiction by assumption 2.5). Thus, z. 3) As examples of models with respect to assumptions 2.2)-2.5) for problem 1.1), we can give the following. i) Set Ax, ξ) 1/px)) ξ px), ax, ξ) ξ px) 2 ξ, where px) 2. Then we obtain the px)-laplace operator div u px) 2 u). ) px)/2 ii) Set Ax, ξ) 1/px))[ 1 ξ ) 2 1], ax, ξ) 1 ξ 2 px) 2)/2 ξ, where px) 2. Then we obtain the generalized mean curvature operator 1 div u 2 ) ) px) 2)/2 u. As the exponent px) appearing in 2.3) and 2.5) depends on the variable x, we must work with Lebesgue and Sobolev spaces with variable exponents. We define the Lebesgue space with variable exponent L p ) ) as the set of all measurable functions u : R for which the convex modular ρ p ) u) : u px) dx is finite. If the exponent is bounded, i.e., if p <, then the expression u p ) : inf {λ > : ρ p ) u/λ) 1} defines a norm in L p ) ), called the Luxembourg norm. The space L p ) ),. p ) ) is a separable Banach space. Moreover, if 1 < p p <, then L p ) ) is uniformly convex, hence reflexive, and its dual space is isomorphic to L p ) ), where 1 px) 1 p x) 1. Finally, we have the Hölder type inequality. uv dx 1 p 1 p ) u p ) v p ), 2.6) for all u L p ) ) and v L p ) ). Now, let { } W 1,p ) ) : u L p ) ) : u L p ) ), which is a Banach space equipped with the norm u 1,p ) u p ) u ) p ). The space W 1,p ) ), u 1,p ) ) is a separable and reflexive Banach space. Next, we define W 1,p ) ) as the closure of C ) in W 1,p ) ) under the norm u : u ) p ). The space W 1,p ) ), u ) is a separable and reflexive Banach space. For the interested reader, more details about Lebesgue and Sobolev spaces with variable exponent can be found in [12] see also [16]). An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by the modular ρ p ) of the space L p ) ). We have the following result cf. [15]).

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 5 Lemma 2.2. If u n, u L p ) and p <, then the following properties hold: 1) u p ) > 1 u p p ) ρ p )u) u p p ) ; 2) u p ) < 1 u p p ) ρ p )u) u p p ) ; 3) u p ) < 1 respectively 1; > 1) ρ p ) u) < 1 respectively 1; > 1); 4) u n p ) respectively ) ) ρ p ) u n ) respectively ); 5) ρ p ) u/ u p ) 1. Following [3], we extend a variable exponent p : [1, ) to [, T ] by setting pt, x) : px) for all t, x). We also consider the generalized Lebesgue space L p ) ) { } u : R measurable such that ut, x) px) dx, t) <, endowed with the norm u L p ) ) : inf { λ > : ut, x) } px) dx, t) < 1, λ which shares the same properties as L p ) ). We now recall the main result of [19] for the study of 1.4). We first recall the definition of the weak and entropy solutions of 1.4). Definition 2.3. A weak solution of 1.4) is a function u W 1,1 ) such that a, u) L 1 loc )) N and ax, u) ϕ dx uϕ dx fx)ϕ dx, 2.7) for all ϕ C ). W 1,p ) ). A weak energy solution is a weak solution such that u Definition 2.4. A measurable function u is an entropy solution to problem 1.4) if, for every t >, T t u) W 1,p ) ) and ut t u ϕ) dx ax, u) T t u ϕ) dx fx)t t u ϕ) dx, 2.8) for all ϕ W 1,p ) ) L ). Now, we recall the two main results in [19]. Theorem 2.5 [19, Theorem 3.2]). Assume that 2.1)-2.5) hold and f L ). Then there exists a unique weak energy solution of 1.4). Theorem 2.6 [19, Theorem 4.3]). Assume that 2.1)-2.5) hold and f L 1 ). Then there exists a unique entropy solution to problem 1.4). Remark 2.7. Theorems 2.5 and 2.6 were generalized by Bonzi and Ouaro see [8, Theorem 3.2 and 4.3]). According to [8, Theorem 3.2], [19, Theorem 3.2 ] hold for f L p ) ).

6 S. OUARO, A. OUÉDRAOGO EJDE-217/32 3. Equivalence between entropy and renormalized solutions Let T k denote the truncation function at height k, that is { s if s k, T k s) k signs) if s > k. For the notion of entropy solution to problem 1.1), we will use the primitive of the truncation function at height k denoted by Θ k : R R such that { r r 2 /2 if r k, Θ k r) T k s) ds k r k2 2 if r k. It is obvious that Θ k r) and Θ k r) k r. We denote T 1,p ) ) { u :, T ] R measurable ; T k u) L p, T ; W 1,p ) )), with T k u) L p ) ) ) N, for every k > }. Next, we define the weak gradient of a measurable function u T 1,p ) ). The proof follows from [4, Lemma 2.1] due to the fact that W 1,p ) ) W 1,p ). Proposition 3.1. For every measurable function u T 1,p ) ), there exists a unique measurable function ν : R N, which we call the weak gradient of u and denote ν u, such that T k u) νχ { u <k}, almost everywhere in and for every k >, where χ E denotes the characteristic function of a measurable set E. Moreover, if u belongs to L 1, T ; W 1,1 )), then ν coincides with the gradient of u. The notion of the weak gradient allows us to give the following definitions of entropy and renormalized solutions to problem 1.1). We define the spaces: V { f L p, T ; W 1,p ) )) : f L p ) ) }, E { ϕ V L ) : ϕ t V L 1 ) }. According to [2], we have E C[, T ]; L 1 )). Definition 3.2. An entropy solution to problem 1.1) is a function u T 1,p ) ) L ) such that the mapping [, T ] t Θ k u φ)t, x) dx is a.e. equal to a continuous function for all k > and all φ E, and Θ k u φ)t ) dx Θ k u φ)) dx φ t T k u φ) dx dt ax, u) T k u φ) dx dt ft k u φ) dt dx, for all k > and φ E. 3.1)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 7 Definition 3.3. A function u T 1,p ) ) L ) is a renormalized solution to problem 1.1) if the following conditions are satisfied: i) lim n {t,x) :n ut,x) n1} ii) for all S in W 2, R) such that S has a compact support, u px) dt dx ; 3.2) t Su) div [ S u)ax, u) S u)ax, u) u ] fs u) in D ), 3.3) Su)) Su ) in L 1 ). 3.4) Remark 3.4. Using the fact that for any function ϕ V L ), there exists functions ϕ n D) that converge strongly to ϕ in V and weak-* in L ), we see that in 3.1) and 3.3) we cannot only use the test functions in D), but also functions in V L ). In fact, we can replace 3.3) by Su), ϕ T [ S u)ax, u) ϕ S u)ax, u) u)ϕ ] dx dt t T 3.5) fs u)ϕ dx dt, where, denotes the duality pairing between V L 1 ) and V L ). To find more estimates for entropy solutions and also to get useful a priori estimates of approximate solutions to the equation 5.2) below, the following integration by parts formula plays a crucial role. Lemma 3.5. Let f : R R be a continuous piecewise C 1 function such that f) and f is zero outside a compact set of R. Let us denote F s) s fr) dr. If u V is such that u t V L 1 ) and if ψ C ), then we have u t, fu)ψ F ut ))ψt ) dx F u))ψ) dx ψ t F u) dx dt, where, denotes the duality pairing between V L 1 ) and V L ). The proof of the above lemma follows the same lines as the proof of [14, Lemma 7.1]; we omit it. Next, we have a result showing the equivalence between entropy and renormalized solutions of 1.1). Theorem 3.6. A function u is an entropy solution of 1.1) if and only if it is a renormalized solution. The proof of the above theorem is the same as in constant exponent case; see [14]. 4. Properties of entropy solutions In this section, we prove the existence of mild solutions of 1.1) satisfying an L 1 - comparison principle. A classical method to prove that consists in approximating 1.1) for >, by an implicit time-discretization u i u i 1 t i t i 1 div ax, u i) f i in D ), for i 1,..., n, u i W 1,p ) ) L ), 4.1)

8 S. OUARO, A. OUÉDRAOGO EJDE-217/32 where n N, t < t 1 < < t n T and fi L ), i 1,..., n such that n t i ft) fi L 1 )dt, max i1,...,n t i t i 1), i1 t i 1 T t n, u u L 1 ) as. The function u is piecewise constant, defined by u u i on t i 1, t i], i 1,..., n; u ) u. This method is actually the method of nonlinear semigroup theory. Naturally, we are led to give the following concept. Definition 4.1. A mild solution of 1.1) is a function u C[, T ]; L 1 )) which is the uniform limit of the piecewise constant function u. The main result of this section is the following. Theorem 4.2. For any u, f) L 1 ) L 1 ), there exists a unique mild solution u of 1.1). Moreover, the following contraction principle holds: for any t T, if u resp. û) is a mild solution of 1.1) with respect to u, f) L 1 ) L 1 ) resp. û, ˆf) L 1 ) L 1 )), then ut) ût) L1 ) u û L1 ) t fs) ˆfs) L ds. 1 ) According to the nonlinear semigroups theory see [6]), the preceding result is, essentially, a consequence of the result of Proposition 4.3 below. Before stating the proposition, we need to recall some definitions. Let A be a possibly) multivalued nonlinear operator in L 1 ) that is A : L 1 ) PL 1 )); as usual, A is identified with its graph {u, v) L 1 ) L 1 ); v Au}. The operator A is called accretive if u û 1 u û σv ˆv) 1, for any u, v), û, ˆv) A, σ > ; i.e., for any σ >, the resolvent of A, I σa) 1, is a single-valued operator and a contraction in the L 1 norm. The operator A is called T -accretive if u û) 1 u û) σv ˆv) 1, for any u, v), û, ˆv) A, σ > ; equivalently, if v ˆv) v ˆv), {u>û} {uû} for any u, v), û, ˆv) A. Finally, the operator A is called m-accretive resp. m T - accretive) if A is accretive resp. T -accretive) and, moreover, RI σa) L 1 ), for any σ > cf. [6] ). Proposition 4.3. There exists an operator such that A {u, f) L 1 ) L 1 ); u is an entropy solution of 1.4)} i) A is T -accretive and even completely accretive, cf. [5]); ii) RI σa) L 1 ), for any σ > ; iii) DA) L 1 ).

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 9 Proof. i) Let u resp. û) be a weak solution of 1.4) for f resp. ˆf) L ). We use 1 k T ku û) as test function in 2.7) for k > to get upon addition u û) 1 k T ku û) dx ax, u) ax, û)) u û) dx f ˆf) 1 k T ku û) dx. { u û <k} Letting k tend to and using assumption 2.4), we obtain u û) dx f ˆf) sign u û) dx f ˆf) dx f ˆf) sign u û) dx {uû} [ u û), f ˆf) ], 4.2) 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 1 g 1 [g, h] lim h dx h sign λ λ g) dx, {g} with r R sign r), the usual sign-function which is equal to 1 on, ), equal to 1 on, ) and equal to for r. Now, let f, ˆf L 1 ) and u, û be two entropy solutions of 1.4) with f and ˆf as data respectively. Then [19], there exist u n, f n ) and û n, ˆf n ) such that u n, û n are weak solutions of 1.4) with f n, ˆf n L ) as data and such that: u n u and û n û in measure, f n f and ˆf n ˆf in L 1 ), as n approaches. According to [19], u n u a.e. in, û n û a.e. in. By setting f n T n f), ˆf n T n ˆf) and using 4.2), we have u n û n ) dx f n ˆf n ) sign u û) dx T n f) sign u û) dx T n ˆf) sign u û) dx f 1 ˆf 1 <. Therefore, by Fatou s lemma, we deduce that u û) dx lim inf u n û n ) dx. 4.3) n From 4.2), we have u n û n ) dx [ u n û n ), f n ˆf n ) ]. Note also that [, ] is upper semi-continuous which gives [ lim sup un û n ), f n ˆf n ) ] [ u û), f ˆf) ]. 4.4) n Finally, we use 4.3) and 4.4) to obtain u û) dx [ u û), f ˆf) ].

1 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Assertion ii) is a direct consequence of [19, Theorem 4.3]. iii) As L ) L 1 ), we will prove that L ) DA). 1. Let α >, and f L ). We denote u α : I αa) 1 f. Then u α, 1 α f u α)) A. As f L ) then, according to Theorem 2.5, u α is a weak energy solution of 1.4). Let s take φ D) as a test function in 2.7) to obtain α ax, u α ) φ dx u α φ dx fx)φ dx. 4.5) The following Lemma provides L -a priori estimates of a solution u and is crucial for the next of the proof. Lemma 4.4. Let u be a weak energy solution of 1.4), then u s C f s, for 1 s. Proof. The proof is rather classical see. [19]). For the sake of completeness, let us recall the arguments. For p P { p C R); p 1, supp p is compact, / supp p }, we take pu α ) as a test function in 4.5) to obtain pu α )fx) dx α ax, u α ) pu α ) dx pu α )u α dx [ α ax, uα ) ax, ) ] u α p u α ) dx α ax, ) u α p u α ) dx 4.6) pu α )u α dx α ax, ) u α p u α ) dx pu α )u α dx pu α )u α dx by the divergence formula). Next, we choose p such that pk) k s 2 k for 1 s < in 4.6) to obtain u α s 2 u α f dx u α s dx. 4.7) By Hölder inequality, from 4.7) we obtain u α s dx f s uα s 1) s dx which gives As f L ), then 4.8) implies u α f. ) 1/s u α s f s. 4.8) Now, let us come back to the proof of Proposition 4.3. We take u α as a test function in 4.5) to obtain α ax, u α ) u α dx u 2 α dx fx)u α dx 4.9) u α q f p.,

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 11 Then, by Lemma 4.4 and 2.5), from 4.9) we deduce that α u α px) dx C f p f q < because f L )). 4.1) Now, by using the Hölder type inequality, for all φ D), we have α ax, u α ) φdx C 1 α jx) uα px) 1) φ dx C α j p ) φ p ) C α u α px) 1) p ) φ p ) ) C max α 1 1 p αρp ) u α )) 1 p 1, α 1 p αρp ) u α )) 1 p Cα. 4.11) According to 4.1), from 4.11) we deduce that α ax, u α ) φdx as α. 4.12) From 4.5) by using 4.12) we obtain u α f as α, in D ). 4.13) Note also that u α ) α> is uniformly bounded by Lemma 4.4, then up to a subsequence, u α f in L p ), for all 1 < p < and a.e. in. Now, u α p f p for all 1 < p < by Lemma 4.4, then by the Lebesgue dominated convergence theorem, we deduce that As is bounded, 4.14) implies u α f as α, in L p ), 1 < p <. 4.14) Therefore, by 4.15), we deduce that DA) L 1 ). u α f in L 1 ) as α. 4.15) By Proposition 4.3, the nonlinear operator A is m-accretive in L 1 ). Then, by the general theory of nonlinear semigroups see [6]) we conclude that the abstract evolution problem corresponding to 1.1) admits a unique mild solution u C[, T ]; L 1 )) for any initial datum u DA). L 1 ) and any right-hand side f L 1, T ; L 1 )). Lemma 4.5. Let u be an entropy solution to problem 1.1), then u L,T ;L 1 )) f L 1 ) u L 1 ), 4.16) T k u) L p ) ) k max { f L1 ) u L1 )) 1/p, f L1 ) u L1 ) ) 1/p }. 4.17) Proof. Step 1: Proof of 4.17). Taking φ as a test function in 3.1), we obtain Θ k u)t ) dx Θ k u ) dx ax, u) T k u) dx dt 4.18) ft k u) dx dt.

12 S. OUARO, A. OUÉDRAOGO EJDE-217/32 By the definition of Θ k, we have Θ k u). Using hypothesis 2.5), inequality 4.18) becomes 1 T k u) px) dx dt Θ k u ) dx ft k u) dx dt C k u dx ft k u) dx dt ) k u dx f dx dt, then, according to Lemma 2.2, we deduce that T k u) L p ) ) k max { f L 1 ) u L 1 )) 1/p, f L 1 ) u L 1 ) ) 1/p }. Step 2: Proof of 4.16). In the following, we use the function S n : R R defined by: S n s) Note that S n satisfies s h n r)dr, where h n s) 1 T 1 s T n s)). 4.19) S n r) S n T n1 r)), S n L R) 1, supp S n [ n 1), n 1], S n 4.2) 1 [ n 1, n] 1 [n,n1]. ). Then θ is continuous on [, ), θ 1 Let t 1, T ) and θ t) 1 t t1) on [, t 1 ], θ on [t 1, ) and θ is linear on [t 1, t 1 ]. Using ϕ 1 k T ku)θ as a test function in 3.3)since entropy and renormalized solutions are equivalent) and taking S S n, we obtain 1 k T 1 k 1 k T 1 k T θ S n u)) t T k u) dx dt T S nu)ax, u) T k u)θ )dx dt fs nu)t k u)θ dx dt Since S ns) for s / [n, n 1], we can write S nu)ax, u) ut k u)θ dx dt. 4.21) S nu)ax, u) ut k u) S nu)ax, T n1 u)) T n1 u))t k u) L 1 ). Since θ 1 [,t1] and is bounded by 1 as, using Lebesgue dominated convergence theorem in equality 4.21), we obtain 1 k t1 1 k 1 k t1 t1 t1 n u)) t T k u) dx dt S 1 k S nu)ax, u) ut k u) dx dt fs nu)t k u) dx dt. S nu)ax, u) T k u)) dx dt 4.22)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 13 Let n M. We have T k u) T k S n u)) since S n s) s on [ M, M], S n s) M and signs n s)) signs) outside [ M, M]), S n u))t 1 ) ut 1, ) in L 1 ), S n u ) u in L 1 ) and S nu) 1 a.e. in as n. Since S ns) 1 and S ns) only if s [n, n 1], using 2.3) we can write t1 S nu)ax, u) ut k u) dx dt k ax, u) u dx dt {n u n1} k C 1 jx) u px) 1) u dx dt k {n u n1} C 1 jx) u px) 1) u 1 {n u n1} dx as n. Passing to the limit in 4.22) as n, we obtain 1 t1 u t T k u) dx dt 1 t1 ax, T k u)) T k u)) dx dt k k 1 t1 ft k u) dx dt, k for all t 1, T ). By 2.5), from 4.23), we obtain 1 t1 u t T k u) dx dt 1 t1 ft k u) dx dt. k k Letting k in the inequality above, we obtain t1 t1 u t sign u) dx dt f sign u) dx dt. which implies ut 1, ) L1 ) f L1 ) u L1 ), for all t 1, T ) i.e. u L,T ;L 1 )) f L1 ) u L1 ). This completes the proof. 4.23) Now, for any continuous and monotonic function ψ, we define the proper lower semi-continuous and convex or upper semi-continuous and concave function B ψ s) s ψr) dr. To prove the existence of weak solutions, we need an energy estimate similar to the one given in [1, Lemma 1.5]. Lemma 4.6. Let ψ C,1 R) be monotone, let u be a measurable function such that u L p, T ; W 1,p ) )). Then B ψ u) L, T ; L 1 )) and, for almost every t [, T ], B ψ ut))ξt) dx B ψ u )ξ) dx t u t ψu)ξ dx dt t B ψ u)ξ t dx dt 4.24)

14 S. OUARO, A. OUÉDRAOGO EJDE-217/32 for any ξ C,1 ) such that ψu)ξ L 2, T ; W 1,1 )). For the proof of the above lemma, see the proof of [9, Lemma 4]. By a weak solution of 1.1) we understand a solution in the sense of distributions that belongs to the energy space, i.e., u V { f L p, T ; W 1,p ) )); f L p ) ) }, u t div ax, u) f in D ), u, ) u. To complete this section we prove the following proposition. 4.25) Proposition 4.7. Assume that 2.1)-2.5) hold, u L ), f L ) and u is the unique mild solution of 1.1). Then u is a weak solution of 1.1). Proof. For i, 1,..., n, let u i be the unique weak energy solution of We have f i u i 1 I A)u i. ax, u i) ϕdx u i u i 1 ϕdx fi ϕdx, 4.26) for all ϕ W 1,p ) ). Taking ϕ u i t i 1, t i ) and summing up the inequalities over i 1,..., n, we obtain n i1 t i t i 1 t i n i1 t i 1 u i u i 1 u i dx dt fi u i dx dt. as test function in 4.26), integrating over n i1 t i t i 1 ax, u i) u i dx dt By 2.5) and as B Id is convex, from 4.27) we deduce that n t i B Id u i ) B Idu i 1 ) n t i 1 dx dt i1 t i 1 i1 t C i 1 n t i fi u i dx dt. i1 t i 1 u i px) dx dt 4.27) Consequently, if we set t i t i 1, then f t) fi and u t) u i for t t i 1, t i ], i 1,..., n; u ) u. It follows that [ ] B Id u T )) B Id u )) dx 1 T u px) dx dt C T f u dx dt. As B Id u T )) B Id u )), u, f L ), we obtain T T u px) dx dt C u p dx dt C. 4.28) Using the Poincaré inequality with constant exponent, we deduce that u ) > is uniformly bounded in L p, T ; W 1,p ) )). So, there exists a subsequence still denoted u ) >, such that u u in L p, T ; W 1,p ) )) as, 4.29)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 15 u u in L p ) ) ) N as. 4.3) Since u ) > is uniformly bounded in L p ) ) ) N, by 2.3) we deduce that ax, u ) ) > is uniformly bounded in L p ) ) ) N and then we can assume that From 4.26), we have ax, u )) ϕ dx ax, u ) Φ in L p ) ) ) N as. 4.31) u t) u t ) ϕ dx f t)ϕdx, 4.32) for all ϕ W 1,p ) ). Then, taking ψ W 1,1, T ; W 1,1 ) L )) E, ψt ) as a test function in 4.32), we obtain T T We have T T T ax, u t)) ψt) dx dt f t)ψt) dx dt. u t) u t ) ψt) dx dt u t) T ψt) dx dt u t) ψt) dx dt u t) ψt) dx dt T T T T T T u t) u t ) ψt) dx dt 4.33) u t ) ψt) dx dt u s) ψs ) dx ds where s t ) T T u s) ψs ) dx ds ψt ) ψt) u t) dx dt u, t)ψt) dx dt, ut)ψ t dx dt u t) ψt) dx dt u s) ψs ) dx ds T u t)ψt) dx dt T u ψ) dx dt as, where u t) u for t. Therefore, taking limit in 4.33) as, we obtain T T Φ ψ dx dt uψ t dx dt u ψ) dx dt T 4.34) ft)ψ dx dt. Thus, to complete the proof of Proposition 4.7, we only need to show that Φ ax, u). To do so, we apply the Minty-Browder s method. Firstly, we prove that lim sup ax, u ). u dx dt Φ u dx dt. 4.35)

16 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Using 4.27), we have T ax, u ) u dx dt [ ] T B Id u T )) B Id u ) dx f u dx dt. 4.36) Since B Id u T )), then by Fatou s lemma, we have lim inf B Idu T )) dx lim inf B Id u T )) dx. 4.37) Because of the lower semi-continuity of B Id, we have B Id ut )) dx lim inf B Idu T )) dx. 4.38) Inequalities 4.37) and 4.38) imply B Id ut )) dx lim inf B Id u T )) dx, i.e. lim inf B Id u T )) dx B Id ut )) dx. Then, passing to the limit in 4.36) as and according to Lemma 4.6 we have T lim sup ax, u ) u dx dt f u t, u. [ B Id ut )) B Id u)) ] T dx fu dx dt 4.39) Now, we prove that ax, u). ξ dx dt Φ ξ dx dt, 4.4) for any ξ L p, T ; W 1,p ) )). By the monotonicity of a, for any ρ L p, T ; W 1,p ) )), ax, ρ). u ρ) dx dt ax, u ). u ρ) dx dt. 4.41) Since u is a weak energy solution of fi u i 1 IA)u i then, by [19, Proposition 4.11], u converges in measure to u. We can then extract a subsequence such that u u a.e. in. Then according to 2.3), we may apply Lebesgue dominated convergence theorem and pass to the limit in 4.41) as to obtain lim inf ax, u ). u ρ) dx dt ax, ρ). u ρ) dx dt. 4.42) Combining 4.39) and 4.42), we have f u t, u ρ for all ρ L p, T ; W 1,p ) )). ax, ρ). u ρ) dx dt,

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 17 Choosing ρ u σξ, σ R, ξ L p, T ; W 1,p ) )), we obtain f u t, σξ σ ax, u σξ)). ξ dx dt. 4.43) Dividing inequality 4.43) by σ >, resp. σ < and passing to the limit with σ, resp. σ, we obtain f u t, ξ ax, u). ξ dx dt, 4.44) for any ξ L p, T ; W 1,p ) )). By 4.34), we have T Φ ψ dx dt T f u t, ψ. T uψ t dx dt u ψ) dx fψ dx dt 4.45) Combining 4.44) and 4.45) yields 4.4). To conclude, we pass to the limit in 4.33) as to obtain T T fφ dx dt uφ t dx dt uφ))dx T 4.46) ax, u) φ dx dt, for all φ E L ). Hence u is a weak solution of 1.1). Our aim is to prove that this weak solution is also an entropy solution of 1.1). The proof of this result consists of two main steps. Firstly, we prove uniform a-priori-estimates in certain Bochner spaces as well as in appropriate variable exponent Lebesgue spaces for u and u. Secondly, we pass to the limit in the entropy relation as. 5. Existence and uniqueness of an entropy solution Theorem 5.1. Let 2.1)-2.5) hold. Let u L 1 ), f L 1 ). There exists a unique entropy solution for 1.1). The proof of the above theorem is done in several steps. 5.1. A priori estimates. As u L 1 ), f L 1 ) and L is dense in L 1, then we can find two sequences of functions ) f > L ) and u, )> L ) strongly converging respectively to f and u such that f L1 ) f L1 ), u, L1 ) u L1 ). 5.1) Now, let u be a weak solution to problem 1.1) with f and u, as data, i.e. T T f φ dx dt u φ t dx dt u, φ, )dx T 5.2) ax, u ) φ dx dt, for all φ E L ).

18 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Lemma 5.2. The estimates in Lemma 4.5 hold with u replaced by u, and all the constants are independent of, i.e. u L,T ;L 1 )) f L 1 ) u L 1 ), 5.3) T k u ) L p ) ) k max { f L1 ) u L1 )) 1/p, f L1 ) u L1 ) ) 1/p }. 5.4) The proof of the above lemma is similar to that of Lemma 4.5. 5.2. Basic convergence results. The a priori estimates in lemmas 4.5 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 u as : )> u u a.e. in, 5.5) for all k >. T k u ) T k u) in L p ) )) N, 5.6) T k u ) T k u)in L p, T ; W 1,p ) )) 5.7) Proof. Proof of 5.5). Let u 1 and u 2 be two weak solutions of problem 1.1). Choosing θ T 1 u 1 u 2 ) as a test function corresponding to u 1 and θ T 1 u 2 u 1 ) as a test function corresponding to u 2, we obtain T θ u 1 ) t T 1 u 1 u 2 ) dx dt and T T T T T θ ax, u 1 ) T 1 u 1 u 2 ) dx dt θ f 1 T 1 u 1 u 2 ) dx dt θ u 2 ) t T 1 u 2 u 1 ) dx dt θ ax, u 2 ) T 1 u 2 u 1 ) dx dt θ f 2 T 1 u 2 u 1 ) dx dt. Adding 5.8) and 5.9), then by using 2.4) and letting approach zero we have t1 u 1 u 2 ) t T 1 u 1 u 2 ) dx dt t1 t1 t1 ax, u1 ) ax, u 2 ) ) T 1 u 1 u 2 ) dx dt f2 f 1 ) T1 u 1 u 2 ) dx dt f2 f 1 ) T1 u 1 u 2 ) dx dt. 5.8) 5.9) 5.1)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 19 From 5.1) we deduce that Θ 1 u 1 u 2 )t 1 ) dx Θ 1 u,1 u,2 ) dx f 2 f 1 L 1 ) u,1 u,2 L1 ) f 2 f 1 L1 ) : a 1 2. 5.11) By the definition of Θ 1, we have [u 1 u 2 )t 1)] 2 2 if u 1 t 1 ) u 2 t 1 ) < 1 Θ 1 u 1 u 2 )t 1 ) u 1 u 2 )t 1 ) if u 1 t 1 ) u 2 t 1 ) 1. On the set { u 1 u 2 1}, we have u 1 u 2 )t1) 2 u 1 t 1 ) u 2 t 1 ). Then, from 5.11) we deduce { u 1 u 2 <1} u 1 u 2 ) 2 t 1 ) 2 Θ 1 u 1 u 2 )t 1 ) dx a 1 2. Using Hölder inequality, u 1 t 1 ) u 2 t 1 ) dx u 1 t 1 ) u 2 t 1 ) dx { u 1 u 2 <1} { u 1 u 2 <1} u 1 t 1 ) u 2 t 1 ) dx dx { u 1 u 2 1} 2 { u 1 u 2 1} u 1 t 1 ) u 2 t 1 ) 2 dx) 1/2 meas) 1/2 2a 1 2 u 1 t 1 ) u 2 t 1 ) dx 2 meas)) 1/2 a 1/2 1 2 2a 1 2. 5.12) Since ) f and u >, )> are convergent respectively in L1 ) and L 1 ), we have a 1 2 for 1, 2. Thus from 5.12) we deduce that u is a Cauchy )> sequence in C[, T ]; L 1 )) and u converges to u in C[, T ]; L 1 )). Then we find an a.e. convergent subsequence still denoted by u )> ) in such that u u a.e. in. The proof of 5.5) is complete. Proof of 5.6) and 5.7). By 5.4), the sequence T k u ) ) is bounded in > L p ) ) ) N ; hence the sequence Tk u ) ) 1,p ) is bounded in W > ). Then, up to a subsequence we can assume that for any k >, T k u ) ) converges weakly to > σ k in W 1,p ) ) and so T k u ) ) > converges strongly to σ k in L p ). By 5.5), we have u u a.e. in. As for k >, T k is continuous, then T k u ) T k u) a.e. in and σ k T k u) a.e. in, which yields 5.7). Using also the boundedness of Tk u ) ) > in L p ) ) ) N, we can find a subsequence still denoted by u ) > ) from u ) > such that T k u ) converges weakly to T k u) in L p ) ) ) N, i.e. 5.6) holds.

2 S. OUARO, A. OUÉDRAOGO EJDE-217/32 5.3. Strong convergence. We start by recalling a suitable time-regularization procedure, which was first introduced by Landes see [17]) and employed by several authors to solve nonlinear time dependent problems with L 1 or measure data see e.g. [7]). We denote this time regularized function to T n u) by T n u)) µ, with µ >. It is defined as the unique solution T n u)) µ L p, T ; W 1,p ) )) L ), with T n u)) µ L p ) ) ) N, of the equation with the initial condition where w µ t T n u)) µ µt n u)) µ T n u)) in D ), 5.13) is a sequence of functions such that Following [17] we can prove that T n u)) µ t T n u)) µ t w µ in, 5.14) w µ W 1,p ) ) L ), w µ L ) n w µ T nu ) a.e. in as µ, 1 µ wµ as µ. W 1,p ) ) L p, T ; W 1,p ) )) L ), T n u)) µ L ) n, T n u)) µ T n u) a.e. in, weak-* in L ) and strongly in L p, T ; W 1,p ) )). To continue our proof of Theorem 5.1, we need the following result. Proposition 5.4. For all k > we have: 5.15) 5.16) i) ax, T k u )) ax, T k u)) in L p ) ) ) N, ii) T k u ) T k u) a.e. in, iii) ax, T k u )) T k u ) ax, T k u)) T k u) strongly in L 1 ) and a.e. in, iv) T k u ) T k u) in L p ) ) ) N. Proof. i) The sequence ax, T k u )) ) > is bounded in L p ) ) ) N according to 2.3). We can extract a subsequence such that ax, T k u )) ζ k in L p ) ) ) N. We have to show that ζ k ax, T k u)) a.e. in. To this end, we take a subsequence u ) > such that u u almost everywhere in. For h > 2k, we introduce the function w T 2k u T h u ) T k u ) T k u) ) ), µ where T k u) ) µ is the approximation of T ku) defined in 5.13). The use of w as a test function to prove the strong convergence of truncations was first introduced in the stationary case in [18], then adapted to parabolic equations in [2]. The advantage in working with w is that since w u T h u ) T k u ) T k u) ) ) χ µ E, with E { u T h u )T k u ) T k u) ) 2k }, in particular we have w µ if u > h 4k. Thus the estimate on T k u ) in L p, T ; W 1,p ) )) appearing

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 21 in Lemma 5.3 implies that w is bounded in L p, T ; W 1,p ) )). Then by the almost everywhere convergence of u to u as, we deduce that w T 2k u T h u) T k u) T k u) ) µ ) 5.17) in L p, T ; W 1,p ) )) and a.e. in. In the following, we set M h 4k, moreover we will denote by w, µ, h) all quantities possibly different) such that lim lim h µ lim sup w, µ, h). 5.18) Similarly we will write only w) or w, µ), to mean that the limits are made only on the specified parameters. Choosing w as a test function in 5.2) we have T Notice that T T 2k T T u ) t w dx dt f w dx dt T ax, u ) w dx dt T f f T 2k u T h u ) T k u ) T k u)) µ ) dx dt ft 2k u T h u ) T k u ) T k u)) µ ) dx dt f w dx dt. 5.19) T f f dx dt ft 2k u T h u ) T k u ) T k u)) µ ) dx dt. Since f is strongly compact in L 1 ), using 5.5), the definition of T k u) ) µ and the Lebesgue dominated convergence theorem, we have lim lim lim h µ T f w dx dt T lim ft 2k u T h u) dx dt. h Thus, recalling the notation introduced in 5.18), we have proven that T f w dx dt w, µ, h). 5.2) Let us estimate the second term in 5.19). Since w if u > M h 4k, we have T ax, u ) w dx dx T ax, T M u )) w dt dt.

22 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Next we split the integral in the sets { u k} and { u > k}, so that we have, recalling that h > 2k, T ax, T M u )) T 2k u T h u ) T k u ) T k u)) µ ) dx dt { u k} { u >k} { u >k} ax, u ) u T k u)) µ ) dx dt ax, T M u )) u T h u )) dx dt ax, T M u )) T k u)) µ dx dt : I 1 I 2 I 3. 5.21) Let us estimate I 2. Since u T h u ) if u h, using 2.3), Remark 2.1 and Young inequality, we obtain I 2 C { u >k} {h u M} {h u M} {h u M} {h u M} C {h u M} {h u M} ax, T M u )) u T h u )) dx dt {h u M} ax, u ) u dx dt C 1 jx) u px) 1 ) u dx dt C 1 jx) u dx dt C 1 p jx) p x) dx dt C 1 u px) dx dt C 1 u px) dx dt C 1 p jx) p x) dx dt. {h u M} {h u M} C 1 u px) dx dt C 1 p u px) dx dt 5.22) The functions jt, x) and ) u > are bounded in 1,p ) Lp, T ; W )) and in L p, T ; W 1,p ) )) respectively, and meas{h u h4k} converges uniformly to zero as h tends to infinity with respect to. Then, passing to the limit in 5.22) as and h respectively, and using Lebesgue dominated convergence theorem, we obtain I 2 w, h). For I 3, let us remark that, since u ) > is bounded in Lp, T ; W 1,p ) )), 2.3) implies that ax, T M u )) ) > is bounded in L p ) ) ) N. The almost everywhere convergence of u to u, as, implies that T k u) χ { u >k} strongly converges to zero in L p, T ; W 1,p ) )). So that, by the Lebesgue dominated

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 23 convergence theorem, we have lim sup Thus, we obtain I 3 { u >k} { u >k} { u >k} w) { u >k} ax, T M u )) T k u) dx dt. ax, T M u )) T k u)) µ dx dt ax, T M u )) T k u) dx dt ax, T M u )) T k u)) µ T k u)) dx dt { u >k} ax, T M u )) T k u)) µ T k u)) dx dt. Using the fact that ax, T M u )) ) is bounded in L p ) ) ) N > and thanks to 5.16), we can apply the Lebesgue dominated convergence theorem to obtain ax, T M u )) T k u)) µ T k u)) dx dt w, µ), { u >k} therefore we conclude that I 3 w, µ). Then from 5.21), according to the fact that I 2 and I 3 converge to zero, we obtain T ax, u ) w dx dt 5.23) ax, u ) u T k u)) µ ) dx dt w, µ, h). { u k} Putting together 5.19), 5.2) and 5.23) we have T u ) t w dx dt ax, u ) u T k u)) µ ) dx dt w, µ, h). { u k} For the first term of 5.24), we can apply [2, Lemma 2.1] to obtain T u ) t w dx dt w, µ, h). Hence 5.24) becomes { u k} 5.24) ax, u ) u T k u)) µ ) dx dt w, µ, h). 5.25) Since T k u)) µ T k u) strongly in L p ) ) ) N as µ, we deduce from 5.25) that T ax, T k u )) T k u ) T k u))) dx dt w, µ, h). 5.26)

24 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Therefore, passing to the limit in 5.26) as tends to zero, µ and h tend to infinity respectively, we deduce that T lim sup ax, T k u )) T k u ) T k u))) dx dt. 5.27) Now, let ϕ D) and λ R. Using 5.27) and 2.4), we obtain T λ lim ax, T k u )) ϕ dx dt T lim sup ax, T k u )) [ T k u ) T k u) λϕ ] dx dt T lim sup ax, T k u) λϕ)) [ T k u ) T k u) λϕ ] dx dt λ T ax, T k u) λϕ)) ϕ dx dt. 5.28) Dividing 5.28) by λ > and by λ < respectively, passing to the limit with λ it follows that T T ax, T k u )) ϕ dx dt ax, T k u) ϕ dx dt. lim This means that for all k >, T ζ k ϕ dx T Hence ζ k ax, T k u)) a.e. in and we have ax, T k u) ϕ dx dt. ax, T k u )) ax, T k u)) in L p ) ) ) N. ii) From 5.26), we have T ax, T k u )) ax, T k u))) T k u ) T k u))) dx dt T ax, T k u)) T k u ) T k u))) dx dt w, µ, h). 5.29) The weak convergence of T k u ) to T k u) in L p ) ) ) N allows to conclude that T lim sup ax, T k u)) T k u ) T k u))) dx dt. Therefore, passing to the limit in 5.29) as tends to zero, µ and h tend to infinity respectively, we deduce that T lim sup ax, T k u )) ax, T k u))) T k u ) T k u))) dx dt. 5.3) Now, set g t, x) [ ax, u ) ax, u) ] [ T k u ) T k u) ]. g t, x) strongly in L 1 ) as. Up to a subsequence, g t, x) a.e. in, which means that there exists ω such that meas ω) and g t, x) in \ω.

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 25 Let t, x) \ω. Using assumptions 2.5) and2.3), it follows that the sequence Tk u t, x)) ) > is bounded in R RN and so we can extract a subsequence which converges to some θ in R R N. Passing to the limit in the expression of g t, x), it follows that [ ax, θ) ax, T k u)) ] [θ T k u) ] and it yields θ T k u) for all t, x) \ω. As the limit does not depend on the subsequence, the whole sequence T k u t, x)) ) > converges to θ in R RN. This means that T k u ) T k u) a.e. in. iii) The continuity of ax, ξ) with respect to ξ R R N gives us Therefore, ax, T k u )) ax, T k u)) a.e. in. ax, T k u )) T k u ) ax, T k u)) T k u) a.e. in. Setting z ax, T k u )) T k u ) and z ax, T k u)) T k u), we have and as z >, z z a.e. in, z L 1 ), z dx dt z dx dt z z dx dt 2 z z ) dx dt z z) dx dt and z z ) z, it follows by using the Lebesgue dominated convergence theorem that z z dx dt, which implies lim ax, T k u )) T k u ) ax, T k u)) T k u) strongly in L 1 ) and a.e. in. To prove iv), we need the following lemmas. Lemma 5.5 [15]). Let u, u n L p ) ), n 1, 2,.... Then the following statements are equivalent to each other: 1) lim n u n u p ) ; 2) lim n ρ p ) u n u) ; 3) u n converges to u in in measure and lim n ρ p ) u n ) ρ p ) u). Next we have a Lebesgue generalized convergence theorem. Lemma 5.6. Let f n ) n N be a sequence of measurable functions and f a measurable function such that f n f a.e. in. Let g n ) n N L 1 ) such that for all n N, f n g n a.e. in and g n g in L 1 ). Then f n dx f dx. Now, set f T k u ) px), f T k u) px), g ax, T k u )) T k u ) and g ax, T k u)) T k u). We have: f is a sequence of measurable functions, f is a measurable function and according to ii), f f a.e. in.

26 S. OUARO, A. OUÉDRAOGO EJDE-217/32 Using iii), we have g ) > L 1 ), g g a.e. in, g g in L 1 ) and using 2.5), we have f Cg. Then, by Lemma 5.6, we have f dx dt f dx dt, which is equivalent to say T k u ) px) dx dt T k u) px) dx dt. We deduce from ii) that the sequence T k u ) ) > converges to T ku) in in measure. Then, by Lemma 5.5 we deduce that T k u ) T k u) px) dx dt, lim which is equivalent to saying that T k u ) T k u) in L p ) ) ) N. 5.4. Existence of entropy solutions. For a given a, k > defines the function T k,a s) T a s T k s)). s k signs) if k s < k a, T k,a s) a signs) if s k a, if s k. Let u be a weak solution of 1.1). Using θ T k,a u ) as a test function in 5.2) and letting goes to zero, we find t1 t1 u ) t T k,a u ) dx dt ax, u ) T k,a u ) dx dt t1 5.31) f T k,a u ) dx dt. We have t1 t1 t1 t1 u ) t T k,a u ) dx dt { u >k} u ) t T a u T k u )) dx dt { u >k} { u >k} u ) t T a u k) dx dt u k) t T a u k) dx dt Θ a u k)t 1 ) dx Using 2.5) and 5.32), from 5.31) we obtain Θ a u k)t 1 ) dx { u >k} 1 C t1 {k u ka} f T k,a u ) dx dt, u px) dx dt { u, >k} { u, >k} Θ a u, k) dx. Θ a u, k) dx 5.32)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 27 which yields {k u ka} u px) dx dt C f dx dt { u >k} Recalling that u u a.e. in, we have { u, >k} ) u, dx. lim meas{t, x) : u > k} uniformly with respect to. k 5.33) Therefore, passing to the limit in 5.33) with and k tending to zero and infinity respectively, we conclude that u px) dx dt. lim k {t,x) :k u ka} Choosing a 1, we obtain the renormalized condition 3.2), i.e., u px) dx dt. lim k {t,x) :k u k1} Now, let ϕ D) with ϕ., T ) and S in W 2, R) which is piecewise C 1 satisfying that supp S [ M, M] for some M >. Taking S u )ϕ as a test function in 5.2), it yields T T u ) t S u )ϕ dx dt ax, u ) S u )ϕ) dx dt T 5.34) f S u )ϕ dx dt. We have u ) t S u )ϕ Su ) ) ϕ and t S u )ϕ) S u ) ϕ S u ) u ϕ. Then, equality 5.34) becomes T Su ) ) T ϕ dx dt S u t )ax, u ) ϕ dx dt T T S u )ax, u ) u ϕ dx dt f S u ) ϕ dx dt. 5.35) We consider the first term on the left-hand side of 5.35). Since S is continuous, 5.5) implies that Su ) converges to Su) a.e. in and weakly in L ). Then Su )) t converges to Su)) t in D ) as, that is T Su ) ) T t ϕ dx dt Su)) t ϕ dx dt. For the other terms on the left-hand side of 5.35), as supp S [ M, M], we have S u )ax, u ) S u )ax, T M u )), S u )ax, u ) u S u )ax, T M u )) T M u ). Using 5.5) and Proposition 5.4, we have S u )ax, T M u )) S u)ax, T M u)) in L p ) ) ) N,

28 S. OUARO, A. OUÉDRAOGO EJDE-217/32 S u )ax, T M u )) T M u ) S u)ax, T M u)) T M u) in L 1 ). For the right-hand side of 5.35), thanks to the strong convergence of f, we have f S u ) fs u) in L 1 ). Therefore, we can pass to the limit in 5.35) as to obtain T T Su) )t ϕ dx dt S u)ax, u) ϕ dx dt T T S u)ax, u) uϕ dx dt fs u) ϕ dx dt. Employing the integration by parts formula for the evolution term, we obtain T ) Su) ϕ dx dt t SuT, x))ϕt, x) dx T Su )ϕ, x) dx Therefore, we deduce from 5.36) that T Su )ϕ, x) dx T T T Su )ϕ, x) dx Su)ϕ) t dx dt Su)ϕ) t dx dt since ϕt, x) ). Su)ϕ) t dx dt [ S u)ax, u) ϕ S u)ax, u) uϕ ] dx dt fs u)ϕ dx dt. 5.36) 5.37) This complete the proof of the existence of a renormalized solution, and then of the entropy solution cf. Theorem 3.6). 5.5. Uniqueness of the entropy solution. Now, we prove the uniqueness of the entropy solution. By Theorem 3.6, it is enough to prove the uniqueness of the renormalized solution. Let u and v be two renormalized solutions for problem 1.1). Let S n be defined as in 4.19). We choose T k Sn u) S n v) ) as a test function in both the equations solved by u and v. Subtracting the equations, we have T T T T Sn u) S n v) ) t T k Sn u) S n v) ) dx dt S n u)ax, u) S nv)ax, v) ) T k Sn u) S n v) ) dx dt S nu)ax, u) u S nv)ax, v) v ) T k Sn u) S n v) ) dx dt f S nu) S nv) ) T k Sn u) S n v) ) dx dt. 5.38)

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 29 We set J 2 J 1 T T J T Sn u) S n v) ) t T k Sn u) S n v) ) dx dt S n u)ax, u) S nv)ax, v) ) T k Sn u) S n v) ) dx dt S nu)ax, u) u S nv)ax, v) v ) T k Sn u) S n v) ) dx dt T J 3 f S nu) S nv) ) T k Sn u) S n v) ) dx dt. We estimate J, J 1, J 2 and J 3 one by one. Recalling the definition of Θ k r), J can be written as J Θ k Sn u) S n v) ) T ) dx Θ k Sn u) S n v) ) ) dx. Because u and v have the same initial condition, and by the properties of Θ k, we obtain J Θ k Sn u) S n v) ) T ) dx. 5.39) We deal with J 1 splitting it as below J 1 { S nu) S nv) k} { u n, v n} { S nu) S nv) k} { u n, v >n} u S n v) dx dt { S nu) S nv) k} { u >n} S n u) S n v) dx dt : J 1 1 J 2 1 J 3 1. ax, u) ax, v) ) u v) dx dt ax, u) S n v)ax, v) ) S n u)ax, u) S nv)ax, v) ) Since { S n u) S n v) k, u > n} { u > n, v > n k}, we have, using the fact that S nt) if t > n 1 and S nt) 1: J1 3 ax, u) u dx dt {n u n1} {n u n1} {n k v n1} {n u n1} {n k v n1} {n k v n1} ax, v) v dx dt. ax, u) v dx dt ax, v) u dx dt 5.4) Using assumption 2.3) and Young s inequality, from the first integral in the righthand side of 5.4), we obtain ax, u) u dx dt {n u n1} {n u n1} C 1 jt, x) u px) 1 ) u dx dt

3 S. OUARO, A. OUÉDRAOGO EJDE-217/32 C {n u n1} {n u n1} {n u n1} {n u n1} C 1 jt, x) u dx dt C 1 p jx) p x) dx dt C 1 u px) dx dt u px) dx dt C {n u n1} {n u n1} {n u n1} C 1 u px) dx dt C 1 p u px) dx dt jx) p x) dx dt. Function jx) is bounded in L p 1,p ), T ; W )) and meas{n u n 1} converges uniformly to zero as n tends to infinity. Using the condition 3.2), we can conclude that lim ax, u) u dx dt. n {n u n1} Similarly, we prove that all the other integrals in the right-hand side of 5.4) converge to zero as n. Thus J1 3 converges to zero. Changing the roles of u and v, the same arguments prove that J1 2 also converges to zero. We use Fatou s lemma to obtain lim inf J ) 1 ax, u) ax, v) u v) dx dt. 5.41) n { u v k} Let us study the limit of J 2 now. We have T J 2 S nu)ax, u) ut k Sn u) S n v) ) dx dt T S nv)ax, v) vt k Sn v) S n u) ) dx dt : J2 1 J2 2. By symmetry between J2 1 and J2 2, it is sufficient to prove that J2 1 tends to zero. Since S ns) 1 and S ns) only if s [n, n 1], using 2.3) we can write J 1 2 k ax, u) u dx dt k k {n u n1} {n u n1} We conclude that C 1 jx) u px) 1 ) u dx dt C 1 jx) u px) 1) u 1 {n u n1} dx dt as n. lim J 2. 5.42) n Let us recall that by definition of S n we have that S n converges to 1 for every s in R. Then fs nu) S nv)) strongly in L 1 ) as n. Using the Lebesgue dominated convergence theorem, we deduce that lim J 3. 5.43) n

EJDE-217/32 NONLINEAR PARABOLIC PROBLEMS 31 Putting together 5.39), 5.41), 5.42) and 5.43), from 5.38), we obtain that as n tends to infinity, ) ax, u) ax, v) u v) dx dt { u v k} and then letting k gœs to infinity recall that u and v are finite a.e. in ), we deduce that ) ax, u) ax, v) u v) dx dt. The strict monotonicity assumption 2.4) then implies that u v a.e. in. Then, let ξ n T 1 Tn1 u) T n1 v) ). We have ξ n L p, T ; W 1,p ) )) and, since u v a.e. in, on { u n 1, v n 1} { u > n 1, v > n 1} ξ n 1 { u Tn1v) 1} u on { u n 1, v > n 1} 1 { v Tn1u) 1} v on { u > n 1, v n 1}. But, if s > n 1, t n 1 and t T n1 s) 1, then n t n 1, which implies ξ n px) dx dt u px) dx dt v px) dx dt {n u n1} {n v n1} as n. Then, ξ n in L p, T ; W 1,p ) )), and thus in D ) as n. Since ξ n T 1 u v) a.e. as n and remains bounded by 1, we also have ξ n T 1 u v) in D ). Hence, T 1 u v), i.e. u v on. Therefore we obtain the uniqueness of the renormalized solution to 1.1), and then the uniqueness of the entropy solution. Acknowledgments. The authors want to express their deepest thanks to the anonymous referees for their comments and suggestions on this article. References [1] H. W. Alt, S. Luckhaus; uasi-linear elliptic-parabolic differential equations, Math. Z., 183 1983), 311-341. [2] S. N. Antontsev, J. F. Rodrigues; On stationary thermo-rheological viscous flows, Annal. del. Univ. de Ferrara. 52 26), 19-36. [3] M. Bendahmane, P. Wittbold, A. Zimmermann; Renormalized solutions for a nonlinear parabolic equation with variable exponents and L 1 data, J. Diff. Equ., 249 21), 1483-1515. [4] Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez; An L 1 theory of existence and uniqueness of nonlinear elliptic equations, Ann Scuola Norm. Sup. Pisa., 22, No. 2 1995), 24-273. [5] Ph. Bénilan, M. G. Crandal; Completely accretive operators, Semigroup theory and evolution equations, Proc. 2nd Int. Conf., Delft / Neth. 1989, Lect. Notes Pure Appl. Math. 135, 1991), 41-75. [6] Ph. Bénilan, M. G. Crandal, A. Pazy; Evolution equations governed by accretive operators, forthcoming book. [7] D. Blanchard, F. Murat; Renormalized solutions of nonlinear parabolic problems with L 1 data: Existence and Uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 6) 1997), 1137-1152.

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