Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

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1 Communications in Matematics ) 55 7 Copyrigt c 217 Te University of Ostrava 55 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran Abstract. In tis work, we study te existence and uniqueness of weak solutions of fourt-order degenerate parabolic equation wit variable exponent using te difference and variation metods. 1 Introduction Te study of differential equations involving variable exponent conditions is a new and interesting topic in recent years. Te interest in studying suc problems is stimulated by teir applications in elastic mecanics, fluid dynamics, nonlinear elasticity, electroreological fluids etc. In particular, parabolic equations involving te px)-laplacian appeared in te field of image restoration in [15], [19] and electroreological fluids wic are caracterized by teir ability to cange te mecanical properties under te influence of te exterior electromagnetic field in [24]. Furter, porous medium type equation wit variable exponents is also studied in [4]. Tese pysical problems are facilitated by te development of Lebesgue and Sobolev spaces wit variable exponent. Recently, parabolic and elliptic equations wic involves variable exponents as studied well in te literature, for example, see [1], [6], [7], [8], [5], [18], [19], [29] and also te references tere in. Tis paper is devoted to study te existence and uniqueness of weak solutions of te following fourt-order parabolic equation wit variable exponents: u t + div u px) 2 u ) = f div g, x, t) Q, u Γ = u Γ =, ux, ) = u x), x, were we denote te cylinder Q, T ], te lateral surface Γ, T ] and p: 1, ) is a continuous function called te variable exponent) and is 21 MSC: 35K55, 35K65. Key words: px)-laplacian, Weak solution, Variable exponents. DOI: /cm )

2 56 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran a bounded open domain of R N wit smoot boundary and T is a given positive number. Based on te pysical consideration, as usual 1) is supplemented wit te natural boundary conditions and te initial value condition. We assume tat u H 1 ), f L p x), T ; L p x)) )), g L p x) Q)) N. 2) Wen N = 1 and px) = pconstant), 1) is a generalized tin film equation, see [2], wic as been extensively studied recently. Xu and Zou [25] obtained te existence and uniqueness of weak solutions for suc a kind of generalized tin film equation. Also wen px) = 2, 1) is known as te Can-Hilliard equation see [12]) wic occurs in science and engineering. Indeed tis equation also forms a base for te metods used to improve te sarpness of vague images in image analysis. Calderon and Kwembe [13] used te Can-Hilliard equation to model te long range effect of insects dispersal. Recently King [2] derived te omogeneous equation of type 1) for te case wen p > 1 wic is relevant to capillary driven flows of tin films of power-law fluids, were ux, t) denotes te eigt from te surface of te oil to te surface of te solid. Zang and Zou [28] establised te existence, uniqueness and long-time beavior of weak solutions for fourt-order degenerate parabolic equation wit variable exponents of nonlinearity. Te exponent p is related to te reological properties of te liquid: p = 2 corresponds to a Newtonian liquid, wereas p 2 emerges wen considering power-law liquids. Wen p > 2, te liquid is said to be sear-tinning. Xu and Zou [26] studied te stability and regularity of weak solutions for a generalized tin film equation for te corresponding omogeneous equation of type 1) wit p as a constant. In tis connection, Buveneswari et al. [1] establised te existence of weak solutions for p-laplacian equation. Bertsc et al. [9] proved te existence of weak solutions for a class of fourt-order degenerate equation. Moreover te existence, uniqueness and qualitative properties of solutions of 1) wic are related to constant case ave been studied in [2], [3], [2], [22], [23], [25] and references terein. Bowen et al. [11] investigated similiarity solutions of te tin film equation. As far as we know, tere are few papers concerned wit te fourt-order nonlinear parabolic equation involving multiple anisotropic exponents. It is not a trivial generalization of similar problems studied in te constant case. Te main difficulties in studying te problem are caused by te complicated nonlinearities of omogeneous equation of type 1) and te lack of a maximum principle for fourt-order equations. Due to te degeneracy, problem 1) does not admit classical solutions in general. Te paper is organized as follows: In section 2, we introduce some basic results regarding te variable exponent spaces and notations. In section 3, we introduce te suitable time-independent equation of original parabolic equation and are proving tat tere exists a weak solution for time-independent equation. Furter, using tis result, we establis te existence of solutions of original equation. Finally, in section 4, we prove tat te solutions obtained are unique. 2 Preliminaries In tis section, we recall some basic definitions, inequalities and te properties of te generalized Lebesgue and Sobolev spaces wit variable exponents. However,

3 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 57 for more detailed teory and te proofs of te following results, one can refer [16], [28]. 2.1 Variable Exponent Spaces Set { C + ) = p C); min x } px) > 1. For any p C + ), we define p + = sup px) and p = inf px). We define te x x Lebesgue space wit variable exponent L p ) ) as te set of all measurable functions u: R suc tat u px) dx < endowed wit te norm { u p ) = inf λ > ; ux) λ px) dx 1 called te Luxemburg norm. Te space L p ) wit te above norm is a separable and reflexive Banac space. Te dual space of L p ) ) is isometric to L p ) ) 1 were px) + 1 p x) = 1 and p x) is te conjugate of px). Te variable exponent p: [1, ) can be extended upto Q T by setting pt, x) := px) for all t, x) Q T ; we may also consider te generalized Lebesgue space L p ) Q T ) as te set of all measurable functions u: Q T R suc tat Q T ux, t) px) dx dt <, endowed wit te norm { u p ) = inf λ > ; Q T ux, t) λ px) } } dx dt 1, wic also as te same properties as tose of L p ) ). For any positive integer k, te variable exponent Sobolev space is given by W k,p ) = { u L p ) ); D α u L p ) ) }, endowed wit te norm u W k,p ) = α k D α u L p ). Te exponent px) is log-hölder continuous function, tat is, px) py) c logx y) for all x, y wit x y < 1 2 wit some constant c. Ten te smoot functions are dense in variable exponent Sobolev spaces and te spaces W 1,p ) ) are te completion of te C ) wit respect to te norm W ). For more 1,p ) details, see [29]. Lemma 1. 1) Te space L p ) is a separable, uniform convex Banac space and its conjugate space is L p ) 1 ) were px) + 1 p x) = 1. For any u Lp ) ) and v L p ) ), we ave uv dx 1 p + 1 ) p ) u px) v p x) 2 u px) v p x).

4 58 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran 2) If p 1, p 2 C + ), p 1 x) p 2 x) for any x. Ten tere exists a continuous embedding L p2x) ) L p1x) ) wose norm does not exceed + 1. Lemma 2. If we denote Pu) = u px) dx, for every u L p ) ), ten min { u p px), } u p+ px) Pu) max{ u p px), u p+ px) }. Lemma 3. 1) W k,p ) ) is a separable and reflexive Banac space. 2) For u W 1,p ) ) wit p C + ) satisfying log-hölder continuity, te inequality, u L p ) ) c u L p ) ) olds, were te positive constant c depends on p and. 3) For u W 1,p ) ) wit p C + ), 1 p p + N, te Sobolev imbedding W 1,p ) ) L r ) ) olds ) for any measurable function r : [1, ) suc tat ess lim Npx) x 1 px) rx). 3 Existence of Weak Solutions In tis section, first we define te weak solutions of te degenerate parabolic problem 1). Furter we introduce suitable time-independent equation of 1) and using te existence of solutions of time-independent equation, we establis te existence of weak solutions of te original parabolic equation 1). Definition 1. [28] A function u is called a weak solution of te fourt-order parabolic equation 1) if te following conditions old true, tat is, i) u C [, T ]; L 2 ) ) L, T ; H 1 ) ) L p, T ; W 2,px) ) ) wit u L p, T ; W 1,px) ) ) and u L px) Q) ) N, ii) For any ϕ C 1 Q) wit ϕ, T ) =, we ave T u x)ϕx, ) dx [ uφt + u px) 2 u ϕ ] dx dτ T = T fϕ dx dτ + g ϕ dx dτ. 3) Remark 1. Let u be a weak solution of 1). If px) satisfies te log-hölder continuity condition, ten u W 1,px) ) H 1 ) W 1,px) ) W 1,1 ) and tus u W 1,px) ) = W 1,px) ). By using te approximation tecnique, we ave, for eac t [, T ] and every ϕ C 1 Q), uϕ dx t t [ uϕt + u px) 2 u ϕ ] dx dτ = t fϕ dx dτ + t g ϕ dx dτ. 4)

5 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 59 Remark 2. Since C ) is dense in W 1,px) ) due to te log-hölder continuity condition, we can coose u as a test function in 3) and 4). Indeed we may use te Steklov averages [v] x, t) = 1 t+ vx, τ) dτ, t [f] x, t) = 1 [g] x, t) = 1 t+ t fx, τ) dτ, t+ gx, τ) dτ. t of te function vx, t) to replace te corresponding function and ten pass to te limits. Terefore we obtain from 4) an energy type estimate 1 2 ut) 2 L 2 ) + t u px) dxdτ = 1 2 u 2 L 2 ) + x) C f p + L p x),t ;L p ) x) )) C g p x) L p x) Q) Lemma 4. Suppose tat px) > 1. Ten, for every v W 2,px) ) W 1,px) ), 5) 6) v W 2,px) C v px). 7) were positive constant C > depends only on p, N,. Proof. Using te definition of te space W 2,px), Lemma 3 and te teorem in [27], we get v W 2,px) ) C v px) + v px) ), 8) were C > is a constant. To prove te desired result of te lemma, we want to sow tat v W 1,px) ) C v px). 9) Suppose we assume tat 9) is not true, ten tere exists a sequence {v n } n=1 in te space W 2,px) ) W 1,px) ) suc tat v n W 1,px) ) > n v n px). 1) Witout loss of generality, we assume tat v n W 1,px) ) = 1. Ten it follows from 8) and 1), v n W 2,px) ) C, v n px) 1 n. Now consider a subsequence still denoted by {v n } n=1) and a function v W 2,px) ) W 1,px) ) suc tat v n v weakly in W 2,px) ), wic implies tat v n v strongly in W 1,px) ). Terefore we get v W 1,px) ) = 1. 11) On te oter and, by te weak convergence of second derivative of v n, we get, v px) lim inf n v n px) =, wic implies tat v =. Since v W 1,px) ), we conclude tat v = a.e. in wic contradicts 11). Tis completes proof of te teorem.

6 6 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran Now let n be a positive integer and = T n. Ten consider te following timediscrete problem of 1) u k u k 1 + u k px) 2 u k ) = [f] k 1)) div [g] k 1)) in, u k = u k =, k = 1, 2,..., n were [f], [g] are respectively as defined in 5). on. 12) Clearly, from te definition, [f] ) L p x)) ) and [g] ) L p x) ) ) N were p is te Sobolev conjugate exponent of p given by Np p N p if p < N, = q if p = N, q N, + ), 13) + if p > N. Before we prove te existence of weak solutions of 12), first we establis te existence of weak solutions of te following elliptic problem, tat is, te case k = 1 in 12), u u + div u px) 2 u ) = [f] ) div [g] ), in, u = u =, k = 1 on. Now we consider te space W = { u H 1 ) W 2,px) ) u W 1,px) ) } wit te norm u W = u H 1 ) + u W 2,px) ) + u 1,px) W verify tat W is a Banac space. ) } 14). It is easy to Definition 2. A function u W is called a weak solution of te problem 14), if, for any ϕ C 1 ) W 1,px) ), we ave u u ) ϕ dx u px) 2 u ϕ dx = [f] )ϕ dx div [g] )ϕ dx. 15) Teorem 1. Under te assumptions of u H 1 ), f L p, T ; L p x)) ) and g L p x) Q) ) N, tere exists at least one weak solution for 14). Proof. Consider te variational problem, min {Ju) u W } were te functional J is defined by Ju) = 1 u 2 1 dx + 2 px) u px) dx [f] ) u dx [g] ) u dx, 16)

7 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 61 were f L p, T ; L p x)) ) ), g L p x) Q) ) N are known functions. Now we establis tat Ju) as a minimizer u 1 x) in W. Terefore te function u 1 is a weak solution of te corresponding Euler-Lagrange equation of Ju) wic is 14) in te case k = 1. Using Lemma 1 wit Hölder s and Young s inequalities, we ave [f] ) u dx ) 1 ) [f] ) p x)) p x)) 1 dx u p x) p x) dx [f] ) L p x)) ) u L p x) ) [f] ) L p x)) ) C u L px) ) ɛ u px) L px) ) + Cɛ) [f] ) p x) L p x)) ). 17) were C is a small positive constant. Similarly we get [g] ) u dx ɛ u px) L px) ) + Cɛ) [g] x), 18) ) p L p x) ) were C is a small positive constant. For sufficiently small ɛ, we get Ju) 1 u 2 1 dx + 2 px) u px) dx + ɛ u px) L px) ) + Cɛ) [f] ) p x) + L p x)) ) ɛ u px) L px) ) + Cɛ) [g] x) ) p 1 u 2 1 dx + 2 px) u px) dx L p x) ) + 2ɛ u px) L px) ) + Cɛ) [f] x) ) p + Cɛ) [g] x) L p x)) ) ) p 1 { } 1 u 2 dx + min 2 p +, 2ɛ u min{px),p} dx L p x) ) + Cɛ) [f] ) p x) L p x)) ) + Cɛ) [g] ) p x) L p x) ). 19) Recalling u W 2,px) ) W 1,px) ) and Lemma 3, we ave Since u W 1,px) ), we get Terefore u W 2,px) ) C u px). 2) u px) C u px). 21) u W u H 1 ) + C u px) + u 1,px) W ) u H 1 ) + C u px) + u 1,px) W ) u H 1 ) + C u px) + C u px) ] C [ u H 1 ) + u px). 22)

8 62 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran Hence 22) assures tat Ju) +, if u W +. On te oter and, Ju) is clearly weakly lower semi continuous on W. So it follows from te critical point teory, see [14], tere exists u 1 W suc tat Ju 1 ) = inf w W Jw). Terefore, from te above equation, we conclude tat te function u 1 is a weak solution of te corresponding Euler-Lagrange equation of Ju). ) Teorem 2. Under assumptions of u H 1 ), f L p, T ; L p x)), g L p x) Q) ) N and px) C+ ), px) satisfies te log-hölder continuity condition, te initial-boundary value problem 1) admits a unique weak solution. Proof. Construct a suitable approximation solution sequence {u } for te parabolic problem 1). For k = 1, from te above teorem, tere exists a weak solution u 1 W. Continuing te same procedures, one find weak solutions u k W of 12), k = 2,..., n. It follows tat, for every η W, 1 u k u k 1 ) η dx u k px) 2 u k ) η dx = [f] k 1)) η dx + [g] k 1)) η dx. 23) Take η = u k as a test function in 23) and using Young s inequality, we get 1 2 u k 2 L 2 ) + u k px) dx 1 2 u k 1 2 L 2 ) From te above, we get + C [f] k 1)) p x) L p x)) ) + C [g] k 1)) p x) L p x) ). 24) k 1 u k 2 L 2 ) u 2 L 2 ) + C [f] i) p x) L p x)) ) were k 1 i= i= k 1 [f] i) p L p x)) ) k 1 i= = i+ i= i k k 1 [g] i) p x) L p x) ) = i= i k k 1 + C [g] i) p x), 25) L p x) ) i= fτ) p x) L p x)) ) dτ fτ) p x) L p x)) ) dτ, i+ gτ) p x) L p x) ) dτ gτ) p x) L p x) ) dτ.

9 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 63 For every = T n, we define u x), t =, u 1 x), < t,,, u x, t) = u j x), j 1) < t j,,, u n x), n 1) < t n = T. 26) For eac t, T ], tere exists some k {1, 2,..., n} suc tat t k 1), k]. Now by 25), we ave u t) 2 L 2 ) u 2 L 2 ) + C k + C k gτ) p L p x) ) dτ u 2 L 2 ) + C T Hence te above inequality sows tat T + C gτ) p x) dτ. L p x) ) fτ) p x) L p x)) ) dτ fτ) p x) L p x)) ) dτ u 2 L,T ;L 2 )) C, 27) were C > is a constant. Summing up te inequalities in 24), we obtain 1 n 2 u k 2 L 2 ) u k px) dx n u k 1 2 L 2 ) + C [f] k 1)) p x) L p x)) ) + C [g] k 1)) p x) L p x) ) 1 2 u 2 L 2 ) u n 1 2 L 2 ) + C + C T T fτ) p x) L p x)) ) dτ gτ) p x) dτ. 28) L p x) )

10 64 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran From te above inequality, we get T u px) dx dt = u k px) dx u 2 L 2 ) + C T T + C gτ) p dτ L p x) ) C, were C > is a constant. By Lemma 2 we ave T fτ) p L p x)) ) dτ { } T min u p+ px), u p px) dt u px) dx dt C. Tus we conclude tat u L px) Q)) N and u L p,t ;L px) )) C. 29) Employing te same tecnique as in te proof of 22), we obtain u L,T ;H 1)) + u L p,t ;W 2,px) )) + u Lp C. 3),T ;W 1,px) )) Terefore, from 29) and 3), tere exists a subsequence u wic also denoted by u ) suc tat u u, weakly in L, T ; H 1 )), u u, weakly in L p, T ; W 2,px) )), u u, weakly in L px) Q)) N, u px) 2 u ζ, weakly in L p ), T ; L p x) )), wic follows see [17]) tat u L,T ;H 1 )) + u L p,t ;W 2,px) )) + u L p,t ;W 1,px) )) C. Next we prove tat te function u is a weak solution of problem 1). For eac ϕ C 1 Q) wit ϕ, T ) = and ϕx, t) Γ = and for every k {1, 2,..., n}, we solve te equation η k x) = ϕx, k) to find a function η k W and let it be a test function in 23) to ave 1 u k u k 1 )ϕx, k) dx u k px) 2 u k ϕx, k) dx [ = [f] x, k 1))ϕx, k) dx + [g] x, k 1)) ϕx, k) ] dx.

11 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 65 Summing up all te equalities and using te definition of u x, t), we ave n 1 ϕx, k) ϕx, k + 1)) u x, k) dx u x)ϕx, ) dx u px) 2 u )x, k) ϕx, k) dx [ = [f] x, k 1))ϕx, k) + [g] x, k 1)) ϕx, k) ] dx. 31) From te above convergence results and ϕ C 1 Q), we ave u px) 2 ) u x, k) ϕx, k) dx Since and + k = k 1) T u px) 2 u ) x, τ) ϕx, τ) dx dτ u px) 2 u ) x, τ) ϕx, k) ϕx, τ)) dx dτ [f] x, k 1))ϕx, k) dx = [g] x, k 1)) ϕx, k) dx = T k as, we obtain from 31), we get, as, k 1) k ζ ϕx, τ) dxdτ, as. k 1) fx, τ)ϕx, k) dx dτ T fϕ dx dτ gx, τ) ϕx, k) dx dτ T g ϕ dx dτ, T u ϕ T t d xdτ u x)ϕx, ) dx ζ ϕx, τ) dx dτ T = fϕ + g ϕ) dx dτ 32)

12 66 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran Te above equation proves tat u t ) Lp, T ; W 1,p x) ) ). For some larger integer s suc tat W 1,p x) ) H s ) we obtain and it follows [3] tat u t Lp ), T ; H s ) ), u C [, T ]; H s ) ). For eac ɛ > and all t, t [, T ], by 3), tere exists a positive number δ > suc tat δ ut) ut ) L2 ) ɛ 2. From te compact imbedding relation H 1 ) L 2 ) H s ), we ave, for all t, t [, T ], ut) ut ) L 2 ) δ ut) ut ) H 1 ) + Cδ) ut) ut ) H s ) δ ut) ut ) L2 ) + Cδ) ut) ut ) H s ) ɛ 2 + Cδ) ut) ut ) H s ), were te first inequality is guaranteed by Lemma 5.1 in Capter 1 of [21]. Hence we proved tat u C [, T ]; L 2 ) ). Finally we sow tat ζ = u px) 2 u, a.e in Q to prove te existence of weak solutions. Considering u as a test function in 32), we ave u 2 L 2 ) ut ) 2 L 2 ) 2 T ζ u dx dτ T = [f u + g u] dx dτ. 33) Denote Au = u px) 2 u and, by te monotonicity assumption of te operator, ζ px) 2 ζ η px) 2 η ) ζ η), for all ζ, η R N, we ave Auk Avτ) ) u k vτ) ) dx, 34) for eac k = 1, 2,..., n and every v L p, T ; W 2,px) ) ) wit v L p 1,px), T ; W ) ). Considering u k as a test function in 23), we ave 1 u k u k 1 ) u k dx Au k u k dx = [f] k 1)) u k dx + [g] k 1)) u k dx. 35)

13 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents 67 From 34), we obtain 1 u k u k 1 ). u k dx Au k vτ) dx Av u k vτ) dx [f] k 1)) u k dx [g] k 1)) u k dx. 36) Now, by Young s inequality, we obtain u k u k 1 ) u k dx u k 1 2 L 2 ) u k 2 L 2 ). 37) 2 Integrating 35) over k 1), k) and using te above result, we get u k 1 2 L 2 ) u k 2 L 2 ) 2 k k 1) k k 1) Au k v dx dτ Av u k v) dx dτ. Summing up te above inequalities for k = 1, 2,..., n, we obtain u 2 L 2 ) u T ) 2 T L 2 ) Au. v dx dτ 2 T T Av u v)dx dτ f u g u ) dx dτ. Passing to limits as, we get T u 2 L 2 ) ut ) 2 L 2 ) 2 T Av u v) dx dτ ζ v dx dτ T f u g u) dx dτ. 38) Combining 38) wit 33), we ave T ζ Av) u v) dx dτ 39) We coose v = u λw for any λ >, w L px) Q)) N in te above inequality to ave T ζ Au λw)) w dx dτ. Passing to limits as λ + and, using Lebesgue s dominated convergence teorem, we obtain T ζ Au) ψ dx dτ, ψ L px) Q)) N. Hence we conclude tat ζ = Au, a.e. in Q.

14 68 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran 4 Uniqueness of Weak Solutions Teorem 3. Te solutions of te given degenerate parabolic fourt-order equation 1) are unique. Proof. Suppose tere exist two weak solutions u and v of problem 1). Remark 1, we ave Using u v)ϕ dx t t [ u v)ϕt + u px) 2 u v px) 2 v ) ϕ ] dx ds =. Coosing u v) as a test function in te above equality and using Remark 2, we ave, for every t, T ), + t u v 2 t) dx 2 [ u px) 2 u v px) 2 v ] u v) dx ds =. 4) Since te two terms on te left-and side are nonnegative, we ave u = v a.e. in Q, Since u v = on Γ, we conclude u v = a.e. in Q, wic implies u = v a.e. in Q. Tus we obtain te uniqueness of weak solutions. Acknowledgement Te autors wis to tanks te referees for useful comments and suggestion wic led to improvement in te quality of te paper. Furter, te work of te first autor is supported by te DST-SERB Early Career Award File No. ECR/2l6/624. References [1] B. Andreianov, M. Bendamane, S. Ouaro: Structural stability for variable exponent elliptic problems, I: Te px)-laplacian kind problems. Nonlinear Anal ) [2] L. Ansini, L. Giacomelli: Sear-tinning liquid films: macroscopic and asymptotic beavior by quasi-self-similar solutions. Nonlinearity 15 22) [3] L. Ansini, L. Giacomelli: Doubly nonlinear tin-film equations in one space dimension. Arc. Ration. Mec. Anal ) [4] S.N. Antontsev, S.I. Smarev: A model porous medium equation wit variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 6 25) [5] S. Antontsev, S. Smarev: Elliptic equations wit anisotropic nonlinearity and nonstandard growt conditions. Handbook of Differential Equations: Stationary Partial Differential Equations 3 26) 1 1. [6] S. Antontsev, S. Smarev: Parabolic equations wit anisotropic nonstandard growt conditions. Internat. Ser. Numer. Mat ) [7] S. Antontsev, S. Smarev: Blow-up of solutions to parabolic equations wit nonstandard growt conditions. J. Comput. Appl. Mat )

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16 7 Lingeswaran Sangerganes, Arumugam Gurusamy, Krisnan Balacandran [29] C. Zang, S. Zou: Renormalized and entropy solutions for nonlinear parabolic equations wit variable exponents and L 1 data. J. Differential Equations ) [3] S. Zou: A priori L -estimate and existence of weak solutions for some nonlinear parabolic equations. Nonlinear Anal. 42 2) Autor s address: Lingeswaran Sangerganes, Krisnan Balacandran, Department of Humanities and Sciences, National Institute of Tecnology, Goa 43 41, India. sangerganes a nitgoa.ac.in, sangerganes a gmail.com Arumugam Gurusamy, Department of Matematics, Baratiar University, Coimbatore , India. guru.poy a gmail.com, kb.mat.bu a gmail.com Received: 3 January, 217 Accepted for publication: 6 February, 217 Communicated by: Olga Rossi