Weak Solutions to Nonlinear Parabolic Problems with Variable Exponent

Transcription

1 International Journal of Mathematical Analysis Vol. 1, 216, no. 12, HIKARI Ltd, Weak Solutions to Nonlinear Parabolic Problems with Variable Exponent Ahmed Jamea Département de Mathématiques Centre Régional des Métiers de l Education et de Formation El Jadida, Morocco Copyright c 216 Ahmed Jamea. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we prove existence and uniqueness of weak solutions to nonlinear parabolic problems with variable exponent u t p(x)u + α(u) = f in ], T [, with Neumann-type boundary conditions. Our approach is based essentially on time discretization technique by Euler forward scheme. Mathematics Subject Classification: 35B35, 35D3, 35K55, 35J62 Keywords: Existence, nonlinear parabolic problem, Neumann-type boundary, weak solution, semi-discretization, variable exponent, uniqueness 1 Introduction In this article we consider the nonlinear Neumann problems with variable exponent u t p(x)u + α(u) = f in Q T := ], T [, p 2 u Du η = on Σ T :=], T [, (1) u(., ) = u in.

2 554 Ahmed Jamea where p(.) is a continuous function defined on with p(x) > 1 for all x, is a connected open bounded set in R d, d 3, with a connected Lipschitz boundary, and η is the unit outward normal on. T is a fixed positive real number and α is a non decreasing continuous function on R. The operator p(x) u = div ( u p(x) 2 u ) is called p(x)-laplacian, which becomes p-laplacian when p(x) p (a constant) and Laplacian when p(x) 2. In the particular case where p(.) = constant, the existence and uniqueness of the solutions to problem (1) have been intensively studied by many authors, we refer for example the reader to the bibliography [4, 6, 7, 9] and references therein. In this paper we study existence, uniqueness and stability questions by Euler forward scheme, we apply here a time discretization of given continuous problem (1), we recall that the Euler forward scheme has been used by several authors while studying time discretization of nonlinear parabolic problems. In recent years, there are a lot of interest in the study of various mathematical problems with variable exponent (see for example [13, 15, 19, 2] and references therein), the problems with variable exponent are interesting in applications and raise many difficult mathematical problems, some of the models leading to these problems of this type are the models of motion of electrorheological fluids, the mathematical models of stationary thermo-rheological viscous flows of non-newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porousmedium, we refer the reader for example to [1, 15, 17] and references therein for more details for more details. 2 Preliminaries and Notations In this section, we give some notations and definitions and some results that we use in this work. Let be a measurable connected open bounded set in R d, d 3, we write C + () = {continuous function p(.) : R + such that 1 < p < p + < }, where p = min x p(x) and p + = max p(x). x Let p(.) C + (), we define the Lebesgue space with variable exponent L p(.) () by L p(.) () = {u : R measurable : u(x) p(x) dx < },

3 Nonlinear parabolic problems with variable exponent 555 endowed with the Luxemburg norm { u p(.) = u L p(.) () = inf λ >, u(x) λ p(x) dx 1 }. The space ( ) L p(.) (),. p(.) is a reflexive Banach space, uniformly convex and its dual space is isomorphic to L p (.) () where = 1 (see [12]). p(.) p (.) Proposition 2.1 (Hölder type inequality [12]) Let p(.) and p (.) are two elements of C + 1 () such that + 1 = 1, for any u L p(.) () and p(.) p (.) v L p (.) () we have ( u.vdx ) u p(.) u p p (.). On the space L p(.) (), we also consider the function ρ p(.) : L p(.) () R defined by ρ p(.) (u) = ρ L ()(u) = u(x) p(x) dx. p(.) The connection between ρ p(.) and. p(.) is established by the next result. p Proposition 2.2 (Fan and Zhao [12]) a) Let u an element of L p(.) (), we have i) u p(.) < 1 (respectively 1) ρ p(.) (u) < 1 (respectively 1). ii) u p(.) = α ρ p(.) (u) = α (when α ). iii) If u p(.) < 1 then u p + p(.) ρ p(.)(u) u p p(.). iv) If u p(.) > 1 then u p p(.) ρ p(.)(u) u p + p(.). b) For a sequence (u n ) n N L p(.) () and an element u L p(.) (), the following statements are equivalent i) lim n u n = u in L p(.) (), ii) lim n ρ p(.) (u n u) =, iii) u n u in measure in. The variable exponent Sobolev space W 1,p(.) () consists of all u L p(.) () such that the absolute value of the gradient is in L p(.) (). Let the norm u 1,p(.) = u p(.) + u p(.). The space ( W 1,p(.) (),. 1,p(.) ) is a separable and reflexive Banach space.

4 556 Ahmed Jamea Lemma 2.3 ([6]) For ξ, η R N and 1 < p <, we have 1 p ξ p 1 p η p ξ p 2 ξ(ξ η). (2) Lemma 2.4 (Y. Fu [14]) Let p(.) an element of L () and let u W 1,p(.) (), there exists a constant C depends only on such that u p(x) C u p(x). (3) Lemma 2.5 ([11]) Let p, p two reals numbers such that p > 1, p > 1 and = 1, we have 1 p + 1 p ξ p 2 ξ η p 2 η p C { (ξ η)( ξ p 2 ξ η p 2 η) } β 2 { ξ p + η p } 1 β 2, ξ, η R d, where β = 2 if 1 < p 2 and β = p if p 2. Remark 2.6 Hereinafter, c i, (i N) are positive constants independent of N. 3 The semi-discrete problem In this section, we discretize the problem (1) by Euler forward scheme and study the questions of existence and uniqueness to this discretized problems. Firstly, we suppose the following hypotheses (H 1 ) α is a non decreasing continuous real function on R, surjective such that α() = and α(x) C x, where C is an positive constant. (H 2 ) f L (Q T ) and u L () W 1,p(.) (). (H 3 ) p(.) L (). By Euler forward scheme, we discretize the problem (1), we obtain the following problems U n p(x) U n + α(u n ) = f n + U n 1 in, (P n) DU n p 2 U n = on, η U = u in. where N = T, < < 1, 1 n N and f n (.) = 1 n (n 1) f(s,.)ds, in,

5 Nonlinear parabolic problems with variable exponent 557 Definition 3.1 An weak solution to discretized problems (Pn) is a sequence (U n ) n N such that U = u and U n is defined by induction as an weak solution to problem u p(x) u + α(u) = f n + U n 1 in, p(x) 2 u u = on. η i.e. U n L () W 1,p(.) () and ϕ W 1,p(.) (), >, we have U n ϕdx + U n p(x) 2 U n ϕdx + α(u n )ϕdx = (f n + U n 1 )ϕdx. (4) Theorem 3.2 Let hypotheses (H 1 ), (H 2 ) and (H 2 ) be satisfied, the problem (Pn) has a unique weak solution (U n ) n N and for all n = 1,..., N, U n L () W 1,p(.) (). Proof. For n = 1, we pose U = U 1, we rewrite problem (3) as p(x) U + α(u) = f 1 + u in, p(x) 2 U DU η = on, (5) By hypothesis (H 2 ), the function g = f 1 + u is an element of L () and the function b(s) = s + α(s) is a non decreasing continuous real function on R, surjective such that b() =, therefore, according to [3], the problem (5) has a unique weak solution U 1 in L () W 1,p(.) (). By induction, we deduce in the same manner that the problem (Pn) has a unique weak solution (U n ) n N such that n = 1,..., N, U n L () W 1,p(.) (). 4 Some stability results In this section, we give some a priori estimates for the discrete weak solution (U n ) 1 n N which we use later to derive convergence results for the Euler forward scheme. Theorem 4.1 Let hypotheses (H 1 ) (H 2 ) (H 3 ) be satisfied. Then, there exists a positive constant C(u, f, g) depending on the data but not on N such that for all n = 1,..., N, we have (1) U n 2 2 C(u, f, g), (2) U i U i C(u, f, g), (3) U i p 1,p(.) C(u, f, g).

6 558 Ahmed Jamea Proof. For (1) Let k > and 1 n N, we take ϕ = U n k U n as test function in equality (4) we obtain U n k+2 dx + U n p(x) 2 ( ) U n k U n dx + α(u n ) U n k U n dx This implies that = (f n + U n 1 ) U n k U n dx. (6) U n k+2 k+2 c 1 U n k+1 k+1 + U n 1 k+2 U n k+1 k+2. (7) If U n k+2 =, we get immediately the result (1). If U n k+2, inequality (7) becomes U n k+2 c 1 + U n 1 k+2 (8) T c 2 + U. (9) Taking the limit as k, we deduce the result (1). For (2) Let 1 i N, we take ϕ = U i as test function in equality (4) we obtain (U i U i 1 )U i dx + U i p(x) dx + α(u i )U i dx = (f i + U i 1 )U i dx.(1) It follows that 1 2 U i U i U i U i U i p(x) dx c 4 U i 2. (11) Now, summing (11) from i = 1 to n and using the stability result (1), we obtain 1 2 U n U U i U i U i p(x) dx c 5. (12) This implies the stability result (2). For (3) We pose s = { i 1; 2;...; N : U i 1,p(x) 1 }, we have U i p 1,p(.) i s U i p 1,p(.) + i s U i p 1,p(.). (13) By applying lemma (2.4), we get U i p 1,p(.) T + c 5 ρ p(.) ( U i ). (14) i s And, by inequality (12), we deduce the stability result (3).

7 Nonlinear parabolic problems with variable exponent 559 Theorem 4.2 Let hypotheses (H 1 ) (H 2 ) (H 3 ) be satisfied. Then, there exists a positive constant C(u, f, g) depending on the data but not on N such that for all n = 1,..., N, we have (1) α(u i ) 1 C(u, f, g), (2) lim k (3) U i p(x) C(u, f, g), k {U i k} U i U i 1 1 C(u, f, g). Proof. For (1) and (2). Let k >, we define the following function { s if s k, T k (s) := k sign (s) if s > k, where 1 if s >, sign (s) := if s =, 1 if s <. In equality (4), we take ϕ = T k (U i ) as a test function and dividing this equality by k, taking limits when k goes to, we get U i 1 + α(u i ) 1 + lim U i p(x) f i 1 + U i 1 1. (15) k k {U i k} Summing (15) from i = 1 to n, we obtain the stability results (1) and (2). For (3). Taking ϕ = T (U i U i 1 ) in equality (4) and dividing this equality by, we obtain by applying Lemma (2.3) that (U i U i 1 ) T (U i U i 1 ) dx + i ( 1 p(x) U i p(x) 1 p(x) U i 1 p(x) ) dx α(u i ) 1 + f i 1, (16) where i = { U i U i 1 }. Summing inequality (16) from i = 1 to n, we get by applying the stability result (1) that (U i U i 1 ) T (U i U i 1 ) dx 1 U p(x) dx + c 6. (17) p Then, we let approach in the above inequality, we deduce the stability result (3).

8 56 Ahmed Jamea 5 Convergence and existence result In this section and from the above results, we build a weak solution of problem (1) and then show that this solution is unique. Firstly, we start with giving the weak formulation of nonlinear parabolic problem (1). Definition 5.1 A measurable function u : Q T R is an weak solution to nonlinear parabolic problems (1) in Q T if u(., ) = u in, u C(, T ; L 2 ()) L p (, T ; W 1,p(.) ()), u t L2 (Q T ), u ( L p(.) (Q T ) ) d and for all ϕ C 1 (Q T ), we have T u T T t ϕdxdt + u p(x) 2 u ϕdxdt + α(u)ϕdxdt = (18) T fϕdxdt. Now, we state our main result of this work. Theorem 5.2 Let hypotheses (H 1 ) (H 2 ) (H 3 ) be satisfied. Then the nonlinear parabolic problem (1) has a unique weak solution. Proof. Existence. Let us introduce a piecewise linear extension (called Rothe function) { un () := u, u N (t) := U n 1 + (U n U n 1 ) (t tn 1 ), t ]t n 1, t n ], n = 1,..., N in, (19) and a piecewise constant function { un () := u, u N (t) := U n t ]t n 1, t n ], n = 1,..., N in, (2) where t n := n. By theorem 3.2, for any N N, the solution (U n ) 1 n N unique. Thus, u N and u N are uniquely defined. of problems (3) is Lemma 5.3 Let hypotheses (H 1 ) (H 2 ) (H 3 ) be satisfied. Then, there exists a positive constant C(T, u, f, g) not depending on N such that for all

9 Nonlinear parabolic problems with variable exponent 561 N N, we have (1) u N u N 2 L 2 (Q T ) 1 N C(T, u, f, g), (2) u N L (,T,L 2 ()) C(T, u, f, g), (3) u N L (,T,L 2 ()) C(T, u, f, g), (4) u N L p (,T,W 1,p(.) ()) C(T, u, f, g), (5) α(u N ) L 1 (Q T ) C(T, u, f, g), 2 (6) u N t C(T, u, f, g). L 2 (Q T ) Proof of lemma (5.3). For (1). We have T u N u N 2 L 2 (Q T ) = u N u N 2 dxdt i=n t n t n 1 1 N C(T, u, f, g). ( ) t U n U n 1 2 n 2 t dxdt And in the same manner, we show the results (2), (3), (4) and (5). For (6). In the weak formulation (4) we take ϕ = U n U n 1 and summing this equality from i = 1 to N, we get by applying lemma (2.3) and hypotheses (H 1 ) and (H 2 ) that i=n This implies that u N t (U i U i 1 ) 2 i=n dx + 2 L 2 (Q T ) ( 1 p(x) U i p(x) 1 ) p(x) U i 1 p(x) dx i=n c 7 U i U i 1 1. (21) 1 p i=n U p(x) + c 8 U i U i 1 1. (22) Then, we apply lemma (4.2) and hypothesis (H 2 ) we obtain the result (6). This finish the proof of lemma (5.3). Now, using the tow results (2) and (3) of lemma (5.3), the sequences (u N ) N N and (u N ) N N are uniformly bounded in L (, T, L 2 ()), therefore there exists two elements u and v in L (, T, L 2 ()) such that u N u in L (, T, L 2 ()), (23)

10 562 Ahmed Jamea u N v in L (, T, L 2 ()). (24) And from the result (1) of lemma (5.3), it follows that u v. Furthermore, by lemma (5.3) and hypothesis (H 2 ), we have that u N t u t in L2 (Q T ), (25) α(u N ) α(u) in L 1 (Q T ), (26) u N u in L p (, T, W 1,p(.) ()), (27) ( d u N p(x) 2 u N u p(x) 2 u weakly in L p (.) (Q T )). (28) On the other hand, we have by lemma (5.3) and Aubin-Simons compactness result that u N u in C(, T, L 2 ()). (29) Now, we prove that the limit function u is a weak solution of problem (1). Firstly, we have u N () = U = u for all N N, then u(,.) = u. Secondly, Let ϕ C 1 (Q T ), we rewrite (5.3) in the forms T where u N t ϕdxdt + T = u N p(x) 2 u N ϕdxdt + T T α(u N )ϕdx f N ϕdxdt, (3) f N (t, x) = f n (x), t ]t n 1, t n ], n = 1,..., N. taking limits as N in (3) and using the above results, we deduce that u is a weak solution of nonlinear parabolic problem (1). Uniqueness. Let u and v tow weak solutions of nonlinear parabolic problem (1), in equality (18) we take ϕ = u v, as test functions, we get T (u v) (u v)dxdt+ t T ( u p(x) 2 u v p(x) 2 v ) (u v)dxdt T + (α(u) α(v)) (u v)dx =. (31) So u(x, ) = v(x, ) for all x, then, according to hypothesis (H 1 ), we obtain that u v

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