This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Rothe-Galerkin's method for nonlinear integrodifferential equations Boundary Value Problems 2012, 2012:10 doi:10.1186/1687-2770-2012-10 Abderrazek Chaoui (razwel2004@yahoo.fr) Assia Guezane-Lakoud (a_guezane@yahoo.fr) ISSN 1687-2770 Article type Research Submission date 29 September 2011 Acceptance date 8 February 2012 Publication date 8 February 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/10 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Boundary Value Problems © 2012 Chaoui and Guezane-Lakoud ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rothe–Galerkin’s method for a nonlinear integro differential equation Abderrazek Chaoui ∗ and Assia Guezane-Lakoud Laboratory of advanced materials, Badji Mokhtar University, Annaba, Algeria ∗ Corresponding author: Email address: razwel2004@yahoo.fr, a guezane@yahoo.fr Abstract In this article we propose approximation schemes for solving nonlinear initial boundary value problem with Volterra operator. Existence, uniqueness of solution as well as some regularity results are obtained via Rothe–Galerkin method. Keywords: Rothe’s method; a priori estimate; integrodifferential equation; Galerkin method; weak solution. Mathematics Subject Classification 2000: 35k55; 35A35; 65M20. 1 1 Introduction The aim of this work is the solvability of the following equation ∂ t β (u) − ∂ t a (u) − ∇d (t, x, u, ∇a (u)) + K (u) = f (t, x, u) (1.1) where (t, x) ∈ (0, T ) × Ω = Q T , with the initial condition β (u (0, x)) = β (u 0 (x)) , x ∈ Ω (1.2) and the boundary condition u (t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω. (1.3) The memory operator K is defined by K (t) u, v = Ω t 0 k (t, s) g (s, x, ∇u (s, x)) ∇v (t, x) dsdx. (1.4) Let us denote by (P), the problem generated by Equations (1.1)–(1.3). The problem (P) has relevant interest applications to the porous media equation and to integro-differential equation modeling memory effects. Several problems of thermoelasticity and viscoelasticity can also be reduced to this type of problems. A variety of problems arising in mechanics, elasticity theory, molecular dynamics, and quantum mechanics can be described by doubly nonlinear problems. The literature on the subject of local in time doubly nonlinear evolution equations is rather wide. Among these contributions, we refer the reader to [1] where the authors studied the convergence of a finite volume scheme for the numerical solution for an elliptic–parabolic 2 equation. Using Rothe method, the author in [2] studied a nonlinear degenerate parabolic equation with a second-order differential Volterra operator. In [3] the solutions of nonlinear and degenerate problems were investigated. In general, existence of solutions for a class of nonlinear evolution equations of second order is proved by studying a full discretization. The article is organized as follows. In Section 2, we specify some hypotheses, precise sense of the weak solution, then we state the main results and some Lemmas that needed in the sequel. In Section 3, by the Rothe–Galerkin method, we construct approximate solutions to problem (P). Some a priori estimates for the approximations are derived. In Section 4, we prove the main results. 2 Hypothesis and mean results To solve problem (P), we assume the following hypotheses: (H 1 ) The function β : R −→ R is continuous, nondecreasing, β (0) = 0, β (u 0 ) ∈ L 2 (Ω) and satisfies |β (s)| 2 ≤ C 1 B ∗ (a (s)) + C 2 , ∀s ∈ R. (H 2 ) a : R −→ R is continuous, strictly increasing function, a (0) = 0 and a (u 0 ) ∈ H 1 0 (Ω). (H 3 ) d : (0, T ) × Ω × R × R N −→ R N is continuous, elliptic i.e., ∃d 0 > 0 such that d (t, x, z, ξ) ξ ≥ d 0 |ξ| p for ξ ∈ R N and p ≥ 2, strongly monotone i.e., (d (t, x, η, ξ 1 ) − d (t, x, η, ξ 2 )) (ξ 1 − ξ 2 ) ≥ d 1 |ξ 1 − ξ 2 | p for ξ 1 , ξ 2 ∈ R N , d 1 > 0 and satisfies |d (t, x, z, ξ)| ≤ C 1 + |ξ| p−1 + (B ∗ (a (z))) p−1 p for any (t, x) ∈ (0, T ) × Ω, ∀z ∈ R, ξ ∈ R N . 3 (H 4 ) f : (0 , T ) × Ω × R −→ R is continuous such that |f (t, x, z)| ≤ C 1 + (B ∗ (a (z))) p−1 p for any (t, x) ∈ (0, T ) × Ω, ∀z ∈ R. The functions g and k given in (1.4) satisfy the following hypotheses (H 5 ) and (H 6 ), respectively: (H 5 ) g : (0, T ) × Ω × R N −→ R N is continuous and satisfies |g (t, x, ξ)| ≤ C 1 + |ξ| p−1 and |g (t, x, ξ 1 ) − g (t, x, ξ 2 )| ≤ d 1 |ξ 1 − ξ 2 | p−1 . (H 6 ) k : (0, T) × (0, T) −→ R is weak singular, i.e.|k (t, s)| ≤ |t − s| −γ ω (t, s) for 0 ≤ γ ≤ 1 p and the function ω : [0, T] × [0, T ] −→ R is continuous. (H 7 ) For p = 2, we have |d (t, x, η 1 , ξ 1 ) − d (t, x, η 2 , ξ 2 )| ≤ C (|a (η 1 ) − a (η 2 )| + |ξ 1 − ξ 2 |) and |f (t, x, η 1 ) − (t, x, η 2 )| ≤ C |a (η 1 ) − a (η 2 )| where (t, x) ∈ (0, T ) × Ω, η 1 , η 2 ∈ R, ξ 1 , ξ 2 ∈ R N . As in [3] we define the function B ∗ by B ∗ (s) := β a −1 (s) s − s 0 β a −1 (z) dz for s ∈ {y ∈ R : y = a (z) , z ∈ R} . We are concerned with a weak solution in the following sense: Definition 1 By a weak solution of the problem (P) we mean a function u : Q T −→ R such that: 4 (1) β (u) ∈ L 2 (Q T ) , ∂ t (β (u) − a (u)) ∈ L q ((0, T ) , W −1,q (Ω)) , a (u) ∈ L p (0, T ) , W 1,p 0 (Ω) , a (u) ∈ L ∞ ((0, T ) , H 1 0 (Ω)) . (2) ∀v ∈ L p (0, T ) , W 1,p 0 (Ω) , v t ∈ L 2 ((0, T ) , H 1 0 (Ω)) and v (T ) = 0 we have − Q T β (u) ∂ t vdxdt − Q T ∇a (u) ∇∂ t vdxdt + Q T d (t, x, u, a (u)) ∇vdxdt + Q T β (u 0 ) v t dxdt + Q T ∇a (u 0 ) ∇v t dxdt + T 0 K (u) , v dt = Q T f (t, x, u) vdxdt. (2.1) The main result of this article is the following theorem. Theorem 2 Under hypotheses (H 1 ) − (H 6 ), there exists a weak solution u for problem (P) in the sense of Definition 1. In addition, if (H 7 ) is also satisfied, then u is unique. The proof of this theorem will be done in the last section. In the sequel, we need the following lemmas: Lemma 3 [3] Let J : R N −→ R N be continuous and for any R > 0, (J (x) , x) ≥ 0 for all |x| = R. Then there exists an y ∈ R N such that y = 0, |y| ≤ R and J (y) = 0. Lemma 4 [4] Assume that ∂ t (β (u) − a (u)) ∈ L q ((0, T ) , W −1,q (Ω)) , a (u) ∈ L p (0, T ) , W 1,p 0 (Ω) , a (u) ∈ L ∞ ((0, T ) , H 1 0 (Ω)) , B ∗ ∈ L ∞ ((0, T ) , L 1 (Ω)) , β (u 0 ) ∈ L 2 (Ω) and a (u 0 ) ∈ H 1 0 (Ω). Then for almost all t ∈ (0, T ), we have t 0 (∂ t (β (u) − a (u)) , a (u)) dt = Ω B ∗ (a (u (t))) dx 5 + 1 2 Ω |∇a (u (t))| 2 dx − Ω B ∗ (a (u 0 )) dx − 1 2 Ω |∇a (u 0 )| 2 dx. 3 Discretization scheme and a priori estimates To solve problem (P) by Rothe–Galerkin method, we proceed as follows. We divide the interval I = [0, T] into n subintervals of the length h = T n and denote u i = u (t i ), with t i = ih, i = 1, , n, then problem (P) is approximated by the following recurrent sequence of time-discretized problems 1 h (β (u i ) − β (u i−1 )) − 1 h (a (u i ) − a (u i−1 )) − ∇d (t i , x, u i−1 , a (u i )) −f (t i , x, u i−1 ) + K (ˆu i−1 ) = 0 (3.1) u i (x) = 0 on ∂Ω where ˆu i−1 = u j−1 , t ∈ [t j−1 , t j ) , j = 1, , i − 1 u i−1 , t ∈ [t i−1 , T ] Hence, we obtain a system of elliptic problems that can be solved by Galerkin method. Let ϕ 1 , . . . , ϕ m , . . . be a basis in W 1,p 0 (Ω) and let V m be a subspace of W 1,p 0 (Ω) generated by the m first vectors of the basis. We search for each m ∈ N ∗ the functions {u m i } n i=1 such that a (u m i ) = m k=1 a m ik e k and satisfying Ω 1 h β (u m i ) − β u m i−1 ξdx + Ω 1 h ∇a (u m i ) − ∇a u m i−1 ∇ξdx + Ω d t i , x, u m i−1 , a (u m i ) ∇ξdx + K ˆu m i−1 , ξ − Ω f t i , x, u m i−1 ξdx = 0 (3.2) 6 Remark 5 In what follows we denote by C a nonnegative constant not depending on n, m, j and h. Theorem 6 There exists a solution u m i in V m of the family of discrete Equation (3.2). Proof. We pro ceed by recurrence, suppose that u m 0 is given and that u m i−1 is known. Define the continuous function J hm : R m −→ R m by: J hm (r) = 1 h Ω (β (v) e j + ∇a (v) ∇e j ) dx − 1 h Ω β u m i−1 e j (3.3) +∇au m i−1 ∇e j dx + Ω d t i , x, u m i−1 , a (v) ∇e j dx + K ˆu m i−1 , ξ − Ω f t i , x, u m i−1 e j dx where a (v) = m j=1 r j e j . We shall prove that J hm satisfies the following estimates J hm (r) r ≥ 1 h Ω B ∗ (a (v)) + 1 2 |∇a (v)| 2 dx − 1 h Ω B ∗ a u m i−1 + 1 2 ∇a u m i−1 2 dx +d 0 Ω |∇a (v)| p dx − Cδ Ω |∇a (v)| p dx −C (δ) .C (γ) i k=1 h Ω ∇u m k−1 p dx −Cδ 0 Ω |∇a (v)| p dx − C (δ 0 ) Ω f t i , x, u m i−1 q dx ≥ C Ω |∇a (v)| 2 dx + C Ω |∇a (v)| p dx − C (3.4) 7 Indeed, from hypothesis (H 1 ) and the definition of B ∗ we deduce 1 h Ω β (v) − β u m i−1 a (v) dx ≥ 1 h Ω B ∗ (a (v)) dx − 1 h Ω B ∗ a u m i−1 dx, (3.5) the hypotheses on a and d imply Ω d t i , x, u m i−1 , a (v) ∇a (v) dx ≥ d 0 Ω |∇a (v)| p dx, (3.6) using the identity 2 (x, x − y) = x 2 − y 2 + x − y 2 , (3.7) we obtain 1 h Ω ∇a (v) − ∇a u m i−1 ∇a (v) dx ≥ 1 2h Ω |∇a (v)| 2 dx − 1 2h Ω ∇a u m i−1 2 dx, applying Holder and δ−inequalities to the integral operator, it yields K ˆu m i−1 , a (v) ≤ C δ Ω t i 0 k (t i , s) g s, x, ∇ˆu m i−1 ds q ∇dx (3.8) +Cδ Ω |∇a (v)| p dx the first integral in (3.8) can be estimated as t i 0 k (t i , s) g s, x, ∇ˆu m i−1 ds ≤ i k=1 h ∇u m k−1 p 1 q i k=1 h (t i − t k ) −γp 1 p + C. (3.9) 8 Since γ < 1 p , then i k=1 h (t i − t k ) −γp ≤ 1 1 − γp =: C (γ) for the function f we have Ω f t i , x, u m i−1 a (v) dx ≤ Cδ Ω |∇a (v)| p dx + C (δ) Ω B ∗ (a (u m i )) dx + C. (3.10) Therefore (3.4) holds. Then for |r| big enough, J hm (r) r ≥ 0. Taking into account that J hm is continuous, Lemma 3 states that J hm has a zero. Since the function a is strictly increasing then there exists v = u m i solution of (3.2). Now we derive the following estimates. Lemma 7 There exists a constant C > 0 such that max 1≤i≤n Ω B ∗ (a (u m i )) dx ≤ C, (3.11) max 1≤i≤n Ω |∇a (u m i )| 2 dx ≤ C, (3.12) n i=1 h Ω |∇a (u m i )| p dx ≤ C. (3.13) Proof. Testing Equation (3.2) with the function a (u m i ) , then summing on i it yields j i=1 Ω 1 h β (u m i ) − β u m i−1 a (u m i ) dx + j i=1 Ω 1 h ∇a (u m i ) − ∇a u m i−1 ∇a (u m i ) dx + j i=1 Ω d t i , x, u m i−1 , a (u m i ) ∇a (u m i ) dx 9 [...]... Eymard, R, Gutnic, M, Hilhorst, D: The finite volume method for an elliptic parabolic equation Acta Math Univ Comenianae 17(1), 181–195 (1998) [2] Slodicka, M: An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator J Comput Appl Math 168, 447–458 (2004) [3] Jager, W, Kacur, J: Solution of doubly nonlinear and degenerate parabolic problems by... τ, x)) − a (um (t, x))) dx ≤ Cτ n n (3.28) for k = 0, n − 1 and τ ∈ (kh, (k + 1) h) Remark 11 (1) Corollary 10 and hypothesis (H3 ) imply dn t, x, um (t, x) , a (um ) n,h n Lq (QT )N ≤C (2)From Equation (3.2) we get ∂h (β (um ) − n a (um )) n Lq ((0,T ),H −1,q (Ω)) ≤C (3) The estimate of B ∗ in Corollary 10 and hypothesis (H1 ) give β (um ) n L2 (QT ) ≤C (4) For the memory operator we have K um ˆn−1... b (um ) −→ b (u) a.e in L2 (QT ) According to n n,h the hypothesis (H4 ) we get fn t, x, um n,h Lq (QT ) ≤ C, and consequently fn um −→ f (u) n,h in Lq (QT ) For B ∗ we can easily prove that B ∗ (u) ∈ L∞ ((0, T ) , L1 (Ω)) Based on the 15 foregoing points, Equation (3.2) involves T T z, v dt + 0 χ vdxdt + T µ, v dt = 0 QT fn (t, x, u) vdxdt 0 (4.1) Ω Rewriting the discrete derivative with respect... that n−k β um − β um j+k j h j=1 a um − a um j+k j dx ≤ chk, (3.20) Ω n−k a um − j+k h j=1 a um j 2 dx ≤ chk (3.21) Ω Proof Summing Equation (3.2) for i = j + 1, j + k, choosing a um − a um as test j j+k 11 function, then summing the resultant equations for j = 1 , n − k, we get n−k 1 β um − β um j+k j h j=1 Ω n−k 1 h + a um − j+k a um − a u m j+k j 2 a um j dx dx j=1 Ω n−k j+k d ti , x, um , a... the question of convergence and existence From Corollary 10, Remark 11 and Kolomogorov compactness criterion, one can cite the following: 14 Corollary 12 There exist subsequences with respect to n and m for (um ) that we will note n again (um ) such that n a (um ) n a (um ) n 1,p α in Lp (0, T ) , W0 (Ω) β (um ) n ∂h (β (um ) − n 1 α in L∞ (0, T ) , H0 (Ω) b in L2 (QT ) a (um )) n z in Lq (0, T ) , H... dx (3.24) n n C +C a (um )|p i | i=1 Ω n f ti , x, um i−1 q B ∗ (a (um )) dx, i dx + C (3.25) i=1 Ω n B ∗ (a (um )) dx i dx ≤ C + C i=1 Ω (3.26) i=1 Ω The operator K can be estimated as previously Therefore we get n−k j=1 Ω 1 β um − β um j+k j h n−k 1 h + j=1 Ω a u m − a um j+k j a um − j+k n a um j 2 dx dx ≤ (3.27) n | a (um )|p i B ∗ (a (um )) dx i dx + C i=1 Ω i=1 Ω n | a (um )|p dx i +C (γ) i=1 Ω... ) and (H6 ), the operator memory can be estimated as τ K um , a (um ) − a (v m ) dt ˆn−1 n n 0 τ C ≤ δ t | a (um ) − n 0 Ω a (v m )|p dxdt n 0 +Cδ a (um ) − a (v m ) n n 18 p 1,p Lp ((0,T ),W0 (Ω)) +C For fn we have τ fn t, x, um (a (um ) − a (v m )) dxdt ≤ Cε, n,h n n 0 Ω regrouping the estimates of all terms of Equation (4.3) we obtain τ | a (um ) − n (d1 − Cδ) 0 a (v m )|p dxdt n Ω t τ | a (um )... Gronwall lemma we get s a u1 − 0 1 a u2 2 dxdt = 0 Ω 2 consequently u ≡ u This achieves the Proof of Theorem 2 20 2 dxdτ dt Acknowledgments The authors would like to express their thanks to the referees for their helpful comments and suggestions This work was supported by the National Research Project (PNR, Code8/u160/829) Competing interests The authors declare that they have no competing interests Authors’... ti , x, um a (um ) dx = 0 i−1 i − i=1 (3.14) i=1 Ω From the definition of B ∗ we obtain j i=1 Ω 1 β (um ) − β um i−1 i h a (um ) dx ≥ i B ∗ um dx − j Ω B ∗ (um ) dx 0 (3.15) Ω Using the identity (3.7) for the second integral in (3.14), we get j i=1 Ω ≥ 1 h a (um ) − i 1 2 2 a um j a um i−1 dx − 1 2 a (um ) dx i | a (um )|2 dx 0 Ω (3.16) Ω The hypotheses on d imply j j d ti , x, um , i−1 a (um ) i a... W, Kacur, J: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes Math Model Numer Anal 29(5), 605–627 (1995) [4] Showalter, RE: Monotone operators in Banach space and nonlinear partial differential equations Math Surv Monogr 49, 113–142 (1996) [5] Brezis, H: Analyse Fonctionnelle, Th´orie et Applications Masson, Paris (1983) e 22 . as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Rothe-Galerkin's method for nonlinear integrodifferential equations Boundary. freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For. studied the convergence of a finite volume scheme for the numerical solution for an elliptic–parabolic 2 equation. Using Rothe method, the author in [2] studied a nonlinear degenerate parabolic equation