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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.frAssia Guezane

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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

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© 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

integrodifferential equation Abderrazek Chaoui∗ and Assia Guezane-Lakoud

Laboratory of advanced materials, Badji Mokhtar University, Annaba, Algeria

Keywords: Rothe’s method; a priori estimate; integrodifferential equation; Galerkinmethod; weak solution

Mathematics Subject Classification 2000: 35k55; 35A35; 65M20

1 Introduction

The aim of this work is the solvability of the following equation

∂ t β (u) − ∂ t 4a (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 (u0(x)) , x ∈ Ω (1.2)and the boundary condition

The literature on the subject of local in time doubly nonlinear evolution equations israther wide Among these contributions, we refer the reader to [1] where the authors studiedthe convergence of a finite volume scheme for the numerical solution for an elliptic–parabolic

equation Using Rothe method, the author in [2] studied a nonlinear degenerate parabolicequation with a second-order differential Volterra operator In [3] the solutions of nonlinearand degenerate problems were investigated In general, existence of solutions for a class ofnonlinear 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, precisesense of the weak solution, then we state the main results and some Lemmas that needed inthe 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

To solve problem (P), we assume the following hypotheses:

(H1) The function β : R −→ R is continuous, nondecreasing, β (0) = 0, β (u0) ∈ L2(Ω)

and satisfies |β (s)|2 ≤ C1B ∗ (a (s)) + C2, ∀s ∈ R.

(H2) a : R −→ R is continuous, strictly increasing function, a (0) = 0 and a (u0) ∈ H1

0(Ω)

(H3) d : (0, T ) × Ω × R × R N −→ R N is continuous, elliptic i.e., ∃d0 > 0 such that

d (t, x, z, ξ) ξ ≥ d0|ξ| p for ξ ∈ R N and p ≥ 2, strongly monotone i.e.,

(d (t, x, η, ξ1) − d (t, x, η, ξ2)) (ξ1− ξ2) ≥ d1|ξ1− ξ2| p for ξ1, ξ2 ∈ R N , d1 > 0 and satisfies

(H4) f : (0, T ) × Ω × R −→ R is continuous such that

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:

(1) β (u) ∈ L2(Q T ) , ∂ t (β (u) − 4a (u)) ∈ L q ((0, T ) , W −1,q (Ω)) , a (u) ∈ L p¡

(0, T ) , W01,p(Ω)¢,

a (u) ∈ L ∞ ((0, T ) , H1

0 (Ω)) (2) ∀v ∈ L p¡

Theorem 2 Under hypotheses (H1) − (H6), there exists a weak solution u for problem (P)

in the sense of Definition 1 In addition, if (H7) is also satisfied, then u is unique.

The proof of this theorem will be done in the last section In the sequel, we need thefollowing 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 6= 0, |y| ≤ R and J (y) = 0.

Lemma 4 [4] Assume that ∂ t (β (u) − 4a (u)) ∈ L q ((0, T ) , W −1,q (Ω)) , a (u)

Ω

|∇a (u0)|2dx.

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

Hence, we obtain a system of elliptic problems that can be solved by Galerkin method

Let ϕ1, , ϕ m , be a basis in W01,p (Ω) and let V m be a subspace of W01,p(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 ) =Pm k=1 a m

ik e k and satisfyingZ

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 proceed by recurrence, suppose that u m

0 is given and that u m

i−1 is known Define

the continuous function J hm : Rm −→ R m by:

¢¢

+12

Indeed, from hypothesis (H1) and the definition of B ∗ we deduce

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

Proof Testing Equation (3.2) with the function a (u m

i ) , then summing on i it yields

h¯¯∇u m k−1

Ω

B ∗¡

u m j

¢

dx +1

2Z

¢

as test

function, then summing the resultant equations for j = 1 , n − k, we get

¢

− β¡u m j

¢¢ ¡

a¡u m j+k

¢

− a¡u m j

¢

− a¡u m j

¢

− a¡u m j

Using the estimates of previous Lemma we obtain the desired results.

Notation 9 Let us introduce the step functions

n (t − h, x) , t ∈ [h, T ]

u m n,h (t, x) = u m

k∂ h (β (u m

n ) − 4a (u m

n ))k L q ((0,T ),H −1,q(Ω)) ≤ C (3) The estimate of B ∗ in Corollary 10 and hypothesis (H1) give

kβ (u m n )k L2(Q T)≤ C (4) For the memory operator we have

°

°K¡uˆm n−1

¢°°

L q ((0,T ),H −1,q(Ω)) ≤ C

Now we attend to the question of convergence and existence From Corollary 10, Remark

11 and Kolomogorov compactness criterion, one can cite the following:

Corollary 12 There exist subsequences with respect to n and m for (u m

n ) that we will note

foregoing points, Equation (3.2) involves

In fact, taking in (3.2) the function ξ = a (u m

n ) − a (v m

n) as test function and integrating

on the interval (0, τ ) , where a (v m

n ) ∈ L p ((0, T ) , V m (Ω)) is the approximate of a (u) in

Q τ

f n

¡

t, x, u m n,h¢(a (u m n ) − a (v m n )) dxdt (4.3)Lemma 4 implies

Ω

|∇a (u (τ ))|2dx,

n ) to a (u) in L2(Q T ) , the convergence of a (v m

n)

to a (u) in L p¡

(0, T ) , W01,p(Ω)¢, the continuity of d, the weak convergence of d in L q (Q T)Nand the dominated convergence theorem, we obtain

d n¡t, x, u m n,h , ∇a (u)¢−→ d (t, x, u, ∇a (u)) in L q (Q T)N

In addition to monotonicity of d gives

n)¢ and K¡uˆm

n−1

¢and the almost everywhere

convergences imply that χ = d (t, x, u, ∇a (u)) and µ = K (u) So u is the weak solution of

the problem (P) in the sense of Definition 1

Now we prove the uniqueness of the weak solution We assume that the problem (P) has

two solutions u1 and u2 ∈ L2((0, T ) , H1

0(Ω)) Taking into account that β (u1

0) = β (u2

0) and

[3] Jager, W, Kacur, J: Solution of doubly nonlinear and degenerate parabolic problems byrelaxation schemes Math Model Numer Anal 29(5), 605–627 (1995)

[4] Showalter, RE: Monotone operators in Banach space and nonlinear partial differentialequations Math Surv Monogr 49, 113–142 (1996)

[5] Brezis, H: Analyse Fonctionnelle, Th´eorie et Applications Masson, Paris (1983)