Rothe-Galerkin's method for a nonlinear integrodifferential equation
© Chaoui and Guezane-Lakoud; licensee Springer. 2012
Received: 29 September 2011
Accepted: 8 February 2012
Published: 8 February 2012
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.
Mathematics Subject Classification 2000: 35k55; 35A35; 65M20.
KeywordsRothe's method a priori estimate integrodifferential equation Galerkin method weak solution
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  where the authors studied the convergence of a finite volume scheme for the numerical solution for an elliptic-parabolic equation. Using Rothe method, the author in  studied a nonlinear degenerate parabolic equation with a second-order differential Volterra operator. In  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:
(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 .
(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 for any (t, x) ∈ (0, T) × Ω, ∀z ∈ ℝ, ξ ∈ ℝ N .
for any (t, x) ∈ (0, T) × Ω, ∀z ∈ ℝ.
The functions g and k given in (1.4) satisfy the following hypotheses (H5) and (H6), respectively:
(H5) 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, ξ1) - g (t, x, ξ2)| ≤ d1 |ξ1 - ξ2| p -1.
(H6) k : (0, T) × (0, T) → ℝ is weak singular, i.e. |k (t, s)| ≤ |t - s|-γω(t, s) for and the function ω : [0, T ] × [0, T] → ℝ is continuous.
where (t, x) ∈ (0, T) × Ω, η1, η2 ∈ ℝ, ξ1, ξ2 ∈ ℝ N .
We are concerned with a weak solution in the following sense:
β (u) ∈ L2 (Q T ), ∂ t (β (u) - Δa (u)) ∈ L q ((0, T), W-1,q(Ω)), a (u) ∈ L p ((0, T), , a (u) ∈ L∞ ((0, T), .
- (2)∀v ∈ L p ((0, T), , v t ∈ L2 ((0, T), and v (T) = 0 we have(2.1)
The main result of this article is the following theorem.
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 the
Lemma 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.
3 Discretization scheme and a priori estimates
Hence, we obtain a system of elliptic problems that can be solved by Galerkin method.
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 in V m of the family of discrete Equation (3.2).
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 solution of (3.2). ■
Now we derive the following estimates.
Choosing δ conveniently and applying the discrete Gronwall inequality, we achieve the proof of Lemma 7. ■
Proof. Summing Equation (3.2) for i = j + 1, j + k, choosing as test
Using the estimates of previous Lemma we obtain the desired results. ■
for and τ∈ (kh, (k + 1) h).
4 Convergence results and existence
Now we attend to the question of convergence and existence. From Corollary 10, Remark 11 and Kolomogorov compactness criterion, one can cite the following:
when m, n → ∞.
The weak convergences of and 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.
consequently u1 ≡ u2. This achieves the Proof of Theorem 2.
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).
- Eymard R, Gutnic M, Hilhorst D: The finite volume method for an elliptic parabolic equation. Acta Math Univ Comenianae 1998, 17(1):181–195.MathSciNetGoogle Scholar
- Slodicka M: An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator. J Comput Appl Math 2004, 168: 447–458. 10.1016/j.cam.2003.12.007MATHMathSciNetView ArticleGoogle Scholar
- Jager W, Kacur J: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math Model Numer Anal 1995, 29(5):605–627.MathSciNetGoogle Scholar
- Showalter RE: Monotone operators in Banach space and nonlinear partial differential equations. Math Surv Monogr 1996, 49: 113–142.Google Scholar
- Brezis H: Analyse Fonctionnelle, Théorie et Applications. Masson, Paris; 1983.Google Scholar
This article is published under license to BioMed Central Ltd. 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.