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.
Keywords:Rothe's method; a priori estimate; integrodifferential equation; Galerkin method; weak solution
The aim of this work is the solvability of the following equation
where (t, x) ∈ (0, T) × Ω = QT, with the initial condition
and the boundary condition
The memory operator K is defined by
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.
(H3) d : (0, T) × Ω × ℝ × ℝN → ℝN is continuous, elliptic i.e., ∃d0 > 0 such that d (t, x, z, ξ) ξ ≥ d0 |ξ|p for ξ ∈ ℝN and p ≥ 2, strongly monotone i.e.,
(H4) f : (0, T) × Ω × ℝ → ℝ is continuous such that
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 × ℝN → ℝN 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.
(H7) For p = 2, we have
where (t, x) ∈ (0, T) × Ω, η1, η2 ∈ ℝ, ξ1, ξ2 ∈ ℝN.
As in  we define the function B* by
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 : QT → ℝ such that:
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 : ℝN → ℝN be continuous and for any R > 0, (J (x), x) ≥ 0 for all |x| = R. Then there exists an y ∈ ℝN such that y ≠ 0, |y| ≤ R and J (y) = 0.
Lemma 4 Assume that ∂t(β (u) - Δa(u)) ∈ Lq((0, T), W-1,q(Ω)), a(u) ∈ Lp (0, T), , a(u) ∈ L∞((0, T), , B* ∈ L∞((0, T), L1(Ω)), β(u0) ∈ L2(Ω) and . Then for almost all t ∈ (0, T), we have
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 and denote ui = u (ti), with ti = ih, i = 1, ..., n, then problem (P) is approximated by the following recurrent sequence of time-discretized problems
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.
Indeed, from hypothesis (H1) and the definition of B* we deduce
the hypotheses on a and d imply
using the identity
applying Holder and δ-inequalities to the integral operator, it yields
the first integral in (3.8) can be estimated as
for the function f we have
Therefore (3.4) holds. Then for |r| big enough, Jhm(r) r ≥ 0. Taking into account that Jhm is continuous, Lemma 3 states that Jhm has a zero. Since the function a is strictly increasing then there exists solution of (3.2). ■
Now we derive the following estimates.
Lemma 7 There exists a constant C > 0 such that
From the definition of B* we obtain
Using the identity (3.7) for the second integral in (3.14), we get
The hypotheses on d imply
The memory operator can be estimated as
Using similar steps as in the proof of Theorem 6 we obtain
Applying Poincaré inequality, we get
Substituting inequalities (3.15)-(3.18) in (3.14) it yields
Choosing δ conveniently and applying the discrete Gronwall inequality, we achieve the proof of Lemma 7. ■
Lemma 8 There exists a constant C > 0 independent on m, n, h, i, and j such that
function, then summing the resultant equations for j = 1 . . . , n - k, we get
The third and fifth integrals in (3.22) can be estimated as
From hypotheses on d and f it yields
The operator K can be estimated as previously. Therefore we get
Using the estimates of previous Lemma we obtain the desired results. ■
Notation 9 Let us introduce the step functions
Corollary 10 There exists a constant C independent of n, m, j and h such that
Remark 11 (1) Corollary 10 and hypothesis (H3) imply
(2)From Equation (3.2) we get
(3) The estimate of B* in Corollary 10 and hypothesis (H1) give
(4) For the memory operator we have
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 → ∞.
Proof of Theorem 2. We have to show that the limit function satisfies all the conditions of Definition 1. Using Corollary 10 (third and fourth inequalities) and Kolmogorov compactness criterion [, p. 72] it yields in L2(QT). Since a is strictly increasing then almost everywhere in QT. From the continuity of a it yields almost everywhere in QT and α = a (u), consequently a.e. in L2(QT). Applying Poincaré inequality and the fourth estimate in (3.28) we obtain
then a.e. in QT. Analogously a.e. in L2(QT). According to the hypothesis (H4) we get and consequently in Lq(QT). For B* we can easily prove that B*(u) ∈ L∞((0, T), L1(Ω)). Based on the foregoing points, Equation (3.2) involves
Now we prove that
Lemma 4 implies
From Fatou Lemma we deduce
In addition to monotonicity of d gives
as previously using hypotheses (H5) and (H6), the operator memory can be estimated as
For fn we have
regrouping the estimates of all terms of Equation (4.3) we obtain
Gronwall inequality implies
hence we get
Following the Proof of Theorem 2: From the continuity of d and g it yields
Choosing in (4.4) the test function
and since vs(s) = 0 then integrating by parts it yields
On the other hand, we have
Applying Gronwall lemma we get
consequently u1 ≡ u2. This achieves the Proof of Theorem 2.
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
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).
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). Publisher Full Text