Rothe-Galerkin's method for a nonlinear integrodifferential equation

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.

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

where (t, x) ∈ (0, T) × Ω = Q_T, 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 [1] 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 [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.

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.

(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 × 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.

(H₇) For p = 2, we have

and

where (t, x) ∈ (0, T) × Ω, η₁, η₂ ∈ ℝ, ξ₁, ξ₂ ∈ ℝ^N.

As in [3] 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 : Q_T → ℝ such that:

(1) 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

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

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) - Δa(u)) ∈ L^q((0, T), W^-1,q(Ω)), a(u) ∈ L^p (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

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 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

where

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

Let φ₁, . . . , φ_m, . . . be a basis in and let V_m be a subspace of generated by the m first vectors of the basis. We search for each m ∈ ℕ* the functions such that and satisfying

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).

Proof. We proceed by recurrence, suppose that is given and that is known. Define the continuous function J_hm : ℝ^m → ℝ^m by:

where . We shall prove that J_hm satisfies the following estimates

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

the hypotheses on a and d imply

using the identity

we obtain

applying Holder and δ-inequalities to the integral operator, it yields

the first integral in (3.8) can be estimated as

Since , then

for the function f we have

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.

Lemma 7 There exists a constant C > 0 such that

Proof. Testing Equation (3.2) with the function , then summing on i it yields

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

Proof. Summing Equation (3.2) for i = j + 1, j + k, choosing as test

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

for and τ ∈ (kh, (k + 1) h).

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

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 that we will note again such that

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 [[5], p. 72] it yields in L²(Q_T). Since a is strictly increasing then almost everywhere in Q_T. From the continuity of a it yields almost everywhere in Q_T and α = a (u), consequently a.e. in L²(Q_T). Applying Poincaré inequality and the fourth estimate in (3.28) we obtain

then a.e. in Q_T. Analogously a.e. in L²(Q_T). According to the hypothesis (H₄) we get and consequently in L^q(Q_T). For B* we can easily prove that B*(u) ∈ L^∞((0, T), L¹(Ω)). Based on the foregoing points, Equation (3.2) involves

Rewriting the discrete derivative with respect to t and taking into account in we obtain

∀v ∈ L^p ((0, T), , v_t ∈ L²((0, T), and v(T) = 0. Since v belongs to a dense subspace in L^p ((0, T), and using the second estimate in Remark 11 we get

Now we prove that

In fact, taking in (3.2) the function as test function and integrating on the interval (0, τ), where is the approximate of a(u) in , constant on each interval ((k - 1) h, kh), we obtain

Lemma 4 implies

From Fatou Lemma we deduce

consequently

Taking into account the convergence of to a(u) in L²(Q_T), the convergence of to a(u) in , the continuity of d, the weak convergence of d in L^q(Q_T)^N and the dominated convergence theorem, we obtain

In addition to monotonicity of d gives

as previously using hypotheses (H₅) and (H₆), the operator memory can be estimated as

For f_n 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

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.

Now we prove the uniqueness of the weak solution. We assume that the problem (P) has two solutions u¹ and . Taking into account that and , we get

Choosing in (4.4) the test function

and since v_s(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).

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