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 RotheGalerkin method.
Mathematics Subject Classification 2000: 35k55; 35A35; 65M20.
Keywords:
Rothe's method; a priori estimate; integrodifferential equation; Galerkin method; weak solution1 Introduction
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 integrodifferential 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 ellipticparabolic equation. Using Rothe method, the author in [2] studied a nonlinear degenerate parabolic equation with a secondorder 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 RotheGalerkin 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 .
(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 for any (t, x) ∈ (0, T) × Ω, ∀z ∈ ℝ, ξ ∈ ℝ^{N}.
(H_{4}) 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 (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 and the function ω : [0, T ] × [0, T] → ℝ is continuous.
(H_{7}) For p = 2, we have
and
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
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) ∈ L^{2 }(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 }∈ L^{2 }((0, T), and v (T) = 0 we have
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)  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), L^{1}(Ω)), β(u_{0}) ∈ L^{2}(Ω) and . Then for almost all t ∈ (0, T), we have
3 Discretization scheme and a priori estimates
To solve problem (P) by RotheGalerkin 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 timediscretized problems
Hence, we obtain a system of elliptic problems that can be solved by Galerkin method.
Let φ_{1}, . . . , 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 (H_{1}) and the definition of B* we deduce
the hypotheses on a and d imply
using the identity
we obtain
applying Holder and δinequalities 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, 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
Remark 11 (1) Corollary 10 and hypothesis (H_{3}) imply
(2)From Equation (3.2) we get
(3) The estimate of B* in Corollary 10 and hypothesis (H_{1}) 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:
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 L^{2}(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 L^{2}(Q_{T}). Applying Poincaré inequality and the fourth estimate in (3.28) we obtain
then a.e. in Q_{T}. Analogously a.e. in L^{2}(Q_{T}). According to the hypothesis (H_{4}) we get and consequently in L^{q}(Q_{T}). For B* we can easily prove that B*(u) ∈ L^{∞}((0, T), L^{1}(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 }∈ L^{2}((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 L^{2}(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 (H_{5}) and (H_{6}), 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 u^{1 }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 u^{1 }≡ u^{2}. This achieves the Proof of Theorem 2.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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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