Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Akdim, Youssef; Bennouna, Jaouad; Mekkour, Mounir; Redwane, Hicham

doi:10.4064/am39-1-1

Cited by 14 publications

(9 citation statements)

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“…It is our purpose, in this paper to generalize the result of ( [3], [4], [5], [16]) and we prove the existence of a renormalized solution of (1.1).…”

Section: Introductionmentioning

confidence: 89%

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Existence results for a nonlinear parabolic problems with lower order terms

Aberqi¹,

Bennouna²,

Mekkour³

et al. 2013

ijma

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In this paper, we study the nonlinear parabolic problem: ∂b(x, u) ∂t − div(a(x, t, u, ∇u)) + div(φ(x, t, u)) = f where b(x, u) is unbounded function of u, the term −div a(x, t, u, ∇u) is a Leray-Lions operator and the function φ is a nonlinear lower order and satisfy only the growth condition. The second term f belongs to L 1 (Ω × (0, T)). The main contribution of our work is to prove the existence of a renormalized solution.

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“…It is our purpose, in this paper to generalize the result of ( [3], [4], [5], [16]) and we prove the existence of a renormalized solution of (1.1).…”

Section: Introductionmentioning

confidence: 89%

“…In the case where b(x, u) = u, the existence of renormalized solutions for (1.1) has been established by R.-Di Nardo [16]. For the degenerated parabolic equation with b(x, u) = u, div(φ(x, t, u)) = H(x, t, u, ∇u) and f ∈ L 1 (Q), the existence of renormalized solution has been proved by Y. Akdim and al [5].…”

Section: Introductionmentioning

confidence: 99%

Existence results for a nonlinear parabolic problems with lower order terms

Aberqi¹,

Bennouna²,

Mekkour³

et al. 2013

ijma

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show abstract

“…(Q)) N , we deduce that the right hand side converges to zero as n, m and µ tend to infinity . Since (Ω)) as µ → ∞, it follows that the first and second integrals on the right-hand side of (5.17) converge to zeros as n, m, µ → ∞, using [3] where g(u) = |u| 2+u 4 | is bounded positive continuous function which belongs to L 1 (R). Note that H(x, t, u, ∇u) does not satisfy the sign condition or the coercivity condition.…”

Section: Proof Of Theorem 42mentioning

confidence: 99%

“…The very definition of the sequence ω i µ makes it possible to establish the following lemma. (See [20,6]). For k ≥ 0, we have Proof.…”

mentioning

confidence: 99%

Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

Akdim

Gorch

Mekkour

2016

bspm

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In this article, we study the existence of a renormalized solution for the nonlinear p(x)-parabolic problem associated to the equation:The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on H(x, t, u, ∇u) is with respect to∇u, no growth with respect to u and no sign condition or the coercivity condition.

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“…have been studied by many authors under various conditions on the data in the classical Sobolev spaces (see, e.g., [1,2,11,15]), and by J. Bennouna [9] in the setting of Orlicz spaces.…”

Section: Introductionmentioning

confidence: 99%

Renormalized Solutions for Nonlinear Parabolic Systems in the Lebesgue-Sobolev Spaces with Variable Exponents

Hamdaoui¹,

Bennouna²,

Aberqi³

2018

Z. mat. fiz. anal. geom.

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The existence result of renormalized solutions for a class of nonlinear parabolic systems with variable exponents of the typefor i = 1, 2, is given. The nonlinearity structure changes from one point to other in the domain Ω. The source term is less regular (bounded Radon measure) and no coercivity is in the nondivergent lower order term div(c(x, t)|u(x, t)| γ(x)−2 u(x, t)). The main contribution of our work is the proof of the existence of renormalized solutions without the coercivity condition on nonlinearities which allows us to use the Gagliardo-Nirenberg theorem in the proof.where v is the velocity, [∇v]v is the convective term, π denotes the pressure, S denotes the extra stress tensor, g is the external body force, E is the electric field, and P is the electric polarization. The extra stress tensor is given by S = α 2 1((1 + |D| 2 ) p−1 2 − 1)E ⊗ E + (α 2 1 + α 2 1|E| 2 )(1 + |D| 2 ) p−2 2 ),

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Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Cited by 14 publications

References 0 publications

Existence results for a nonlinear parabolic problems with lower order terms

Existence results for a nonlinear parabolic problems with lower order terms

Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

Renormalized Solutions for Nonlinear Parabolic Systems in the Lebesgue-Sobolev Spaces with Variable Exponents

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