Existence of solutions for some quasilinear parabolic systems with weight and weak monotonicity general data
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Existence of solutions for some quasilinear parabolic systems with weight and weak monotonicity general data El Houcine Rami1
· Elhoussine Azroul1 · Abdelkrim Barbara1
Received: 6 June 2020 / Accepted: 27 June 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract We prove the existence of weak solution u for the nonlinear parabolic systems: ⎧ ⎨ ∂t u − divσ (x, t, u, Du) = v(x, t) + f (x, t, u, Du) + divg(x, t, u) in T u(x, t) = 0 on ∂ × (0, T ) (Q P S)ω ⎩ u(x, 0) = u 0 (x) on
which is a Dirichlet Problem. In this system, v belongs to L p (0, T , W −1, p (, ω∗ , IR m )) and u 0 ∈ L 2 (, ω0 , IR m ), f and g satisfy some standards continuity and growth conditions. We prove existence of a weak solution of different variants of this system under classical 2n regularity for some ps ∈] n+2 ; ∞[, growth and coercivity for σ but with only very mild monotonicity assumptions. Keywords Nonlinear parabolic system · Young measure · The div-curl type inequality Mathematics Subject Classification 35R35 · 58J35
Contents 1 2 3 4 5 6
Introduction . . . . . . . . . . . . . . Functional framework . . . . . . . . . Hypothesis . . . . . . . . . . . . . . . A first parabolic div-curl inequality . . A second parabolic div-curl inequality Passage to the limit for σ . . . . . . .
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El Houcine Rami [email protected] Elhoussine Azroul [email protected] Abdelkrim Barbara [email protected]
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Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, B.P 1796, Atlas Fez, Morocco
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E. H. Rami et al. 7 Galerkin scheme . . . . . . . . 8 Passage to the limit in (QPS)ω 9 Examples . . . . . . . . . . . . References . . . . . . . . . . . . .
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1 Introduction Let be a bounded open domain in IR n , n ≥ 3 and 0 < T < ∞ is given; we recall that T is defined as the cylinder ×(0; T ) this part is devoted to establish Leray-lions existence results for parabolic problems in divergence form of type (QPS)ω. In this paper, the aims of this text is to prove existence results under relaxed monotonicity, in particular under strict quasimonotonicity. The main technical tool we advocate and use throughout the proof are Young measures. By applying a Galerkin schema, we obtain easil
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