Existence Results to a Class of Nonlinear Parabolic Systems Involving Potential and Gradient Terms
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Existence Results to a Class of Nonlinear Parabolic Systems Involving Potential and Gradient Terms B. Abdellaoui, A. Attar, R. Bentifour and E.-H. Laamri Abstract. In this paper, we investigate the existence of solutions to a nonlinear parabolic system, which couples a non-homogeneous reactiondiffusion-type equation and a non-homogeneous viscous Hamilton– Jacobi one. The initial data and right-hand sides satisfy suitable integrability conditions and non-negative. To simplify the presentation of our results, we will consider separately two simplified models : first, vanishing initial data, and then, vanishing right-hand sides. Mathematics Subject Classification. 35B05, 35K15, 35B40, 35K55, 35K65. Keywords. Parabolic System, nonlinear gradient terms, a priori estimates, fixed point Theorem.
1. Introduction The main goal of this paper is to study the existence of solutions to the following non-linear system: ⎧ ut − Δu = v q + f in ΩT = Ω × (0, T ), ⎪ ⎪ ⎪ p ⎪ − Δv = |∇u| + g in ΩT = Ω × (0, T ), v ⎪ t ⎪ ⎨ u=v =0 on ΓT = ∂Ω × (0, T ), . (1.1) (x) in Ω, u(x, 0) = u ⎪ 0 ⎪ ⎪ ⎪ in Ω, v(x, 0) = v0 (x) ⎪ ⎪ ⎩ u, v ≥0 in Ω × (0, T ), where Ω is a bounded domain of IRN , N ≥ 1 and p, q ≥ 1. Here (f, g) and (u0 , v0 ) are non-negative data and satisfy some suitable integrability conditions that we will specify later. Our objective is to find “natural” relation between p, q and the regularity of the data to get the existence of a solution to system (1.1). The first author is partially supported by project MTM2016-80474-P, MINECO, Spain The first three authors are supported by the DGRSDT, Algeria.
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Systems with gradient term appear for instance when considering electrochemical models in engineering and some other models in fluid dynamics. We refer to [23] and [28] for more details and more applications of this class of systems. Before stating our main results, let us begin by recalling some previous results related to our system. Stationary case Concerning existence of solutions, it is well known in some particular cases. For this, we refer to [8,19,20] and [1] where some general existence results were established. Parabolic case In the case of a single equation and under the presence of gradient term, many results of global existence are known. We refer to [2,18] and the references therein. On the other hand, there is an extensive literature devoted to the study and solvability and properties of solutions to the socalled viscous Hamilton–Jacobi equation (HJ). More precisely, let us consider the following Dirichlet and Cauchy problems: ⎧ in ΩT , ⎨ ut − Δu = a|∇u|p + h(x, t) on ΓT , (HJD) u(x, t) = 0 ⎩ in Ω ; u(x, 0) = u0 (x) and
(HJC)
ut − Δu = a|∇u|p + h(x, t) u(x, 0) = u0 (x)
in RN × (0, T ), in RN ,
where a ∈ IR∗ , p ≥ 0. First case If h ≡ 0. In bounded domains, existence and uniqueness results of (HJD) may be found for example in [4,10,24,47,48] and the references therein. A considerable literature has also been devoted to the analogous whole-space Cauchy pro
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