Existence of bounded solutions of some nonlinear parabolic equations

Mokrane, Abdellah

doi:10.1017/s0308210500031188

Cited by 30 publications

(16 citation statements)

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“…Therefore x u # W pq (T) solves (19). In fact we claim that x u is the greatest solution of (19) in K. Indeed let x^# K be any solution of (19).…”

Section: (Z)mentioning

confidence: 92%

“…Note that in Boccardo et al [3], Deuel and Hess [9], and Mokrane [19] it is assumed that , . # L (T_Z).…”

Section: Definitionmentioning

confidence: 99%

“…smallest) solution of (19) in K can be obtained by a monotone iterative process (see Sattinger [21] for semilinear systems and classical solutions). Proposition 6.…”

Section: (Z)mentioning

confidence: 99%

“…The method of upper and lower solutions, turned out to be a powerful tool for the resolution of nonlinear parabolic problems. The works of Boccardo et al [3], Deuel and Hess [9], and Mokrane [19] were based on this method. However the way this method was implemented in these works, is different from our use of the upper and lower solutions in this paper.…”

Section: Introductionmentioning

confidence: 99%

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On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

Avgerinos

Papageorgiou

1996

Journal of Differential Equations

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We consider a very general second order nonlinear parabolic boundary value problem. Assuming the existence of an upper solution . and a lower solution satisfying ., we show that the problem has extremal periodic solutions in the order interval K=[ , .]. Our proof is based on a general surjectivity result for the sum of two operators of monotone type and on truncation and penalization techniques. In addition we use a result of independent interest which we prove here and which says that the pseudomonotonicity property of A(t, } ) can be lifted to its Nemitsky operator. Finally when we impose stronger conditions on the data, we show that the extremal solutions can be obtained with a monotone iterative process.

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“…Therefore x u # W pq (T) solves (19). In fact we claim that x u is the greatest solution of (19) in K. Indeed let x^# K be any solution of (19).…”

Section: (Z)mentioning

confidence: 92%

“…Note that in Boccardo et al [3], Deuel and Hess [9], and Mokrane [19] it is assumed that , . # L (T_Z).…”

Section: Definitionmentioning

confidence: 99%

“…smallest) solution of (19) in K can be obtained by a monotone iterative process (see Sattinger [21] for semilinear systems and classical solutions). Proposition 6.…”

Section: (Z)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

Avgerinos

Papageorgiou

1996

Journal of Differential Equations

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“…In [14], N. Grenon extends the result of [9] to the case where the natural growth of f with respect to |∇u| is at most of order p; but instead of a periodicity condition the author considered an initial one. The proof therein is based on some regularization techniques used in [6,17].…”

Section: Introductionmentioning

confidence: 99%

Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

Hachimi

Maatouk

2017

Arab. J. Math.

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Existence of renormalized solutions to a nonlinear parabolic equation in L¹ setting with nonstandard growth condition and gradient term

Gao

2014

Math Methods in App Sciences

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Introduction Introduction of our problemIn this paper, we consider the following nonlinear parabolic equation with nonstandard growth condition: jp.x/ p.y/j Ä C log jx yj for every x, y 2 satisfying jx yj Ä 1 2 .(1.2) Z. LI AND W. GAOg.x, t, s, /s 0, (1.4) for any .s, / 2 R R N , and for almost every .x, t/ 2 Q T .We remark that the log-Hölder continuity condition in assumption (H1), originally introduced by Zhikov [1], guarantees that smooth functions are dense in the variable exponent Sobolev space [2]. In general, see [3], they are NOT if p.x/ ¤ constant, which is an interesting feature, called Lavrentiev phenomenon. Under this condition, we can define W 1,p.x/ 0 . / as the closure of C 1 0 . / in W 1,p.x/ . /, which assists in studying the initial and boundary value problem for parabolic equations.In recent years, nonlinear partial differential equations with variable exponents have generated a good deal of interest and attention, as can be found in monograph [4] and the reference therein. Maybe, it attributes to the important physical applications in real world. Problem (1.1), or its variants, is from electro-rheological fluids model [5], which is characterized by their ability to undergo significant changes in their mechanical properties when an electric filed exerted from outside. This non-Newtonian fluids also describe some useful evolutional phenomenon in applied thermodynamics [6]. Comparing with the case when p is a constant exponent, Problem (1.1) with variable exponent is a more realistic and precise model to interpret the diffusion process. It is mainly because the space region that presents nonhomogeneous physical characteristic on the exponent p affects the growth on the state variable u, which leads to so called nonstandard growth condition. There are other applications in the calculus of variations [7], elasticity [8], image restoration [9], and so on. In particular, in image processing, the consideration of nonstandard growth condition has more advantages, chief among which is the so called staircase effect. Or more precisely, the study of variational integrals with nonstandard growth not only preserves edges in the picture but also creates edges where there were none in the original image.In 1980s, when DiPerna and Lions investigated Boltzmann equations, they firstly introduced the concept of renormalized solution [10]. Afterwards, it was used in elliptic equations [11], parabolic cases [12], or problems about conservation laws [13]. Problem (1.1) with p a constant and g D 0, or some variants of it, has been widely analyzed in the literature [14,15]. The case p D 2 was investigated in [12], while the case when the initial value u 0 D 0 was studied in [16]; when 1 < p < 2, p is a constant, and g, f , ! F D 0, the existence of renormalized solution was studied by Xu [17]. The authors of more recent studies [18,19] concentrated on the nonlinear PDEs with variable exponents, L 1 data, and g D 0. To the best of our knowledge, only a few papers deal with the renormalized solution in the setting...

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Existence of bounded solutions of some nonlinear parabolic equations

Abstract: SynopsisThis paper proves the existence of (at least) one solution of the following equation:

Cited by 30 publications

References 5 publications

On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

Existence of renormalized solutions to a nonlinear parabolic equation in L¹ setting with nonstandard growth condition and gradient term

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Existence of bounded solutions of some nonlinear parabolic equations

Abstract: SynopsisThis paper proves the existence of (at least) one solution of the following equation:

Cited by 30 publications

References 5 publications

On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

On the Existence of Extremal Periodic Solutions for Nonlinear Parabolic Problems with Discontinuities

Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

Existence of renormalized solutions to a nonlinear parabolic equation in L1 setting with nonstandard growth condition and gradient term

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Existence of renormalized solutions to a nonlinear parabolic equation in L¹ setting with nonstandard growth condition and gradient term