On a class of nonlinear anisotropic parabolic problems

2015 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
M. H. Abdou ◽  
M. Chrif ◽  
S. El Manouni ◽  
H. Hjiaj
Keyword(s):  
Sobolev Space ◽  
Parabolic Problem ◽  
Nonlinear Term ◽  
Parabolic Problems ◽  
Anisotropic Sobolev Space ◽  
Existence Of Weak Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
Sign Conditions ◽  
Image Position

We prove the existence of weak solutions for the strongly nonlinear parabolic problem in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

scholarly journals Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms

2013 ◽  
Vol 143 (6) ◽  
pp. 1185-1208 ◽  
Author(s):  
Rosaria Di Nardo ◽  
Filomena Feo ◽  
Olivier Guibé
Keyword(s):  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Growth Conditions ◽  
Dirichlet Boundary Conditions ◽  
Parabolic Problems ◽  
Renormalized Solutions ◽  
Uniqueness Results ◽  
Nonlinear Parabolic ◽  
Image Position ◽  
Lower Order Terms

We consider a general class of parabolic equations of the typewith Dirichlet boundary conditions and with a right-hand side belonging to L1 + Lp′ (W−1, p′). Using the framework of renormalized solutions we prove uniqueness results under appropriate growth conditions and Lipschitz-type conditions on a(u, ∇u), K(u) and H(∇u).

Existence of solution for quasilinear equations involving local conditions

2019 ◽  
Vol 150 (6) ◽  
pp. 3074-3086
Author(s):  
Patricio Cerda ◽  
Leonelo Iturriaga
Keyword(s):  
Continuous Function ◽  
Nonlinear Term ◽  
Quasilinear Equation ◽  
Existence Of Solution ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Local Conditions ◽  
Existence Of Weak Solutions ◽  
Carathéodory Function ◽  
Image Position

AbstractIn this paper, we study the existence of weak solutions of the quasilinear equation \begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.

scholarly journals A non-existence result for nonlinear parabolic equations with singular measures as data

2009 ◽  
Vol 139 (2) ◽  
pp. 381-392 ◽  
Author(s):  
Francesco Petitta
Keyword(s):  
Laplace Operator ◽  
Lower Order ◽  
Parabolic Equations ◽  
Existence Result ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Singular Measures ◽  
Nonlinear Parabolic ◽  
Image Position ◽  
Lower Order Terms

In this paper we prove a non-existence result for nonlinear parabolic problems with zero lower-order terms whose model iswhere Δp=div(|∇u|p−2∇u) is the usual p-laplace operator, λ is measure concentrated on a set of zero parabolic r-capacity (1<p<r) and q is large enough.

scholarly journals Nonlinear parabolic problems in unbounded domains

1979 ◽  
Vol 82 (3-4) ◽  
pp. 201-209 ◽  
Author(s):  
Guy Mahler
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Unbounded Domains ◽  
Parabolic Problems ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Nonlinear Parabolic Problems ◽  
Existence Of Weak Solutions ◽  
Nonlinear Parabolic

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

scholarly journals A review of mathematical analysis of nematic and smectic-A liquid crystal models

2013 ◽  
Vol 25 (1) ◽  
pp. 133-153 ◽  
Author(s):  
BLANCA CLIMENT-EZQUERRA ◽  
FRANCISCO GUILLÉN-GONZÁLEZ
Keyword(s):  
Liquid Crystal ◽  
Mathematical Analysis ◽  
Parabolic Problems ◽  
Strongly Nonlinear ◽  
Smectic A ◽  
Nonlinear Parabolic ◽  
Time Periodicity ◽  
Temporal Profiles ◽  
Uniaxial Liquid Crystal ◽  
Smectic A Liquid Crystals

We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data

2016 ◽  
Vol 23 (3) ◽  
pp. 303-321 ◽  
Author(s):  
Youssef Akdim ◽  
Abdelmoujib Benkirane ◽  
Mostafa El Moumni ◽  
Hicham Redwane
Keyword(s):  
Growth Condition ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Measure Data ◽  
Renormalized Solution ◽  
Unbounded Function ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
The Right

AbstractWe study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$where the right-hand side belongs to ${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and ${b(x,u)}$ is unbounded function of u, ${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth ${|\nabla u|^{p-1}}$ in ${\nabla u}$. The critical growth condition on g is with respect to ${\nabla u}$ and there is no growth condition with respect to u, while the function ${H(x,t,\nabla u)}$ grows as ${|\nabla u|^{p-1}}$.

scholarly journals Existence of solutions for some strongly nonlinear parabolic problems involving lower order terms in divergence form in Musielak-Orlicz spaces

2019 ◽  
Vol 38 (6) ◽  
pp. 99-126
Author(s):  
Abdeslam Talha ◽  
Abdelmoujib Benkirane
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Orlicz Spaces ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Strongly Nonlinear ◽  
L1 Data ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this work, we prove an existence result of entropy solutions in Musielak-Orlicz-Sobolev spaces for a class of nonlinear parabolic equations with two lower order terms and L1-data.

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