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Volume 27, Issue 1
Existence of Renormalized Solutions for Nonlinear Parabolic Equations

Youssef Akdim, A. Benkirane, M. EL Moumni & Hicham Redwane

DOI: 10.4208/jpde.v27.n1.2

J. Part. Diff. Eq., 27 (2014), pp. 28-49.

Published online: 2014-03

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  • Abstract
We give an existence result of a renormalized solution for a class of nonlinear parabolic equations $$\frac{\partial b(x,u)}{\partial t}-div(a(x,t,u,\nabla u))+g(x,t,u,\nabla u)+H(x,t,\nabla u)=f,\qquad in\; Q_T,$$ where the right side belongs to $L^{p'}(0,T;W^{-1,p'}(Ω))$ and where b(x,u) is unbounded function of u and where $-div(a(x,t,u,∇u))$ is a Leray-Lions type operatorwith growth $|∇u|^{p-1}$ in ∇u. The critical growth condition on g is with respect to ∇u and no growth condition with respect to u, while the function $H(x,t,∇u)$ grows as $|∇u|^{p-1}$.
  • Keywords

Nonlinear parabolic equations renormalized solutions Sobolev spaces

  • AMS Subject Headings

35K10, 47D20, 46E35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

akdimyoussef@yahoo.fr (Youssef Akdim)

abd.benkirane@gmail.com (A. Benkirane)

mostafaelmoumni@gmail.com (M. EL Moumni)

redwane_hicham@yahoo.fr (Hicham Redwane)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-27-28, author = {Akdim , YoussefBenkirane , A.EL Moumni , M. and Redwane , Hicham}, title = {Existence of Renormalized Solutions for Nonlinear Parabolic Equations}, journal = {Journal of Partial Differential Equations}, year = {2014}, volume = {27}, number = {1}, pages = {28--49}, abstract = { We give an existence result of a renormalized solution for a class of nonlinear parabolic equations $$\frac{\partial b(x,u)}{\partial t}-div(a(x,t,u,\nabla u))+g(x,t,u,\nabla u)+H(x,t,\nabla u)=f,\qquad in\; Q_T,$$ where the right side belongs to $L^{p'}(0,T;W^{-1,p'}(Ω))$ and where b(x,u) is unbounded function of u and where $-div(a(x,t,u,∇u))$ is a Leray-Lions type operatorwith growth $|∇u|^{p-1}$ in ∇u. The critical growth condition on g is with respect to ∇u and no growth condition with respect to u, while the function $H(x,t,∇u)$ grows as $|∇u|^{p-1}$.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v27.n1.2}, url = {http://global-sci.org/intro/article_detail/jpde/5124.html} }
TY - JOUR T1 - Existence of Renormalized Solutions for Nonlinear Parabolic Equations AU - Akdim , Youssef AU - Benkirane , A. AU - EL Moumni , M. AU - Redwane , Hicham JO - Journal of Partial Differential Equations VL - 1 SP - 28 EP - 49 PY - 2014 DA - 2014/03 SN - 27 DO - http://doi.org/10.4208/jpde.v27.n1.2 UR - https://global-sci.org/intro/article_detail/jpde/5124.html KW - Nonlinear parabolic equations KW - renormalized solutions KW - Sobolev spaces AB - We give an existence result of a renormalized solution for a class of nonlinear parabolic equations $$\frac{\partial b(x,u)}{\partial t}-div(a(x,t,u,\nabla u))+g(x,t,u,\nabla u)+H(x,t,\nabla u)=f,\qquad in\; Q_T,$$ where the right side belongs to $L^{p'}(0,T;W^{-1,p'}(Ω))$ and where b(x,u) is unbounded function of u and where $-div(a(x,t,u,∇u))$ is a Leray-Lions type operatorwith growth $|∇u|^{p-1}$ in ∇u. The critical growth condition on g is with respect to ∇u and no growth condition with respect to u, while the function $H(x,t,∇u)$ grows as $|∇u|^{p-1}$.
Akdim , YoussefBenkirane , A.EL Moumni , M. and Redwane , Hicham. (2014). Existence of Renormalized Solutions for Nonlinear Parabolic Equations. Journal of Partial Differential Equations. 27 (1). 28-49. doi:10.4208/jpde.v27.n1.2
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