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Volume 29, Issue 3
Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

Y. Akdim, J. Bennouna, A. Bouajaja & M. Mekkour

Anal. Theory Appl., 29 (2013), pp. 234-254.

Published online: 2013-07

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  • Abstract

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form$$ \left\{\begin{array}{ll} \dfrac{\partial b_1(x,u_1)}{\partial t}- \mathop{div}\big(a(x,t,u_1,Du_1)\big)+\mathop{div}\big(\Phi_1(u_1)\big)+ f_1(x,u_1,u_2)= 0  &  \quad\text{in}\ \  Q, \\\dfrac{\partial b_2(x,u_2)}{\partial t}- \mathop{div}\big(a(x,t,u_2,Du_2)\big)+\mathop{div}\big(\Phi_2(u_2)\big)+ f_2(x,u_1,u_2)= 0  & \quad\text{in}\ \  Q, \\\end{array}\right.$$in the framework of weighted Sobolev spaces, where $b(x,u)$ is unbounded function on $u$, the Carathéodory function $a_i$ satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function $\phi_i$ is assumed to be continuous on $\mathbb{R}$ and not belong to $(L^1_{loc}(Q))^N$.

  • AMS Subject Headings

35K45, 35K61, 35K65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

validee.de@uhp.ac.ma (A. Bouajaja)

  • BibTex
  • RIS
  • TXT
@Article{ATA-29-234, author = {Y. Akdim , J. Bennouna , Bouajaja , A. and Mekkour , M.}, title = {Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces}, journal = {Analysis in Theory and Applications}, year = {2013}, volume = {29}, number = {3}, pages = {234--254}, abstract = {

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form$$ \left\{\begin{array}{ll} \dfrac{\partial b_1(x,u_1)}{\partial t}- \mathop{div}\big(a(x,t,u_1,Du_1)\big)+\mathop{div}\big(\Phi_1(u_1)\big)+ f_1(x,u_1,u_2)= 0  &  \quad\text{in}\ \  Q, \\\dfrac{\partial b_2(x,u_2)}{\partial t}- \mathop{div}\big(a(x,t,u_2,Du_2)\big)+\mathop{div}\big(\Phi_2(u_2)\big)+ f_2(x,u_1,u_2)= 0  & \quad\text{in}\ \  Q, \\\end{array}\right.$$in the framework of weighted Sobolev spaces, where $b(x,u)$ is unbounded function on $u$, the Carathéodory function $a_i$ satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function $\phi_i$ is assumed to be continuous on $\mathbb{R}$ and not belong to $(L^1_{loc}(Q))^N$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2013.v29.n3.4}, url = {http://global-sci.org/intro/article_detail/ata/5060.html} }
TY - JOUR T1 - Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces AU - Y. Akdim , AU - J. Bennouna , AU - Bouajaja , A. AU - Mekkour , M. JO - Analysis in Theory and Applications VL - 3 SP - 234 EP - 254 PY - 2013 DA - 2013/07 SN - 29 DO - http://doi.org/10.4208/ata.2013.v29.n3.4 UR - https://global-sci.org/intro/article_detail/ata/5060.html KW - Nonlinear parabolic system, existence, truncation, weighted Sobolev space, renormalized solution. AB -

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form$$ \left\{\begin{array}{ll} \dfrac{\partial b_1(x,u_1)}{\partial t}- \mathop{div}\big(a(x,t,u_1,Du_1)\big)+\mathop{div}\big(\Phi_1(u_1)\big)+ f_1(x,u_1,u_2)= 0  &  \quad\text{in}\ \  Q, \\\dfrac{\partial b_2(x,u_2)}{\partial t}- \mathop{div}\big(a(x,t,u_2,Du_2)\big)+\mathop{div}\big(\Phi_2(u_2)\big)+ f_2(x,u_1,u_2)= 0  & \quad\text{in}\ \  Q, \\\end{array}\right.$$in the framework of weighted Sobolev spaces, where $b(x,u)$ is unbounded function on $u$, the Carathéodory function $a_i$ satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function $\phi_i$ is assumed to be continuous on $\mathbb{R}$ and not belong to $(L^1_{loc}(Q))^N$.

Y. Akdim, J. Bennouna, A. Bouajaja & M. Mekkour. (1970). Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces. Analysis in Theory and Applications. 29 (3). 234-254. doi:10.4208/ata.2013.v29.n3.4
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