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

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 $(L^1_{loc}(Q))^N$.

• Keywords

Nonlinear parabolic system, existence, truncation, weighted Sobolev space, renormalized solution.

35K45, 35K61, 35K65

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@Article{ATA-29-234, author = { and Y. Akdim and and 21648 and and Y. Akdim and and J. Bennouna and and 21649 and and J. Bennouna and and A. Bouajaja and and 21650 and and A. Bouajaja and M. and Mekkour and and 21651 and and M. Mekkour}, 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 - A. Bouajaja , 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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