Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces
Year: 2013
Author: Y. Akdim, J. Bennouna, A. Bouajaja, M. Mekkour
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 3 : pp. 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 $(L^1_{loc}(Q))^N$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2013.v29.n3.4
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 3 : pp. 234–254
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Nonlinear parabolic system existence truncation weighted Sobolev space renormalized solution.