Uniqueness of the renormalized solution of an elliptic-parabolic problem in anisotropic Sobolev-Orlicz spaces

2017 ◽  
Vol 208 (8) ◽  
pp. 1187-1206 ◽  
Author(s):  
F Kh Mukminov
Keyword(s):  
Parabolic Problem ◽  
Orlicz Spaces ◽  
Renormalized Solution

Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces

2019 ◽  
Vol 189 (2) ◽  
pp. 195-219 ◽  
Author(s):  
A. Aberqi ◽  
J. Bennouna ◽  
M. Elmassoudi ◽  
M. Hammoumi
Keyword(s):  
Existence And Uniqueness ◽  
Orlicz Spaces ◽  
Parabolic Problems ◽  
Renormalized Solution

scholarly journals Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

2016 ◽  
Vol 34 (1) ◽  
pp. 225-252 ◽  
Author(s):  
Youssef Akdim ◽  
Nezha El Gorch ◽  
Mounir Mekkour
Keyword(s):  
Parabolic Equation ◽  
Sobolev Space ◽  
Growth Condition ◽  
Parabolic Problem ◽  
Variable Exponent ◽  
Critical Growth ◽  
Renormalized Solutions ◽  
Renormalized Solution ◽  
Coercivity Condition ◽  
Sign Condition

In this article, we study the existence of a renormalized solution for the nonlinear $p(x)$-parabolic problem associated to the equation: $$\frac{\partial b(x,u)}{\partial t} - \mbox{div} (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - \mbox{div}F \;\mbox{in }\;Q= \Omega\times(0,T)$$with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on $ H(x,t,u,\nabla u)$\; is with respect to$ \nabla u$, no growth with respect to $u$ and no sign condition or the coercivity condition.

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