Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities
2016 ◽
Vol 34
(1)
◽
pp. 225-252
◽
Keyword(s):
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.
2014 ◽
Vol 38
(14)
◽
pp. 3043-3062
◽
Keyword(s):
2020 ◽
pp. 257-274
◽
2009 ◽
Vol 22
(4)
◽
pp. 376-392
◽