The existence of renormalized solution for quasilinear parabolic problem with variable exponents and measure data
Abstract
In this paper, the study of the existence of a renormalized solution for quasilinear parabolic
problem with variable exponents and measure data. The model is:
u_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda
\left\vert u\right\vert ^{p(x)-2}u=\mu
\text{ } &
\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[,\\
u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\
u(.,0)=u_{0}(.) & \text{in}\hspace{0.5cm}\Omega,
where $ \lambda>0$ and $ T $ is any positive constant, $ \mu\in\mathcal{M}_{0}(Q) $ is any measure with bounded variation over $ Q=\Omega \times ]0,T[ $.
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