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

Akdim, Youssef; Gorch, Nezha El; Mekkour, Mounir

doi:10.5269/bspm.v34i1.23667

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“…In the degenerate Sobolev-spaces an existence results is shown in [6] without sign condition in H(x, t, u, ∇u).…”

Section: Statement Of the Problemmentioning

confidence: 98%

Existence of entropy solutions in Musielak Orlicz spaces via a sequence of penalized equations

Elmassoudi¹,

Aberqi²,

Bennouna³

2019

bspm

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This paper, is devoted to an existence result of entropy unilateral solutions for the nonlinear parabolic problems with obstacle in Musielak- Orlicz--spaces:$$ \partial_{t}u + A(u) + H(x,t,u,\nabla u) =f + div(\Phi(x,t,u))$$and $$ u\geq \zeta \,\,\mbox{a.e. in }\,\,Q_T.$$Where $A$ is a pseudomonotone operator of Leray-Lions type defined in the inhomogeneous Musielak-Orlicz space $W_{0}^{1,x}L_{\varphi}(Q_{T})$,$H(x,t,s,\xi)$ and $\phi(x,t,s)$ are only assumed to be Crath\'eodory's functions satisfying only the growth conditions prescribed by Musielak-Orlicz functions $\varphi$ and $\psi$ which inhomogeneous and does not satisfies $\Delta_2$-condition. The data $f$ and $u_{0}$ are still taken in $L^{1}(Q_T)$ and $L^{1}(\Omega)$.

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“…In the degenerate Sobolev-spaces an existence results is shown in [6] without sign condition in H(x, t, u, ∇u).…”

Section: Statement Of the Problemmentioning

confidence: 98%