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

  • A. Aberqi
  • J. Bennouna
  • M. ElmassoudiEmail author
  • M. Hammoumi
Article
  • 11 Downloads

Abstract

In this work, we shall be concerned with the existence and uniqueness result to the nonlinear parabolic equations whose prototype is
$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial b( u)}{\partial t} -\varDelta _{M}u - \text{ div } \left( \overline{c}(x,t)\overline{M}^{-1}M\left( \frac{\alpha _{0}}{\lambda }|b(u)|\right) \right) =f &{} \text {in}&{} Q_{T},\\ \displaystyle u(x,t)=0 &{} \text {on} &{} \partial \varOmega \times (0,T),\\ b(u)(t=0)=b(u_{0}) &{} \text {in} &{} \varOmega , \end{array} \right. \end{aligned}$$
where \( -\,\varDelta _{M}u{=}-\,\text{ div } ((1+|u|)^{2}Du\frac{\log (e+Du)}{|Du|})\), \(\overline{c}\in (L^{\infty }(Q_{T}))^{N}\) and \(M(t)= t\log (e+t)\) is an N-function. The data f and \(b(u_{0})\) in \(L^{1}(Q_T)\) and \(L^{1}(\varOmega )\).

Keywords

Nonlinear parabolic equations Orlicz spaces Renormalized solutions Uniqueness 

Mathematics Subject Classification

Primary 47A15 Secondary 46A32 47D20 

Notes

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire LISA, School of Mathematical SciencesSidi Mohammed Ben Abdellah UniversityAtlas FezMorocco
  2. 2.Laboratoire LAMA, FSDMSidi Mohammed Ben Abdellah UniversityAtlas FezMorocco

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