Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form

Mohamed Saad Bouh Elemine Vall; Ahmed Ahmed; Abdelfattah Touzani; Abdelmoujib Benkirane

Mathematica Bohemica (2018)

  • Volume: 143, Issue: 3, page 225-249
  • ISSN: 0862-7959

Abstract

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We prove the existence of solutions to nonlinear parabolic problems of the following type: ∂ b ( u ) ∂ t + A ( u ) = f + div ( Θ ( x ; t ; u ) ) in Q , u ( x ; t ) = 0 on ∂ Ω × [ 0 ; T ] , b ( u ) ( t = 0 ) = b ( u 0 ) on Ω , where b : ℝ → ℝ is a strictly increasing function of class 𝒞 1 , the term A ( u ) = - div ( a ( x , t , u , ∇ u ) ) is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, Θ : Ω × [ 0 ; T ] × ℝ → ℝ is a Carathéodory, noncoercive function which satisfies the following condition: sup | s | ≤ k | Θ ( · , · , s ) | ∈ E ψ ( Q ) for all k > 0 , where ψ is the Musielak complementary function of Θ , and the second term f belongs to L 1 ( Q ) .

How to cite

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Elemine Vall, Mohamed Saad Bouh, et al. "Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form." Mathematica Bohemica 143.3 (2018): 225-249. <http://eudml.org/doc/294395>.

@article{ElemineVall2018,
abstract = {We prove the existence of solutions to nonlinear parabolic problems of the following type: \[ \{\left\lbrace \begin\{array\}\{ll\} \dfrac\{\partial b(u)\}\{\partial t\}+ A(u) = f + \{\rm div\}(\Theta (x; t; u))& \text\{in\}\ Q,\\ u(x; t) = 0 & \text\{on\}\ \partial \Omega \times [0; T],\\ b(u)(t = 0) = b(u\_0) & \text\{on\}\ \Omega , \end\{array\}\right.\} \] where $b\colon \mathbb \{R\}\rightarrow \mathbb \{R\}$ is a strictly increasing function of class $\{\mathcal \{C\}\}^1$, the term \[ A(u) = -\{\rm div\} (a(x, t, u,\nabla u)) \] is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta \colon \Omega \times [0; T]\times \mathbb \{R\}\rightarrow \mathbb \{R\}$ is a Carathéodory, noncoercive function which satisfies the following condition: $\sup _\{|s|\le k\} |\Theta (\{\cdot \},\{\cdot \},s)| \in E_\{\psi \}(Q)$ for all $k > 0$, where $\psi $ is the Musielak complementary function of $\Theta $, and the second term $f$ belongs to $L^\{1\}(Q)$.},
author = {Elemine Vall, Mohamed Saad Bouh, Ahmed, Ahmed, Touzani, Abdelfattah, Benkirane, Abdelmoujib},
journal = {Mathematica Bohemica},
keywords = {inhomogeneous Musielak-Orlicz-Sobolev space; parabolic problems; Galerkin method},
language = {eng},
number = {3},
pages = {225-249},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form},
url = {http://eudml.org/doc/294395},
volume = {143},
year = {2018},
}

TY - JOUR
AU - Elemine Vall, Mohamed Saad Bouh
AU - Ahmed, Ahmed
AU - Touzani, Abdelfattah
AU - Benkirane, Abdelmoujib
TI - Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form
JO - Mathematica Bohemica
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 143
IS - 3
SP - 225
EP - 249
AB - We prove the existence of solutions to nonlinear parabolic problems of the following type: \[ {\left\lbrace \begin{array}{ll} \dfrac{\partial b(u)}{\partial t}+ A(u) = f + {\rm div}(\Theta (x; t; u))& \text{in}\ Q,\\ u(x; t) = 0 & \text{on}\ \partial \Omega \times [0; T],\\ b(u)(t = 0) = b(u_0) & \text{on}\ \Omega , \end{array}\right.} \] where $b\colon \mathbb {R}\rightarrow \mathbb {R}$ is a strictly increasing function of class ${\mathcal {C}}^1$, the term \[ A(u) = -{\rm div} (a(x, t, u,\nabla u)) \] is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta \colon \Omega \times [0; T]\times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory, noncoercive function which satisfies the following condition: $\sup _{|s|\le k} |\Theta ({\cdot },{\cdot },s)| \in E_{\psi }(Q)$ for all $k > 0$, where $\psi $ is the Musielak complementary function of $\Theta $, and the second term $f$ belongs to $L^{1}(Q)$.
LA - eng
KW - inhomogeneous Musielak-Orlicz-Sobolev space; parabolic problems; Galerkin method
UR - http://eudml.org/doc/294395
ER -

References

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