Geodesic
Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form
Mathematica Bohemica, Tome 143 (2018) no. 3, pp. 225-249
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We prove the existence of solutions to nonlinear parabolic problems of the following type: $$ \begin {cases} \dfrac {\partial b(u)}{\partial t}+ A(u) = f + {\rm div}(\Theta (x; t; u)) \text {in}\ Q,\\ u(x; t) = 0 \text {on}\ \partial \Omega \times [0; T],\\ b(u)(t = 0) = b(u_0) \text {on}\ \Omega , \end {cases} $$ where $b\colon \Bbb {R}\to \Bbb {R}$ 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, $\Theta \colon \Omega \times [0; T]\times \Bbb {R}\to \Bbb {R}$ 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 $L^{1}(Q)$.
We prove the existence of solutions to nonlinear parabolic problems of the following type: $$ \begin {cases} \dfrac {\partial b(u)}{\partial t}+ A(u) = f + {\rm div}(\Theta (x; t; u)) \text {in}\ Q,\\ u(x; t) = 0 \text {on}\ \partial \Omega \times [0; T],\\ b(u)(t = 0) = b(u_0) \text {on}\ \Omega , \end {cases} $$ where $b\colon \Bbb {R}\to \Bbb {R}$ 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, $\Theta \colon \Omega \times [0; T]\times \Bbb {R}\to \Bbb {R}$ 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 $L^{1}(Q)$.
DOI : 10.21136/MB.2017.0087-16
Classification : 58J35, 65L60
Keywords: inhomogeneous Musielak-Orlicz-Sobolev space; parabolic problems; Galerkin method 
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     title = {Entropy solutions to parabolic equations in {Musielak} framework involving non coercivity term in divergence form},
     journal = {Mathematica Bohemica},
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Elemine Vall, Mohamed Saad Bouh; Ahmed, Ahmed; Touzani, Abdelfattah; Benkirane, Abdelmoujib. Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form. Mathematica Bohemica, Tome 143 (2018) no. 3, pp. 225-249. doi: 10.21136/MB.2017.0087-16

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