<abstract xmlns="http://www.w3.org/1999/xhtml">We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0015_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mi>u</mi><mo>≥</mo><mi>ψ</mi></mrow></mtd><mtd><mrow><mtext>in</mtext></mrow></mtd><mtd><mrow><msub><mrow><mi>Q</mi></mrow><mi>T</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>+</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mn>1</mn></msup><mrow><mo>(</mo><mrow><msub><mrow><mi>Q</mi></mrow><mi>T</mi></msub></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math><tex-math>$$\left\{ \matrix{ \matrix{ {u \ge \psi } \hfill &amp; {{\rm{in}}} \hfill &amp; {{Q_T},} \hfill \cr } \hfill \cr {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$</tex-math></alternatives></disp-formula>The growths of the monotone vector field <italic>a</italic>(<italic>x</italic>, <italic>t</italic>, <italic>u</italic>, ᐁ<italic>u</italic>) and the non-coercive vector field <italic>g</italic>(<italic>x</italic>, <italic>t</italic>, <italic>u</italic>) are controlled by a generalized nonhomogeneous <italic>N</italic>- function <italic>M</italic> (see (3.3)-(3.6)). The approach does not require any particular type of growth of <italic>M</italic> (neither Δ2 nor ᐁ2). The proof is based on penalization method.</abstract>

We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider{u≥ψinQT,∂b(x,u)∂t-div(a(x,t,u,∇u))=f+div(g(x,t,u))∈L1(QT).$$\left\{ \matrix{ \matrix{  {u \ge \psi } \hfill & {{\rm{in}}} \hfill & {{Q_T},} \hfill \cr } \hfill \cr  {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$The growths of the monotone vector field a(x, t, u, ᐁu) and the non-coercive vector field g(x, t, u) are controlled by a generalized nonhomogeneous N- function M (see (3.3)-(3.6)). The approach does not require any particular type of growth of M (neither Δ2 nor ᐁ2). The proof is based on penalization method.

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Département de Mathématiques, Laboratoire LAMA. Faculté des Sciences Dhar-Mahrez

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces"}

We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity...

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Published Online: May 16, 2019

Page range: 189 - 206

Received: Feb 25, 2019

Accepted: Apr 23, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0015

Keywords
Inhomogeneous Musielak-Orlicz space, Nonlinear Parabolic problems, Entropy solutions, Non-coercive Lower order term

© 2018 Ahmed Aberqi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Published Online: May 16, 2019

Page range: 189 - 206

Received: Feb 25, 2019

Accepted: Apr 23, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0015

KeywordsInhomogeneous Musielak-Orlicz space, Nonlinear Parabolic problems, Entropy solutions, Non-coercive Lower order term

© 2018 Ahmed Aberqi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Inhomogeneous Musielak-Orlicz space, Nonlinear Parabolic problems, Entropy solutions, Non-coercive Lower order term