<abstract xmlns="http://www.w3.org/1999/xhtml">One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure data<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0001_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mi>b</mi><msub><mrow><mrow><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></mrow><mi>t</mi></msub><mo>-</mo><mtext>div</mtext><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi></mrow></mtd><mtd><mrow><mtext>in </mtext><mi>Q</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>T</mi></mrow><mo>)</mo></mrow><mo>×</mo><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></mtd><mtd><mrow><mtext>on</mtext><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>T</mi></mrow><mo>)</mo></mrow><mo>×</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>u</mi></mrow><mn>0</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mtext>in</mtext><mi> </mi><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ {b{{\left( u \right)}_t} - {\rm{div}}\left( {a\left( {t,x,u,\nabla u} \right)} \right) + h\left( u \right) = \mu } \hfill &amp; {{\rm{in}}Q: = \left( {0,T} \right) \times {\rm{\Omega }},} \hfill \cr {u = 0} \hfill &amp; {{\rm{on}}\left( {0,T} \right) \times \partial {\rm{\Omega }},} \hfill \cr {b\left( u \right) = b\left( {{u_0}} \right)} \hfill &amp; {{\rm{in}}\,{\rm{\Omega }},} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>where Ω is an open bounded subset of ℝ<italic>N</italic>, <italic>N</italic> ≥ 2, <italic>T ></italic> 0 and <italic>Q</italic> is the cylinder (0, <italic>T</italic>) × Ω, Σ = (0, <italic>T</italic>) × ∂Ω being its lateral surface, the operator is modeled on the <italic>p−</italic>Laplacian with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0001_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>></mo><mn>2</mn><mo>-</mo><mfrac><mn>1</mn><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math><tex-math>p > 2 - {1 \over {N + 1}}</tex-math></alternatives></inline-formula>, μ is a Radon measure with bounded total variation on <italic>Q</italic>, <italic>b</italic> is a <italic>C</italic>1<italic>−</italic>increasing function which satisfies 0 &lt; <italic>b</italic>0 ≤ <italic>b′</italic>(<italic>s</italic>) ≤ <italic>b</italic>1 (for positive constants <italic>b</italic>0 and <italic>b</italic>1). We assume that <italic>b</italic>(<italic>u</italic>0) is an element of <italic>L</italic>1(Ω) and <italic>h</italic> : ℝ ↦ ℝ is a continuous function such that <italic>h</italic>(<italic>s</italic>) <italic>s</italic> ≥ 0 for every |<italic>s</italic>| ≥ <italic>L</italic> and <italic>L</italic> ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.</abstract>

One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure data{b(u)t-div(a(t,x,u,∇u))+h(u)=μin Q:=(0,T)×Ω,u=0on(0,T)×∂Ω,b(u)=b(u0)in Ω,\left\{ {\matrix{ {b{{\left( u \right)}_t} - {\rm{div}}\left( {a\left( {t,x,u,\nabla u} \right)} \right) + h\left( u \right) = \mu } \hfill & {{\rm{in}}Q: = \left( {0,T} \right) \times {\rm{\Omega }},} \hfill \cr {u = 0} \hfill & {{\rm{on}}\left( {0,T} \right) \times \partial {\rm{\Omega }},} \hfill \cr {b\left( u \right) = b\left( {{u_0}} \right)} \hfill & {{\rm{in}}\,{\rm{\Omega }},} \hfill \cr } } \right.where Ω is an open bounded subset of ℝN, N ≥ 2, T > 0 and Q is the cylinder (0, T) × Ω, Σ = (0, T) × ∂Ω being its lateral surface, the operator is modeled on the p−Laplacian with p>2-1N+1p > 2 - {1 \over {N + 1}}, μ is a Radon measure with bounded total variation on Q, b is a C1−increasing function which satisfies 0 < b0 ≤ b′(s) ≤ b1 (for positive constants b0 and b1). We assume that b(u0) is an element of L1(Ω) and h : ℝ ↦ ℝ is a continuous function such that h(s) s ≥ 0 for every |s| ≥ L and L ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.

On some parabolic problems with measure sources

University of Fez, Faculty of Sciences Dhar El Mahraz

Moroccan Journal of Pure and Applied Analysis

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

{"article-title":"On some parabolic problems with measure sources"}

One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been...

On some parabolic problems with measure sources

Published Online: Nov 08, 2019

Page range: 1 - 21

Received: May 05, 2019

Accepted: Jun 09, 2019

DOI: https://doi.org/10.2478/mjpaa-2019-0001

Keywords
Generalized solutions, Measures and capacities, Nonlinear parabolic equations, Pseudo-monotone operators, Stability

© 2019 Mohammed Abdellaoui, published by Sciendo

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

On some parabolic problems with measure sources

Published Online: Nov 08, 2019

Page range: 1 - 21

Received: May 05, 2019

Accepted: Jun 09, 2019

DOI: https://doi.org/10.2478/mjpaa-2019-0001

KeywordsGeneralized solutions, Measures and capacities, Nonlinear parabolic equations, Pseudo-monotone operators, Stability

© 2019 Mohammed Abdellaoui, published by Sciendo

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

Keywords
Generalized solutions, Measures and capacities, Nonlinear parabolic equations, Pseudo-monotone operators, Stability