<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>

On some parabolic problems with measure sources

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we introduce the notion of strongly Lipschitz up-nuclear operators. Among other results, we prove an analog of the factorization theorem for these classes and characterize their conjugates.</abstract>

Strongly Lipschitz up-Nuclear Operators

<abstract xmlns="http://www.w3.org/1999/xhtml">In this work, we prove a fixed point theorem for mapping with cyclical conditions using comparison function in <italic>b</italic>-Menger spaces. We support our results by an example.</abstract>

Fixed point theorem satisfying cyclical conditions in <italic>b</italic>-Menger spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">A modified system of nonlinear fractional-order differential equations become used to classify humans of various personalities and different Impact Factors of Memory (IFM); with unique set of model parameters. The model was used to interpret and predict the functions of the union of various people with external circumstance(s) and adapted to neighborhood environment in which the statistics collections were achieved to analyze numerous measures affecting marriages, unique challenges in marriage and associated occasions were investigated through the use of questionnaire. Data had been analyzed and the outcomes have been carried out as parameters to validate the model. Adams Predictor-Corrector Method was used to test the chaoticity of the system and it was confirmed via numerical simulations.Numerical simulation outcomes had been presented to reveal the effectiveness of the model and the accuracy of the statements established. The mathematical information implied by the model unveil an underlying mechanism which can give an explanation for couple disruption in relationships that were initially deliberated to remain all the time. Despite the terrible aspects of relationships, some human beings were still satisfied in their intimate members of the family. The study was addressed on a field survey (use of questionnaires) and with the aid of interrogating the members one on one. A feasible path for future work is the choice to attain balance through modelling and to validate the results with the aid of numerical simulations.</abstract>

A Modified Fractional Differential Love Model

<abstract xmlns="http://www.w3.org/1999/xhtml">We present a general finite volume method to solve a coupled Stokes-Darcy problem, we propose two domains corresponding to fluid region and porous region with a physical intersection. At the contact interface between the fluid region and the porous media we impose two conditions; the first one is the normal continuity of the velocity and the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use Schwarz domain decomposition. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the properties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method.</abstract>

A Discrete Duality Finite Volume Method for Coupling Darcy and Stokes Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we introduced a generalization of the raised cosine distribution. We also provided its several distributional properties and characterizations, including percentiles and some applications.</abstract>

On a Generalized Raised Cosine Distribution: Some Properties, Characterizations and Applications

<abstract xmlns="http://www.w3.org/1999/xhtml">We provide sufficient conditions for the existence of periodic solutions for the class of Duffing differential equations<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0007_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>x</mi><mo>″</mo></msup><mo>+</mo><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mi>x</mi><mo>′</mo></msup><mo>+</mo><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mi>x</mi></mrow><mn>3</mn></msup><mo>=</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo></mrow></math><tex-math>x'' + c(t)x' + a(t)x + b(t){x^3} = h(t,x),</tex-math></alternatives></disp-formula>where the functions <italic>a</italic>(<italic>t</italic>), <italic>b</italic>(<italic>t</italic>), <italic>c</italic>(<italic>t</italic>) and <italic>h</italic>(<italic>t</italic>, <italic>x</italic>) are <italic>C</italic>2 and <italic>T</italic>–periodic in the variable <italic>t</italic>.</abstract>

Periodic solutions for a class of Duffing differential equations

<abstract xmlns="http://www.w3.org/1999/xhtml">We prove in weighted Orlicz-Sobolev spaces, the existence of entropy solution for a class of nonlinear elliptic equations of Leray-Lions type, with large monotonicity condition and right hand side <italic>f</italic> ∈ <italic>L</italic>1(Ω).</abstract>

Department of Mathematics, Faculty of Sciences Fez

Department of Mathematics, Faculty of Sciences El Jadida

On a nonlinear elliptic problems having large monotonocity with <italic>L</italic>1-data in weighted Orlicz-Sobolev spaces

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Moroccan Journal of Pure and Applied Analysis

Volume 5 (2019): Issue 1 (June 2019)

On some parabolic problems with measure sources

Strongly Lipschitz up-Nuclear Operators

Fixed point theorem satisfying cyclical conditions in b-Menger spaces

A Modified Fractional Differential Love Model

A Discrete Duality Finite Volume Method for Coupling Darcy and Stokes Equations

On a Generalized Raised Cosine Distribution: Some Properties, Characterizations and Applications

Periodic solutions for a class of Duffing differential equations

On a nonlinear elliptic problems having large monotonocity with L¹-data in weighted Orlicz-Sobolev spaces

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