Journal of Innovative Applied Mathematics and Computational Sciences http://jiamcs.centre-univ-mila.dz/index.php/jiamcs <p>The Journal of Innovative Applied Mathematics and Computational Sciences (JIAMCS) is an online open access, peer-reviewed semiannual international journal published by<span style="font-size: 0.875rem;"> </span>the Institute of Sciences and Technology, University Center Abdelhafid Boussouf, MILA, ALGERIA.</p> <p>The journal publishes high-quality original research papers from various fields related to applied mathematics, scientific computing, and computer science.</p> <p>In particular, it publishes original papers in the following areas:</p> <ul> <li>Differential equations, (ODE’s, PDE’s, integral equations, difference equations, fractional differential equations)</li> <li>Dynamical systems and bifurcation-chaos theory</li> <li>Fractional calculus</li> <li>Operator's theory</li> <li>Mathematical physics</li> <li>Probability and statistics</li> <li>Mathematical modelling and simulation</li> <li>Numerical analysis</li> <li>Fuzzy logic and systems</li> <li>Lattice theory, number theory and cryptography</li> <li>Modern control theory and practice</li> <li>Mathematical optimization</li> <li>Combinatorial optimisation</li> <li>Computer science</li> <li>Graph theory</li> <li>Bioinformatics</li> <li>Artificial intelligence</li> <li>Pattern recognition</li> <li>Computer vision and image processing</li> <li>Naturel language processing</li> <li>Signal processing</li> <li>Communication theory and network security</li> </ul> <p class="wp-block-columns"><strong>Editor-in-chief: </strong></p> <p>Mohammed Salah Abdelouahab, University Center Abdelhafid Boussouf, Mila, Algeria.</p> <p class="wp-block-columns"><strong>Associate Editors-in-Chief</strong></p> <p><strong>–</strong> Haci Mehmet Baskonus, Harran University, Turkey</p> <p><strong>–</strong> Aissa Boulmerka, University Center Abdelhafid Boussouf, Mila, Algeria</p> <p><strong>– </strong> René Lozi, Université Côte d’Azur, Parc Valrose, Nice, France</p> <p><strong>Published by: </strong></p> <p>Institute of Sciences and Technology, University Center Abdelhafid Boussouf, MILA, ALGERIA.</p> <p><strong>Periodicity: </strong></p> <p>Biannually.</p> Institute of Sciences and Technology, University Center Abdelhafid Boussouf , MILA, ALGERIA. en-US Journal of Innovative Applied Mathematics and Computational Sciences 2773-4196 <ul> <li>Authors keep the rights and guarantee the Journal of Innovative Applied Mathematics and Computational Sciences <em><strong> </strong></em>the right to be the first publication of the document, licensed under a <a href="http://creativecommons.org/licenses/by-nc-nd/4.0/" rel="license">Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License</a> that allows others to share the work with an acknowledgement of authorship and publication in the journal.</li> <li>Authors are allowed and encouraged to spread their work through electronic means using personal or institutional websites (institutional open archives, personal websites or professional and academic networks profiles) once the text has been published.</li> </ul> On weighted Calderón-Zygmund singular integrals and applications http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i1_12 <p>This paper studies some weighted norm inequalities related to some Calderon-Zygmund singular integrals. Applications to the Sobolev-Gagliardo-Nirenberg inequality, differential forms, and the potential equation du = f are given.</p> Ahmed Loulit Copyright (c) 2022 Ahmed Loulit https://creativecommons.org/licenses/by-nc-nd/4.0 2022-02-20 2022-02-20 2 1 1 26 Measure of noncompactness for nonlinear Hilfer fractional differential equation with mixed fractional integral boundary conditions in Banach space http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i1_9 <p>The aim of this work is to study the existence of solutions to a class of fractional differential equations with a mixed of fractional integral boundary conditions involving the Hilfer fractional derivative. The proof is based on Monch's fixed point theorem and the technique of measures of noncompactness. Two examples illustrating the main results are also constructed.</p> <p> </p> Maamar Benbachir Abdelatif Boutiara Copyright (c) 2022 Maamar Benbachir, Abdelatif Boutiara https://creativecommons.org/licenses/by-nc-nd/4.0 2022-05-15 2022-05-15 2 1 27 42 Exponential growth of solution for a couple of semi-linear pseudo-parabolic equations with memory and source terms http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i1_23 <p>This work is concerned with coupled semi-linear pseudo-parabolic equations with memory terms in both equations, associated with the homogeneous Dirichlet boundary condition. We show that the solution grows exponentially under specific conditions<br />regarding the relaxation functions and initial energy. In order to prove the result, we use the energy method based on the construction of a suitable Lyapunov function.<br />The most important behavior of the evolution system is the exponential growth phenomena because of its wide range of applications in modern science, such as chemistry, biology, ecology, and other areas of engineering and physical sciences.</p> Amar OUAOUA Wissem BOUGHAMSA Copyright (c) 2022 Amar OUAOUA, Wissem BOUGHAMSA https://creativecommons.org/licenses/by-nc-nd/4.0 2022-05-18 2022-05-18 2 1 43 52 Taylor collocation method for high-order neutral delay Volterra integro-differential equations http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/10 <pre>In this paper, the Taylor collocation method is applied to numerically solve a kth-order neutral linear Volterra integro-differential equation with constant delay and variable coefficients. We also provide a rigorous analysis to estimate the difference between the exact and approximate solution and their derivatives up to order k-1. Numerical examples are included to prove the performance of the presented convergent algorithm.</pre> Hafida Laib Azzeddine Bellour Aissa Boulmerka Copyright (c) 2022 Hafida Laib, Azzeddine Bellour, Aissa Boulmerka https://creativecommons.org/licenses/by-nc-nd/4.0 2022-05-12 2022-05-12 2 1 53 77 The expressions and behavior of solutions for nonlinear systems of rational difference equations http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i1_24 <p>In this paper, we investigate the form of the solutions of the following systems of difference equations of second order</p> <div class="MathJax_Display" style="text-align: center;"><span id="MathJax-Element-1-Frame" class="MathJax" style="text-align: center; position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; 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top: -3.799em; left: 0.579em;"><span id="MathJax-Span-59" class="texatom"><span id="MathJax-Span-60" class="mrow"><span id="MathJax-Span-61" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span id="MathJax-Span-62" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-63" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-64" class="msubsup"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-65" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-66" class="texatom"><span id="MathJax-Span-67" class="mrow"><span id="MathJax-Span-68" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span></span></span><span style="position: absolute; clip: rect(3.229em, 1003.29em, 4.381em, -999.997em); top: -3.281em; left: 50%; margin-left: -1.668em;"><span id="MathJax-Span-69" class="mrow"><span id="MathJax-Span-70" class="msubsup"><span style="display: inline-block; position: relative; width: 1.097em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-71" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-72" class="texatom"><span id="MathJax-Span-73" class="mrow"><span id="MathJax-Span-74" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span><span id="MathJax-Span-75" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">+</span><span id="MathJax-Span-76" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-77" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-78" class="texatom"><span id="MathJax-Span-79" class="mrow"><span id="MathJax-Span-80" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span></span></span></span></span><span id="MathJax-Span-81" class="mo" style="font-family: MathJax_Main;">,</span></span></span></span><span style="position: absolute; clip: rect(2.653em, 1018.67em, 5.014em, -999.997em); top: -2.935em; left: 0em;"><span id="MathJax-Span-95" class="mtd"><span id="MathJax-Span-96" class="mrow"><span id="MathJax-Span-97" class="mfrac"><span style="display: inline-block; position: relative; width: 3.401em; height: 0px; margin-right: 0.118em; margin-left: 0.118em;"><span style="position: absolute; clip: rect(3.344em, 1002.94em, 4.381em, -999.997em); top: -4.663em; left: 50%; margin-left: -1.495em;"><span id="MathJax-Span-98" class="mrow"><span id="MathJax-Span-99" class="msubsup"><span style="display: inline-block; position: relative; width: 1.097em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-100" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-101" class="texatom"><span id="MathJax-Span-102" class="mrow"><span id="MathJax-Span-103" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span><span id="MathJax-Span-104" class="msubsup"><span style="display: inline-block; position: relative; width: 1.904em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-105" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-106" class="texatom"><span id="MathJax-Span-107" class="mrow"><span id="MathJax-Span-108" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span id="MathJax-Span-109" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-110" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span></span></span><span style="position: absolute; clip: rect(3.229em, 1003.29em, 4.381em, -999.997em); top: -3.281em; left: 50%; margin-left: -1.668em;"><span id="MathJax-Span-111" class="mrow"><span id="MathJax-Span-112" class="msubsup"><span style="display: inline-block; position: relative; width: 1.097em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-113" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-114" class="texatom"><span id="MathJax-Span-115" class="mrow"><span id="MathJax-Span-116" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span><span id="MathJax-Span-117" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-118" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-119" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-120" class="texatom"><span id="MathJax-Span-121" class="mrow"><span id="MathJax-Span-122" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span></span></span></span></span><span id="MathJax-Span-123" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-124" class="mspace" style="height: 0em; vertical-align: 0em; width: 0.982em; display: inline-block; overflow: hidden;"></span><span id="MathJax-Span-125" class="msubsup" style="padding-left: 0.176em;"><span style="display: inline-block; position: relative; width: 1.904em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-126" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-127" class="texatom"><span id="MathJax-Span-128" class="mrow"><span id="MathJax-Span-129" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span id="MathJax-Span-130" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span id="MathJax-Span-131" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-132" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-133" class="mfrac" style="padding-left: 0.291em;"><span style="display: inline-block; position: relative; width: 3.401em; height: 0px; margin-right: 0.118em; margin-left: 0.118em;"><span style="position: absolute; clip: rect(3.344em, 1002.94em, 4.381em, -999.997em); top: -4.663em; left: 50%; margin-left: -1.495em;"><span id="MathJax-Span-134" class="mrow"><span id="MathJax-Span-135" class="msubsup"><span style="display: inline-block; position: relative; width: 1.961em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-136" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-137" class="texatom"><span id="MathJax-Span-138" class="mrow"><span id="MathJax-Span-139" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span><span id="MathJax-Span-140" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-141" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-142" class="msubsup"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-143" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-144" class="texatom"><span id="MathJax-Span-145" class="mrow"><span id="MathJax-Span-146" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span></span></span><span style="position: absolute; clip: rect(3.229em, 1003.29em, 4.381em, -999.997em); top: -3.281em; left: 50%; margin-left: -1.668em;"><span id="MathJax-Span-147" class="mrow"><span id="MathJax-Span-148" class="msubsup"><span style="display: inline-block; position: relative; width: 1.097em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-149" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-150" class="texatom"><span id="MathJax-Span-151" class="mrow"><span id="MathJax-Span-152" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span><span id="MathJax-Span-153" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-154" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-155" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-156" class="texatom"><span id="MathJax-Span-157" class="mrow"><span id="MathJax-Span-158" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">n</span></span></span></span></span></span></span></span></span></span><span id="MathJax-Span-159" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-160" class="mtext" style="font-family: MathJax_Main; padding-left: 0.176em;"> </span><span id="MathJax-Span-161" class="mtext" style="font-family: MathJax_Main;"> </span><span id="MathJax-Span-162" class="mtext" style="font-family: MathJax_Main;"> </span><span id="MathJax-Span-163" class="mi" style="font-family: MathJax_Math-italic;">n</span><span id="MathJax-Span-164" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-165" class="mn" style="font-family: MathJax_Main; padding-left: 0.291em;">0</span><span id="MathJax-Span-166" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-167" class="mn" style="font-family: MathJax_Main; padding-left: 0.176em;">1</span><span id="MathJax-Span-168" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-169" class="mo" style="font-family: MathJax_Main; padding-left: 0.176em;">.</span><span id="MathJax-Span-170" class="mo" style="font-family: MathJax_Main; padding-left: 0.176em;">.</span><span id="MathJax-Span-171" class="mo" style="font-family: MathJax_Main; padding-left: 0.176em;">.</span><span id="MathJax-Span-172" class="mo" style="font-family: MathJax_Main; padding-left: 0.176em;">,</span></span></span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation">xn+1=xnyn−1xn+yn,yn+1=xn−1ynxn+yn,xn+1=xnyn−1xn−yn,yn+1=xn−1ynxn−yn, n=0,1,...,</span></span></div> <p><br />where the initial conditions <span id="MathJax-Element-2-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;/math&gt;"><span id="MathJax-Span-173" class="math" style="width: 7.031em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.648em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1005.42em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-174" class="mrow"><span id="MathJax-Span-175" class="msubsup"><span style="display: inline-block; position: relative; width: 1.558em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-176" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-177" class="texatom"><span id="MathJax-Span-178" class="mrow"><span id="MathJax-Span-179" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-180" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-181" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-182" class="mtext" style="font-family: MathJax_Main; padding-left: 0.176em;"> </span><span id="MathJax-Span-183" class="msubsup"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-184" class="mi" style="font-family: MathJax_Math-italic;">x</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-185" class="texatom"><span id="MathJax-Span-186" class="mrow"><span id="MathJax-Span-187" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span></span></span></span><span id="MathJax-Span-188" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-189" class="mtext" style="font-family: MathJax_Main; padding-left: 0.176em;"> </span><span id="MathJax-Span-190" class="msubsup"><span style="display: inline-block; position: relative; width: 1.443em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-191" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-192" class="texatom"><span id="MathJax-Span-193" class="mrow"><span id="MathJax-Span-194" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-195" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-196" class="mtext" style="font-family: MathJax_Main;"> </span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">x−1, x0, y−1 </span></span> and <span id="MathJax-Element-3-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"><span id="MathJax-Span-197" class="math" style="width: 1.443em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.155em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1001.15em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-198" class="mrow"><span id="MathJax-Span-199" class="mtext" style="font-family: MathJax_Main;"> </span><span id="MathJax-Span-200" class="msubsup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-201" class="mi" style="font-family: MathJax_Math-italic;">y</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-202" class="texatom"><span id="MathJax-Span-203" class="mrow"><span id="MathJax-Span-204" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"> y0</span></span> are arbitrary nonzero real numbers.</p> Kholoud N. Alharbi E. M. Elsayed Copyright (c) 2022 Kholoud N. Alharbi, E. M. Elsayed https://creativecommons.org/licenses/by-nc-nd/4.0 2022-06-23 2022-06-23 2 1 78 91 On a viscoelastic plate equation with a polynomial source term and p (x,t)-Laplacian operator in the presence of delay term http://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i1_30 <p>In this paper, the blow-up of solutions for the following Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of <span id="MathJax-Element-5-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-38" class="math" style="width: 3.459em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.768em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.385em, 1002.65em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-39" class="mrow"><span id="MathJax-Span-40" class="texatom"><span id="MathJax-Span-41" class="mrow"><span id="MathJax-Span-42" class="munderover"><span style="display: inline-block; position: relative; width: 0.579em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.323em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-43" class="mi" style="font-family: MathJax_Math-italic;">p</span></span><span style="position: absolute; top: -4.029em; left: 0.118em;"><span id="MathJax-Span-44" class="mo" style="font-family: MathJax_Main;">⃗ </span></span></span></span></span></span><span id="MathJax-Span-45" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-46" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-47" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-48" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.176em;">t</span><span id="MathJax-Span-49" class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">p→(x,t)</span></span>-Laplacian operator in the presence of time delay is obtained</p> <div class="MathJax_Display" style="text-align: center;"><span id="MathJax-Element-6-Frame" class="MathJax" style="text-align: center; position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0394;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0394;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;&amp;#x222B;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0394;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BC;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0394;&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03BC;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0394;&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C4;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-50" class="math" style="width: 43.378em; display: inline-block;"><span style="display: inline-block; position: relative; width: 34.968em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(0.637em, 1034.91em, 3.459em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-51" class="mrow"><span id="MathJax-Span-52" class="msubsup"><span style="display: inline-block; position: relative; width: 1.155em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-53" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-54" class="texatom"><span id="MathJax-Span-55" class="mrow"><span id="MathJax-Span-56" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span id="MathJax-Span-57" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span></span></span></span><span id="MathJax-Span-58" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">+</span><span id="MathJax-Span-59" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 1.27em; height: 0px;"><span style="position: absolute; clip: rect(3.113em, 1000.81em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-60" class="mi" style="font-family: MathJax_Main;">Δ</span></span><span style="position: absolute; top: -4.375em; left: 0.809em;"><span id="MathJax-Span-61" class="texatom"><span id="MathJax-Span-62" class="mrow"><span id="MathJax-Span-63" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span><span id="MathJax-Span-64" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-65" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">+</span><span id="MathJax-Span-66" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 2.71em; height: 0px;"><span style="position: absolute; clip: rect(3.113em, 1000.81em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-67" class="mi" style="font-family: MathJax_Main;">Δ</span></span><span style="position: absolute; top: -3.741em; left: 0.809em;"><span id="MathJax-Span-68" class="texatom"><span id="MathJax-Span-69" class="mrow"><span id="MathJax-Span-70" class="texatom"><span id="MathJax-Span-71" class="mrow"><span id="MathJax-Span-72" class="munderover"><span style="display: inline-block; position: relative; width: 0.406em; height: 0px;"><span style="position: absolute; clip: rect(3.517em, 1000.35em, 4.266em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-73" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">p</span></span><span style="position: absolute; top: -4.087em; left: 0.118em;"><span id="MathJax-Span-74" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">⃗ </span></span></span></span></span></span><span id="MathJax-Span-75" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span id="MathJax-Span-76" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-77" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span id="MathJax-Span-78" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span id="MathJax-Span-79" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">)</span></span></span></span></span></span><span id="MathJax-Span-80" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-81" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-82" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 1.443em; height: 0px;"><span style="position: absolute; clip: rect(2.422em, 1000.92em, 5.014em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-83" class="mo" style="font-family: MathJax_Size2; vertical-align: 0.003em;">∫</span></span><span style="position: absolute; clip: rect(3.344em, 1000.35em, 4.15em, -999.997em); top: -5.066em; left: 1.097em;"><span id="MathJax-Span-84" class="texatom"><span id="MathJax-Span-85" class="mrow"><span id="MathJax-Span-86" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span></span><span style="position: absolute; clip: rect(3.344em, 1000.41em, 4.15em, -999.997em); top: -3.05em; left: 0.579em;"><span id="MathJax-Span-87" class="texatom"><span id="MathJax-Span-88" class="mrow"><span id="MathJax-Span-89" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">0</span></span></span></span></span></span><span id="MathJax-Span-90" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.176em;">g</span><span id="MathJax-Span-91" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-92" class="mi" style="font-family: MathJax_Math-italic;">t</span><span id="MathJax-Span-93" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-94" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.233em;">s</span><span id="MathJax-Span-95" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-96" class="msubsup"><span style="display: inline-block; position: relative; width: 1.27em; height: 0px;"><span style="position: absolute; clip: rect(3.113em, 1000.81em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-97" class="mi" style="font-family: MathJax_Main;">Δ</span></span><span style="position: absolute; top: -4.375em; left: 0.809em;"><span id="MathJax-Span-98" class="texatom"><span id="MathJax-Span-99" class="mrow"><span id="MathJax-Span-100" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span><span id="MathJax-Span-101" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-102" class="mrow" style="padding-left: 0.176em;"><span id="MathJax-Span-103" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-104" class="mi" style="font-family: MathJax_Math-italic;">s</span><span id="MathJax-Span-105" class="mo" style="font-family: MathJax_Main;">)</span></span><span id="MathJax-Span-106" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.176em;">d</span><span id="MathJax-Span-107" class="mi" style="font-family: MathJax_Math-italic;">s</span><span id="MathJax-Span-108" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-109" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-110" class="mi" style="font-family: MathJax_Math-italic;">μ</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-111" class="texatom"><span id="MathJax-Span-112" class="mrow"><span id="MathJax-Span-113" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span><span id="MathJax-Span-124" class="mi" style="font-family: MathJax_Main;">Δ</span><span id="MathJax-Span-125" class="msubsup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-126" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-127" class="texatom"><span id="MathJax-Span-128" class="mrow"><span id="MathJax-Span-129" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span></span></span></span><span id="MathJax-Span-130" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-131" class="msubsup" style="padding-left: 0.233em;"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-132" class="mi" style="font-family: MathJax_Math-italic;">μ</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-133" class="texatom"><span id="MathJax-Span-134" class="mrow"><span id="MathJax-Span-135" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span><span id="MathJax-Span-136" class="mi" style="font-family: MathJax_Main;">Δ</span><span id="MathJax-Span-137" class="msubsup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-138" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-139" class="texatom"><span id="MathJax-Span-140" class="mrow"><span id="MathJax-Span-141" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span></span></span></span></span></span><span id="MathJax-Span-142" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-143" class="mi" style="font-family: MathJax_Math-italic;">t</span><span id="MathJax-Span-144" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">−</span><span id="MathJax-Span-145" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.233em;">τ</span><span id="MathJax-Span-146" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-147" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-148" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.291em;">u</span><span id="MathJax-Span-149" class="msubsup"><span style="display: inline-block; position: relative; width: 2.422em; height: 0px;"><span style="position: absolute; clip: rect(3.056em, 1001.04em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-150" class="mrow"><span id="MathJax-Span-151" class="mo" style="font-family: MathJax_Main;">|</span><span id="MathJax-Span-152" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-153" class="mo" style="font-family: MathJax_Main;">|</span></span></span><span style="position: absolute; top: -4.433em; left: 1.155em;"><span id="MathJax-Span-154" class="texatom"><span id="MathJax-Span-155" class="mrow"><span id="MathJax-Span-156" class="mi" style="font-size: 70.7%; font-family: MathJax_Math-italic;">q</span><span id="MathJax-Span-157" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span id="MathJax-Span-158" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span><span id="MathJax-Span-159" class="mo" style="font-family: MathJax_Main;">.</span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation">utt+Δ2u+Δp→(x,t)u−∫0tg(t−s)Δ2u(s)ds−μ1Δut−μ2Δut(t−τ)=u|u|q−2.</span></span></div> <p><br />Under suitable conditions on <span id="MathJax-Element-7-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;"><span id="MathJax-Span-114" class="math" style="width: 0.579em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1000.46em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-115" class="mrow"><span id="MathJax-Span-116" class="mi" style="font-family: MathJax_Math-italic;">g</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">g</span></span> and the variable exponent of the <span id="MathJax-Element-8-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-117" class="math" style="width: 4.381em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.517em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.385em, 1003.46em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-118" class="mrow"><span id="MathJax-Span-119" class="texatom"><span id="MathJax-Span-120" class="mrow"><span id="MathJax-Span-121" class="munderover"><span style="display: inline-block; position: relative; width: 0.579em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.52em, 4.323em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-122" class="mi" style="font-family: MathJax_Math-italic;">p</span></span><span style="position: absolute; top: -4.029em; left: 0.118em;"><span id="MathJax-Span-123" class="mo" style="font-family: MathJax_Main;">⃗ </span></span></span></span></span></span><span id="MathJax-Span-160" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-161" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-162" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-163" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.176em;">t</span><span id="MathJax-Span-164" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-165" class="mo" style="font-family: MathJax_Main;">−</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">p→(x,t)−</span></span> Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.</p> <p> </p> <p> </p> Ahlem Merah Fatiha Mesloub Copyright (c) 2022 Ahlem Merah, Fatiha Mesloub https://creativecommons.org/licenses/by-nc-nd/4.0 2022-06-30 2022-06-30 2 1 92 107