Electronic Journal of Qualitative Theory of Differential Equations最新文献

{"title":"Cauchy problem for nonlocal diffusion equations modelling Lévy flights","authors":"Chung‐Sik Sin","doi":"10.14232/ejqtde.2022.1.18","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.18","url":null,"abstract":"In the present paper, we study the time-space fractional diffusion equation involving the Caputo differential operator and the fractional Laplacian. This equation describes the Lévy flight with the Brownian motion component and the drift component. First, the asymptotic behavior of the fundamental solution of the fractional diffusion equation is considered. Then, we use the fundamental solution to obtain the representation formula of solutions of the Cauchy problem. In the last, the L 2 -decay estimates for solutions are proved by employing the Fourier analysis technique.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The existence of solutions for the modified ( p ( x )","authors":"G. Figueiredo, C. Vetro","doi":"10.14232/ejqtde.2022.1.39","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.39","url":null,"abstract":"<jats:p>We consider the Dirichlet problem <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mo>−<!-- − --></mml:mo> <mml:msubsup> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>p</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msubsup> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>q</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mspace width=\"1em\" /> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mtext>in </mml:mtext> </mml:mstyle> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=\"1em\" /> <mml:mi>u</mml:mi> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:math> driven by the sum of a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>p</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math>-Laplacian operator and of a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math>-Laplacian operator, both of them weighted by indefinite (sign-changing) Kirchhoff type","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Attractivity of solutions of Riemann–Liouville fractional\u0000 differential equations","authors":"\t\tEquations\t\t\tTao Zhu","doi":"10.14232/ejqtde.2022.1.52","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.52","url":null,"abstract":"Some new weakly singular integral inequalities are established by a new method, which generalize some results of this type in some previous papers. By these new integral inequalities, we present the attractivity of solutions for Riemann–Liouville fractional differential equations. Finally, several examples are given to illustrate our main results.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74630761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability results for the functional differential equations associated to water hammer in hydraulics","authors":"V. Răsvan","doi":"10.14232/ejqtde.2022.1.19","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.19","url":null,"abstract":"There is considered a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy–Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is \"weak\" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin–Krasovskii–LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Expansion of positivity to a class of doubly nonlinear parabolic equations","authors":"E. Henriques","doi":"10.14232/ejqtde.2022.1.15","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.15","url":null,"abstract":"<jats:p>We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>div</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mtext> </mml:mtext> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext>and</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> considering separately the two possible cases <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:math>. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>p</mml:mi> </mml:math>-Laplacian equation.</jats:p>","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global phase portraits of a predator–prey system","authors":"Érika Diz-Pita, J. Llibre, M. V. Otero-Espinar","doi":"10.14232/ejqtde.2022.1.16","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.16","url":null,"abstract":"We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator–prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On some differential equations involving a new kind of variable exponents","authors":"S. Aouaoui","doi":"10.14232/ejqtde.2022.1.23","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.23","url":null,"abstract":"In this paper, we are concerned with some new first order differential equation defined on the whole real axis R . The principal part of the equation involves an operator with variable exponent p depending on the variable x ∈ R through the unknown solution while the nonlinear part involves the classical variable exponent p ( x ) . Such kind of situation is very related to the presence of the variable exponent and has not been treated before. Our existence result of nontrivial solution cannot be reached using standard variational or topological methods of nonlinear analysis and some sophisticated arguments have to be employed.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lyapunov functionals and practical stability for stochastic\u0000 differential delay equations with general decay rate","authors":"\t\tEquations\t\t\tT. Caraballo, Faten Ezzine, M. Hammami","doi":"10.14232/ejqtde.2022.1.60","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.60","url":null,"abstract":"This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79694608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random invariant manifolds and foliations for slow-fast PDEs with\u0000 strong multiplicative noise","authors":"\t\tEquations\t\t\tWenlei Li, Shiduo Qu, S. Shi","doi":"10.14232/ejqtde.2022.1.70","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.70","url":null,"abstract":"This article is devoted to the dynamical behaviors of a class of slow-fast PDEs perturbed by strong multiplicative noise. We will accomplish the existence of random invariant manifolds and foliations, and show exponential tracking property of them. Moreover, the asymptotic approximation for both objects will be presented.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80910638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic behaviour of solutions of quasilinear\u0000 differential-algebraic equations","authors":"\t\tEquations\t\t\tV. H. Linh, N. Nga, N. Tuan","doi":"10.14232/ejqtde.2022.1.43","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.43","url":null,"abstract":"This paper is concerned with the asymptotic behavior of solutions of linear differential-algebraic equations (DAEs) under small nonlinear perturbations. Some results on the asymptotic behavior of solutions which are well known for ordinary differential equations are extended to DAEs. The main tools are the projector-based decoupling and the contractive mapping principle. Under certain assumptions on the linear part and the nonlinear term, asymptotic behavior of solutions are characterized. As the main result, a Perron type theorem that establishes the exponential growth rate of solutions is formulated.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88191435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}