Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.18
Chung‐Sik Sin
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.
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我们考虑Dirichlet问题- Δ p (x) kp u (x)- Δ q (x) K q u (x) = f (x, u (x)，∇u (x)) in Ω， u |∂Ω = 0，由p (x)-拉普拉斯算子和q (x)-拉普拉斯算子的和驱动，它们都被不定的(改变符号的)基尔霍夫型项加权。利用拓扑工具(Galerkin基的性质和Nemitsky映射的性质)建立了弱解和强广义解的存在性。在正Kirchhoff项的特殊情况下，利用伪单调算子的性质，得到了弱解(=强广义解)的存在性。
{"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":"1 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.52
Equations Tao Zhu
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.
{"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":"33 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.60
Equations T. Caraballo, Faten Ezzine, M. Hammami
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.
{"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":"347 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.70
Equations Wenlei Li, Shiduo Qu, S. Shi
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.
{"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":"2620 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.19
V. Răsvan
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.
{"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":"1 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.15
E. Henriques
We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations ∂t(uq)−div(|Du|p−2Du)=0,p>1andq>0 considering separately the two possible cases q+1−p>0 and q+1−p<0. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic p-Laplacian equation.
我们建立了一类双非线性抛物型方程的非负、局部、弱解的正性展开式？t（uq）−div （|D u |p−2 D u）=0，p>1和q>0，分别考虑两种可能的情况q+1−p>0和q+1−p0。证明依赖于DiBenedetto、Gianazza和Vespri对退化和奇异抛物型p-Laplacian方程提出的程序。
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Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.16
Érika Diz-Pita, J. Llibre, M. V. Otero-Espinar
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.
{"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":"1 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.23
S. Aouaoui
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.
{"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":"1 1","pages":""},"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}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.11
Guowei Zhang
In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.
{"title":"Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities","authors":"Guowei Zhang","doi":"10.14232/ejqtde.2022.1.11","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.11","url":null,"abstract":"In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583711","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}
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