We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.
{"title":"Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation","authors":"Ryunosuke Kusaba, Tohru Ozawa","doi":"10.7153/dea-2023-15-13","DOIUrl":"https://doi.org/10.7153/dea-2023-15-13","url":null,"abstract":"We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Riccati equation method is used to establish some new stability criteria for systems of two linear first-order ordinary differential equations. It is shown that two of these criteria in the two dimensional case imply the Routh - Hurwitz's criterion.
{"title":"On the stability of systems of two linear first-order ordinary differential equations","authors":"G. A. Grigorian","doi":"10.7153/dea-2023-15-15","DOIUrl":"https://doi.org/10.7153/dea-2023-15-15","url":null,"abstract":"The Riccati equation method is used to establish some new stability criteria for systems of two linear first-order ordinary differential equations. It is shown that two of these criteria in the two dimensional case imply the Routh - Hurwitz's criterion.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"180 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We consider a second order singular nonlinear partial differential equation of the form ( t ∂ t ) 2 u = F ( t , x , u , ∂ x u , ∂ 2 x u , t ∂ t u , t ∂ t ∂ x u ) , where F is assumed to be continuous in t and holo-morphic with respect to the other variables. Under certain conditions, we prove that the equation has a unique solution that is continuous in t and holomorphic in x .
{"title":"Unique solvability of second order nonlinear totally characteristic equations","authors":"Michael E. Sta. Brigida, Jose Ernie C. Lope","doi":"10.7153/dea-2023-15-14","DOIUrl":"https://doi.org/10.7153/dea-2023-15-14","url":null,"abstract":". We consider a second order singular nonlinear partial differential equation of the form ( t ∂ t ) 2 u = F ( t , x , u , ∂ x u , ∂ 2 x u , t ∂ t u , t ∂ t ∂ x u ) , where F is assumed to be continuous in t and holo-morphic with respect to the other variables. Under certain conditions, we prove that the equation has a unique solution that is continuous in t and holomorphic in x .","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"274 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In the present article, we prove some existence results for a class of implicit Caputo fractional q -difference equations with non instantaneous impulses in Banach spaces. The used techniques rely on the concepts of measure of noncompactness and the use of suitable fi xed point theorems.
{"title":"Implicit Caputo fractional q-difference equations with non instantaneous impulses","authors":"Saïd Abbas, Mouffak Benchohra, Alberto Cabada","doi":"10.7153/dea-2023-15-12","DOIUrl":"https://doi.org/10.7153/dea-2023-15-12","url":null,"abstract":". In the present article, we prove some existence results for a class of implicit Caputo fractional q -difference equations with non instantaneous impulses in Banach spaces. The used techniques rely on the concepts of measure of noncompactness and the use of suitable fi xed point theorems.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper we continue with our investigation of principal and antiprincipal solutions at in fi nity solutions of a dynamic symplectic system. The paper is a continuation of part I appeared in Differential Equations and Applications in 2022, where we have presenteded a theory of genera of conjoined bases for symplectic dynamic systems on time scales and its connections with principal solutions at in fi nity and antiprincipal solutions at in fi nity for these systems together with some basic properties of this new concept on time scales. Here we provide a characterization of all principal solutions of dynamic symplectic system at in fi nity in the given genus in terms of the initial conditions and a fi xed principal solution at in fi nity from this genus. Further, we provide a characterization of all antiprincipal solutions of dynamic symplectic system at in fi nity in the given genus in terms of the initial conditions and a fi xed principal solution at in fi nity from this genus. We also establish mutual limit properties of principal and antiprincipal solutions at in fi nity.
{"title":"Extremal solutions at infinity for symplectic systems on time scales II - Existence theory and limit properties","authors":"Iva Dřímalová","doi":"10.7153/dea-2023-15-11","DOIUrl":"https://doi.org/10.7153/dea-2023-15-11","url":null,"abstract":". In this paper we continue with our investigation of principal and antiprincipal solutions at in fi nity solutions of a dynamic symplectic system. The paper is a continuation of part I appeared in Differential Equations and Applications in 2022, where we have presenteded a theory of genera of conjoined bases for symplectic dynamic systems on time scales and its connections with principal solutions at in fi nity and antiprincipal solutions at in fi nity for these systems together with some basic properties of this new concept on time scales. Here we provide a characterization of all principal solutions of dynamic symplectic system at in fi nity in the given genus in terms of the initial conditions and a fi xed principal solution at in fi nity from this genus. Further, we provide a characterization of all antiprincipal solutions of dynamic symplectic system at in fi nity in the given genus in terms of the initial conditions and a fi xed principal solution at in fi nity from this genus. We also establish mutual limit properties of principal and antiprincipal solutions at in fi nity.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135600139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The general problem of persistence of species, amounts to de fi ne interactions between them ensuring the survival of all the species initially present in the system. It appears that several relevant persistence schemes induce “forbidden sets” of zero measure for topological rea- sons. These peculiarities (without practical consequences) are nevertheless not consistent with certain mathematical de fi nitions of persistence, which are too much restrictive. We come back to de fi nitions of McGehee – Armstrong and their celebrated counter-example to the so-called “competitive exclusion principle”. We develop these concepts in relation with invasion properties of the species in a rather practical and computational framework. Several examples of communities exhibiting persistence without internal rest point (which necessarily exists according to strict persistence de fi nitions) are given, with explicit description of the attractors, forbidden sets and invasion properties. Mechanisms of contamination of these properties (based on elementary cartesian product and structural stability) are given, showing the widespreading nature of these schemes.
{"title":"On persistence and invading species in ecological dynamics","authors":"E. Sanchez-Palencia, J. Françoise","doi":"10.7153/DEA-2021-13-17","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-17","url":null,"abstract":". The general problem of persistence of species, amounts to de fi ne interactions between them ensuring the survival of all the species initially present in the system. It appears that several relevant persistence schemes induce “forbidden sets” of zero measure for topological rea- sons. These peculiarities (without practical consequences) are nevertheless not consistent with certain mathematical de fi nitions of persistence, which are too much restrictive. We come back to de fi nitions of McGehee – Armstrong and their celebrated counter-example to the so-called “competitive exclusion principle”. We develop these concepts in relation with invasion properties of the species in a rather practical and computational framework. Several examples of communities exhibiting persistence without internal rest point (which necessarily exists according to strict persistence de fi nitions) are given, with explicit description of the attractors, forbidden sets and invasion properties. Mechanisms of contamination of these properties (based on elementary cartesian product and structural stability) are given, showing the widespreading nature of these schemes.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"21 1","pages":"297-320"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83292757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The inverse problem for the degenerate parabolic equation is considered. The minor coef fi cient of the equation is a polynomial of the fi rst power with respect to the space variable with unknown time-dependent coef fi cients. The conditions of local in time existence and global uniqueness of the classical solution to this problem are established. The case of weak power degeneration is investigated.
{"title":"Coefficient inverse problem for the degenerate parabolic equation","authors":"N. Huzyk","doi":"10.7153/DEA-2021-13-14","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-14","url":null,"abstract":". The inverse problem for the degenerate parabolic equation is considered. The minor coef fi cient of the equation is a polynomial of the fi rst power with respect to the space variable with unknown time-dependent coef fi cients. The conditions of local in time existence and global uniqueness of the classical solution to this problem are established. The case of weak power degeneration is investigated.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"243-255"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86387591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper we present an extension of the original version of Leggett-Williams fi xed point theorem for a k -set contraction perturbed by an expansive operator. Our approach is applied to prove the existence of non trivial positive solutions for initial value problems (IVPs for short) covering a class two-dimensional nonlinear wave equations.
{"title":"Leggett-Williams fixed point theorem type for sums of operators and application in PDEs","authors":"S. Georgiev, K. Mebarki","doi":"10.7153/DEA-2021-13-18","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-18","url":null,"abstract":". In this paper we present an extension of the original version of Leggett-Williams fi xed point theorem for a k -set contraction perturbed by an expansive operator. Our approach is applied to prove the existence of non trivial positive solutions for initial value problems (IVPs for short) covering a class two-dimensional nonlinear wave equations.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"2012 1","pages":"321-344"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86399889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, by sub-supersolution methods, Karamata regular variation theory and perturbation method, we study the existence, uniqueness and asymptotic behavior of solutions near the boundary to quasilinear elliptic problem where Ω is a bounded domain with smooth boundary in R N ( N (cid:2) 2 ) , 1 < m (cid:3) 2, 0 < q (cid:3) m / ( m − 1 ) . b ∈ C α ( Ω )( α ∈ ( 0 , 1 )) is positive in Ω , and may be vanishing on the boundary, and f ∈ C 1 [ 0 , + ∞ ) , f ( 0 ) = 0, is increase on ( 0 , + ∞ ) and normalized regularly varying at in fi nity with positive index p and p +( q − 1 )( m − 1 ) > 0.
． 本文利用次超解方法、Karamata正则变分理论和摄动方法，研究了拟线性椭圆型问题的边界附近解的存在唯一性和渐近性，其中Ω是R N (N (cid:2) 2)、1 < m (cid:3) 2,0 < q (cid:3) m / (m−1)的光滑边界有界区域。b∈C α (Ω)(α∈(0,1))在Ω上是正的，并且可能在边界上消失，f∈C 1[0， +∞)，f(0) = 0，在(0，+∞)上是递增的，并且在无穷大处归一化规律变化，正指标p且p +(q−1)(m−1)> 0。
{"title":"Existence and boundary behavior of solutions for boundary blow-up quasilinear elliptic problems with gradient terms","authors":"Chunlian Liu","doi":"10.7153/DEA-2021-13-16","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-16","url":null,"abstract":". In this paper, by sub-supersolution methods, Karamata regular variation theory and perturbation method, we study the existence, uniqueness and asymptotic behavior of solutions near the boundary to quasilinear elliptic problem where Ω is a bounded domain with smooth boundary in R N ( N (cid:2) 2 ) , 1 < m (cid:3) 2, 0 < q (cid:3) m / ( m − 1 ) . b ∈ C α ( Ω )( α ∈ ( 0 , 1 )) is positive in Ω , and may be vanishing on the boundary, and f ∈ C 1 [ 0 , + ∞ ) , f ( 0 ) = 0, is increase on ( 0 , + ∞ ) and normalized regularly varying at in fi nity with positive index p and p +( q − 1 )( m − 1 ) > 0.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"47 1","pages":"281-295"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85426643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomain polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.
{"title":"Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomain Polynomial","authors":"Mousumi Datta, U. Habiba, Md. Babul Hossain","doi":"10.12691/IJPDEA-8-1-2","DOIUrl":"https://doi.org/10.12691/IJPDEA-8-1-2","url":null,"abstract":"In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomain polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"14 1","pages":"6-12"},"PeriodicalIF":0.0,"publicationDate":"2020-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87128689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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