Pub Date : 2024-02-20DOI: 10.1186/s13661-024-01833-7
Nahid Barzehkar, Reza Jalilian, Ali Barati
In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.
{"title":"Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives","authors":"Nahid Barzehkar, Reza Jalilian, Ali Barati","doi":"10.1186/s13661-024-01833-7","DOIUrl":"https://doi.org/10.1186/s13661-024-01833-7","url":null,"abstract":"In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927643","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 : 2024-02-08DOI: 10.1186/s13661-024-01832-8
Weichen Zhou, Zhaocai Hao, Martin Bohner
In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem.
{"title":"Existence and multiplicity of solutions of fractional differential equations on infinite intervals","authors":"Weichen Zhou, Zhaocai Hao, Martin Bohner","doi":"10.1186/s13661-024-01832-8","DOIUrl":"https://doi.org/10.1186/s13661-024-01832-8","url":null,"abstract":"In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"28 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769576","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 : 2024-02-08DOI: 10.1186/s13661-024-01830-w
Zhaoyang Yun, Zhitao Zhang
In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system $$ textstylebegin{cases} -Delta u_{1}-lambda _{1} u_{1}=mu _{1} u_{1}^{3}+beta u_{1}u_{2}^{2}+ kappa (x) u_{2}quadtext{in }mathbb{R}^{3}, -Delta u_{2}-lambda _{2} u_{2}=mu _{2} u_{2}^{3}+beta u_{1}^{2}u_{2}+ kappa (x) u_{1}quadtext{in }mathbb{R}^{3}, int _{mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},qquad int _{mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, end{cases} $$ where $a_{1}$ , $a_{2}$ , $mu _{1}$ , $mu _{2}$ , $kappa =kappa (x)>0$ , $beta <0$ , and $lambda _{1}$ , $lambda _{2}$ are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.
{"title":"Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings","authors":"Zhaoyang Yun, Zhitao Zhang","doi":"10.1186/s13661-024-01830-w","DOIUrl":"https://doi.org/10.1186/s13661-024-01830-w","url":null,"abstract":"In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system $$ textstylebegin{cases} -Delta u_{1}-lambda _{1} u_{1}=mu _{1} u_{1}^{3}+beta u_{1}u_{2}^{2}+ kappa (x) u_{2}quadtext{in }mathbb{R}^{3}, -Delta u_{2}-lambda _{2} u_{2}=mu _{2} u_{2}^{3}+beta u_{1}^{2}u_{2}+ kappa (x) u_{1}quadtext{in }mathbb{R}^{3}, int _{mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},qquad int _{mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, end{cases} $$ where $a_{1}$ , $a_{2}$ , $mu _{1}$ , $mu _{2}$ , $kappa =kappa (x)>0$ , $beta <0$ , and $lambda _{1}$ , $lambda _{2}$ are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769797","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}
In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 $$begin{aligned}& -Delta _{p}u+V(x) vert u vert ^{p-2}u-Delta _{p}bigl( vert u vert ^{2alpha}bigr) vert u vert ^{2alpha -2}u= lambda h_{1}(x) vert u vert ^{m-2}u+h_{2}(x) vert u vert ^{q-2}u, & quad xin {mathbb{R}}^{N}, end{aligned}$$ where $Delta _{p}u=operatorname{div}(|nabla u|^{p-2}nabla u)$ , $1< p< N$ , $lambda ge 0$ , and $1< m< p<2alpha p0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({mathbb{R}}^{N})$ provided that $lambda in [0,lambda _{0}]$ .
在这项工作中，我们研究了以下具有参数 α 和凹凸非线性的准线性薛定谔方程的无穷多个解的存在性： 0.1 $$begin{aligned}& -Delta _{p}u+V(x) vert u vert ^{p-2}u-Delta _{p}bigl( vert u vert ^{2alpha}bigr) vert u vert ^{2alpha -2}u= lambda h_{1}(x) vert u vert ^{m-2}u+h_{2}(x) vert u vert ^{q-2}u、& quad xin {mathbb{R}}^{N}, end{aligned}$$ 其中 $Delta _{p}u=operatorname{div}(|nabla u|^{p-2}nabla u)$ , $1< p< N$ , $lambda ge 0$ , and $1< m< p0$ such that Eq.(0.1) 在$W^{1,p}({mathbb{R}}^{N})$ 中有无限多的高能解，前提是$lambda in [0,lambda _{0}]$。
{"title":"Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities","authors":"Lijuan Chen, Caisheng Chen, Qiang Chen, Yunfeng Wei","doi":"10.1186/s13661-023-01805-3","DOIUrl":"https://doi.org/10.1186/s13661-023-01805-3","url":null,"abstract":"In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 $$begin{aligned}& -Delta _{p}u+V(x) vert u vert ^{p-2}u-Delta _{p}bigl( vert u vert ^{2alpha}bigr) vert u vert ^{2alpha -2}u= lambda h_{1}(x) vert u vert ^{m-2}u+h_{2}(x) vert u vert ^{q-2}u, & quad xin {mathbb{R}}^{N}, end{aligned}$$ where $Delta _{p}u=operatorname{div}(|nabla u|^{p-2}nabla u)$ , $1< p< N$ , $lambda ge 0$ , and $1< m< p<2alpha p<q<2alpha p^{*}=frac{2alpha pN}{N-p}$ . The functions $V(x)$ , $h_{1}(x)$ , and $h_{2}(x)$ satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists $lambda _{0}>0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({mathbb{R}}^{N})$ provided that $lambda in [0,lambda _{0}]$ .","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769759","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 : 2024-02-01DOI: 10.1186/s13661-024-01826-6
Sabbavarapu Nageswara Rao, Mahammad Khuddush, Ahmed Hussein Msmali, Abdullah Ali H. Ahmadini
This paper deals with the existence results of the infinite system of tempered fractional BVPs $$begin{aligned}& {}^{mathtt{R}}_{0}mathrm{D}_{mathrm{r}}^{varrho , uplambda} mathtt{z}_{mathtt{j}}(mathrm{r})+psi _{mathtt{j}}bigl(mathrm{r}, mathtt{z}(mathrm{r})bigr)=0,quad 0< mathrm{r}< 1, & mathtt{z}_{mathtt{j}}(0)=0,qquad {}^{mathtt{R}}_{0} mathrm{D}_{ mathrm{r}}^{mathtt{m}, uplambda} mathtt{z}_{mathtt{j}}(0)=0, & mathtt{b}_{1} mathtt{z}_{mathtt{j}}(1)+mathtt{b}_{2} {}^{ mathtt{R}}_{0}mathrm{D}_{mathrm{r}}^{mathtt{m}, uplambda} mathtt{z}_{mathtt{j}}(1)=0, end{aligned}$$ where $mathtt{j}in mathbb{N}$ , $2
{"title":"Infinite system of nonlinear tempered fractional order BVPs in tempered sequence spaces","authors":"Sabbavarapu Nageswara Rao, Mahammad Khuddush, Ahmed Hussein Msmali, Abdullah Ali H. Ahmadini","doi":"10.1186/s13661-024-01826-6","DOIUrl":"https://doi.org/10.1186/s13661-024-01826-6","url":null,"abstract":"This paper deals with the existence results of the infinite system of tempered fractional BVPs $$begin{aligned}& {}^{mathtt{R}}_{0}mathrm{D}_{mathrm{r}}^{varrho , uplambda} mathtt{z}_{mathtt{j}}(mathrm{r})+psi _{mathtt{j}}bigl(mathrm{r}, mathtt{z}(mathrm{r})bigr)=0,quad 0< mathrm{r}< 1, & mathtt{z}_{mathtt{j}}(0)=0,qquad {}^{mathtt{R}}_{0} mathrm{D}_{ mathrm{r}}^{mathtt{m}, uplambda} mathtt{z}_{mathtt{j}}(0)=0, & mathtt{b}_{1} mathtt{z}_{mathtt{j}}(1)+mathtt{b}_{2} {}^{ mathtt{R}}_{0}mathrm{D}_{mathrm{r}}^{mathtt{m}, uplambda} mathtt{z}_{mathtt{j}}(1)=0, end{aligned}$$ where $mathtt{j}in mathbb{N}$ , $2<varrho le 3$ , $1<mathtt{m}le 2$ , by utilizing the Hausdorff measure of noncompactness and Meir–Keeler fixed point theorem in a tempered sequence space.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"292 1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657811","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}
We establish a Liouville-type theorem for a weighted higher-order elliptic system in a wider exponent region described via a critical curve. We first establish a Liouville-type theorem to the involved integral system and then prove the equivalence between the two systems by using superharmonic properties of the differential systems. This improves the results in (Complex Var. Elliptic Equ. 5:1436–1450, 2013) and (Abstr. Appl. Anal. 2014:593210, 2014).
{"title":"Nonexistence of positive solutions for the weighted higher-order elliptic system with Navier boundary condition","authors":"Weiwei Zhao, Xiaoling Shao, Changhui Hu, Zhiyu Cheng","doi":"10.1186/s13661-024-01831-9","DOIUrl":"https://doi.org/10.1186/s13661-024-01831-9","url":null,"abstract":"We establish a Liouville-type theorem for a weighted higher-order elliptic system in a wider exponent region described via a critical curve. We first establish a Liouville-type theorem to the involved integral system and then prove the equivalence between the two systems by using superharmonic properties of the differential systems. This improves the results in (Complex Var. Elliptic Equ. 5:1436–1450, 2013) and (Abstr. Appl. Anal. 2014:593210, 2014).","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"34 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582786","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 : 2024-01-25DOI: 10.1186/s13661-024-01828-4
Sami Baraket, Anis Ben Ghorbal, Rima Chetouane, Azedine Grine
This paper is devoted to the existence of singular limit solutions for a nonlinear elliptic system of Liouville type under Navier boundary conditions in a bounded open domain of $mathbb{R}^{4}$ . The concerned results are obtained employing the nonlinear domain decomposition method and a Pohozaev-type identity.
{"title":"Blow-up solutions for a 4-dimensional semilinear elliptic system of Liouville type in some general cases","authors":"Sami Baraket, Anis Ben Ghorbal, Rima Chetouane, Azedine Grine","doi":"10.1186/s13661-024-01828-4","DOIUrl":"https://doi.org/10.1186/s13661-024-01828-4","url":null,"abstract":"This paper is devoted to the existence of singular limit solutions for a nonlinear elliptic system of Liouville type under Navier boundary conditions in a bounded open domain of $mathbb{R}^{4}$ . The concerned results are obtained employing the nonlinear domain decomposition method and a Pohozaev-type identity.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"12 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582914","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 : 2024-01-25DOI: 10.1186/s13661-024-01829-3
Sami Baraket, Brahim Dridi, Azedine Grine, Rached Jaidane
This article aims to investigate the existence of nontrivial solutions with minimal energy for a logarithmic weighted $(N,p)$ -Laplacian problem in the unit ball B of $mathbb{R}^{N}$ , $N>2$ . The nonlinearities of the equation are critical or subcritical growth, which is motivated by weighted Trudinger–Moser type inequalities. Our approach is based on constrained minimization within the Nehari set, the quantitative deformation lemma, and degree theory results.
本文旨在研究在 $mathbb{R}^{N}$ 的单位球 B 中，$N>2$ 的对数加权 $(N,p)$ 拉普拉斯问题是否存在能量最小的非小解。方程的非线性是临界或亚临界增长，这是由加权特鲁丁格-莫泽型不等式引起的。我们的方法基于内哈里集的约束最小化、定量变形 Lemma 和度理论结果。
{"title":"Least energy nodal solutions for a weighted ((N, p))-Schrödinger problem involving a continuous potential under exponential growth nonlinearity","authors":"Sami Baraket, Brahim Dridi, Azedine Grine, Rached Jaidane","doi":"10.1186/s13661-024-01829-3","DOIUrl":"https://doi.org/10.1186/s13661-024-01829-3","url":null,"abstract":"This article aims to investigate the existence of nontrivial solutions with minimal energy for a logarithmic weighted $(N,p)$ -Laplacian problem in the unit ball B of $mathbb{R}^{N}$ , $N>2$ . The nonlinearities of the equation are critical or subcritical growth, which is motivated by weighted Trudinger–Moser type inequalities. Our approach is based on constrained minimization within the Nehari set, the quantitative deformation lemma, and degree theory results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"101 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560725","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 : 2024-01-23DOI: 10.1186/s13661-024-01827-5
Wenyi Liu, Chengbin Du, Zhiyuan Li
In this work, a new type of the unique continuation property for time-fractional diffusion equations is studied. The proof is mainly based on the Laplace transform and the properties of Bessel functions. As an application, the uniqueness of the inverse problem in the simultaneous determination of spatially dependent source terms and fractional order from sparse boundary observation data is established.
{"title":"New type of the unique continuation property for a fractional diffusion equation and an inverse source problem","authors":"Wenyi Liu, Chengbin Du, Zhiyuan Li","doi":"10.1186/s13661-024-01827-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01827-5","url":null,"abstract":"In this work, a new type of the unique continuation property for time-fractional diffusion equations is studied. The proof is mainly based on the Laplace transform and the properties of Bessel functions. As an application, the uniqueness of the inverse problem in the simultaneous determination of spatially dependent source terms and fractional order from sparse boundary observation data is established.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"60 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560541","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 : 2024-01-22DOI: 10.1186/s13661-024-01822-w
Soon-Yeong Chung, Jaeho Hwang
The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations $$ u_{t}=Delta u+psi (t)f(u),quad text{in }Omega times (0,infty ), $$ under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$ . As a matter of fact, we prove: $$ begin{aligned} & text{there is no global solution for any initial data if and only if } & int _{0}^{infty}psi (t) frac{f (lVert S(t)u_{0}rVert _{infty} )}{lVert S(t)u_{0}rVert _{infty}},dt= infty &text{for every nonnegative nontrivial initial data } u_{0}in C_{0}( Omega ). end{aligned} $$ Here, $(S(t))_{tgeq 0}$ is the heat semigroup with the mixed boundary condition.
本文旨在给出以下半线性抛物方程的全局解 $$ u_{t}=Delta u+psi (t)f(u),quad text{in }Omega times (0,infty ), $$ 在有界域 Ω 上的混合边界条件下存在与不存在的必要条件和充分条件。事实上，几十年来这一直是个悬而未决的问题，即使对于 $f(u)=u^{p}$ 的情况也是如此。事实上，我们证明了： $$ （begin{aligned} & text{there is no global solution for any initial data if and only if } & int _{0}^{infty}psi (t) frac{f (lVert S(t)u_{0}rVert _{infty} )}{lVert S(t)u_{0}rVert _{infty}}、dt= infty &text{ for every nonnegative nontrivial initial data } u_{0}in C_{0}( Omega ).end{aligned} $$ 这里，$(S(t))_{tgeq 0}$ 是具有混合边界条件的热半群。
{"title":"A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains","authors":"Soon-Yeong Chung, Jaeho Hwang","doi":"10.1186/s13661-024-01822-w","DOIUrl":"https://doi.org/10.1186/s13661-024-01822-w","url":null,"abstract":"The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations $$ u_{t}=Delta u+psi (t)f(u),quad text{in }Omega times (0,infty ), $$ under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$ . As a matter of fact, we prove: $$ begin{aligned} & text{there is no global solution for any initial data if and only if } & int _{0}^{infty}psi (t) frac{f (lVert S(t)u_{0}rVert _{infty} )}{lVert S(t)u_{0}rVert _{infty}},dt= infty &text{for every nonnegative nontrivial initial data } u_{0}in C_{0}( Omega ). end{aligned} $$ Here, $(S(t))_{tgeq 0}$ is the heat semigroup with the mixed boundary condition.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139518434","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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