14 results on '"Nonlocal equations"'
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2. Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
- Author
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Jin, Tianling, Xiong, Jingang, Jin, Tianling, and Xiong, Jingang
- Abstract
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations.
- Published
- 2021
3. About the Structure of Attractors for a Nonlocal Chafee-Infante Problem
- Author
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Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, and Valero, José
- Abstract
In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.
- Published
- 2021
4. About the Structure of Attractors for a Nonlocal Chafee-Infante Problem
- Author
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Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, and Valero, José
- Abstract
In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.
- Published
- 2021
5. About the Structure of Attractors for a Nonlocal Chafee-Infante Problem
- Author
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Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, and Valero, José
- Abstract
In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.
- Published
- 2021
6. Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
- Author
-
Jin, Tianling, Xiong, Jingang, Jin, Tianling, and Xiong, Jingang
- Abstract
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations.
- Published
- 2021
7. Local theory for spatio-temporal canards and delayed bifurcations
- Author
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AVITABILE, DANIELE, DESROCHES, MATHIEU, VELTZ, ROMAIN, WECHSELBERGER, MARTIN, AVITABILE, DANIELE, DESROCHES, MATHIEU, VELTZ, ROMAIN, and WECHSELBERGER, MARTIN
- Abstract
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.
- Published
- 2020
- Full Text
- View/download PDF
8. Local theory for spatio-temporal canards and delayed bifurcations
- Author
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AVITABILE, DANIELE, DESROCHES, MATHIEU, VELTZ, ROMAIN, WECHSELBERGER, MARTIN, AVITABILE, DANIELE, DESROCHES, MATHIEU, VELTZ, ROMAIN, and WECHSELBERGER, MARTIN
- Abstract
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.
- Published
- 2020
- Full Text
- View/download PDF
9. Symmetric solutions of evolutionary partial differential equations
- Author
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Bruell, Gabriele, Ehrnstrom, Mats, Geyer, Anna, Pei, Long, Bruell, Gabriele, Ehrnstrom, Mats, Geyer, Anna, and Pei, Long
- Abstract
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from EhrnstrOm et al (2009 Int. Math. Res. Not. 2009 4578-96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases., QC 20171019
- Published
- 2017
- Full Text
- View/download PDF
10. Symmetric solutions of evolutionary partial differential equations
- Author
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Bruell, G. (author), Ehrnström, Mats (author), Geyer, A. (author), Pei, Long (author), Bruell, G. (author), Ehrnström, Mats (author), Geyer, A. (author), and Pei, Long (author)
- Abstract
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from Ehrnström et al (2009 Int. Math. Res. Not. 2009 4578–96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases., Accepted author manuscript, Mathematical Physics
- Published
- 2017
- Full Text
- View/download PDF
11. A Liouville type result for fractional Schrödinger operators in 1D
- Author
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Universitat Politècnica de Catalunya. Departament de Matemàtiques, Cabré Vilagut, Xavier, Felipe Navarro, Juan Carlos, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Cabré Vilagut, Xavier, and Felipe Navarro, Juan Carlos
- Abstract
The aim of this master's thesis is to obtain an alternative and original proof of a Liouville type result for fractional Schrödinger operators in 1D without using a local extension problem, in the spirit of the recent work of Hamel et al. Thanks to this new proof we can extend the Liouville theorem to other nonlocal operators that do not have a local extension problem, being the first time that a result of this kind is proven. First, we introduce Schrödinger operators, the fractional Laplacian and its local extension problem. Then, we present a recent work about a nonlocal and nonlinear problem, where the prior study of fractional Schrödinger operators is needed. We also present the most important motivation for the study of Liouville type results: the conjecture of De Giorgi, and we review some Liouville type results both with local and nonlocal operators. Finally, we give the proof of the main theorems of the thesis.
- Published
- 2017
12. Symmetric solutions of evolutionary partial differential equations
- Author
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Bruell, G. (author), Ehrnström, Mats (author), Geyer, A. (author), Pei, Long (author), Bruell, G. (author), Ehrnström, Mats (author), Geyer, A. (author), and Pei, Long (author)
- Abstract
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from Ehrnström et al (2009 Int. Math. Res. Not. 2009 4578–96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases., Accepted author manuscript, Mathematical Physics
- Published
- 2017
- Full Text
- View/download PDF
13. Boundary regularity estimates for nonlocal elliptic equations in C1 and C1, domains
- Author
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Universitat Politècnica de Catalunya. EDP - Equacions en Derivades Parcials i Aplicacions, Ros-Oton, Xavier, Serra Montolí, Joaquim, Universitat Politècnica de Catalunya. EDP - Equacions en Derivades Parcials i Aplicacions, Ros-Oton, Xavier, and Serra Montolí, Joaquim
- Abstract
“The final publication is available at Springer via http://dx.doi.org/10.1007/s10231-016-0632-1", We establish sharp boundary regularity estimates in C1 and C1,a domains for nonlocal problems of the form Lu=f in O, u=0 in Oc. Here, L is a nonlocal elliptic operator of order 2s, with s¿(0,1). First, in C1,a domains we show that all solutions u are Cs up to the boundary and that u/ds¿Ca(O¯¯¯¯), where d is the distance to ¿O. In C1 domains, solutions are in general not comparable to ds, and we prove a boundary Harnack principle in such domains. Namely, we show that if u1 and u2 are positive solutions, then u1/u2 is bounded and Hölder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in nondivergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators (Caffarelli et al., in Invent Math, to appear)., Peer Reviewed, Postprint (published version)
- Published
- 2017
14. Stable and periodic solutions to nonlinear equations with fractional diffusion
- Author
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Universitat Politècnica de Catalunya. Departament de Matemàtiques, Cabré Vilagut, Xavier, Sanz Perela, Tomás, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Cabré Vilagut, Xavier, and Sanz Perela, Tomás
- Abstract
The aim of this thesis is to study stable solutions to nonlinear elliptic equations involving the fractional Lapacian. More precisely, we study the extremal solution for the problem $(\Delta )^s u = \lambda f(u)$ in $\Omega$, $u \equiv 0 $ in $\R^n \setminus \Omega$, where $\lambda > 0$ is a parameter and $s \in (0,1)$. The main result of this work, which is new, is the following: we prove that when $s=1/2$ and $\Omega = B_1$, then the extremal solution is bounded whenever $n \leq 8$.
- Published
- 2016
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