## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (191.996 recursos)

Differential Integral Equations

1. #### Bifurcation of Space Periodic Solutions in Symmetric Reversible FDEs

Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without $1:k$ resonance). Physical examples motivating considered settings are related to stationary solutions to PDEs with non-local interaction: reversible mixed delay differential equations (MDDEs) and integro-differential equations (IDEs). In the case of $S_4$-symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. Algorithms and computational procedures used in this paper are also included.

2. #### Bifurcation of Space Periodic Solutions in Symmetric Reversible FDEs

Balanov, Zalman; Wu, Hao-Pin
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without $1:k$ resonance). Physical examples motivating considered settings are related to stationary solutions to PDEs with non-local interaction: reversible mixed delay differential equations (MDDEs) and integro-differential equations (IDEs). In the case of $S_4$-symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. Algorithms and computational procedures used in this paper are also included.

3. #### On ordinary differential inclusions with mixed boundary conditions

Bonanno, Gabriele; Iannizzotto, Antonio; Marras, Monica
By means of nonsmooth critical point theory, we prove existence of three weak solutions for an ordinary differential inclusion of Sturm-Liouville type involving a general set-valued reaction term depending on a parameter, and coupled with mixed boundary conditions. As an application, we give a multiplicity result for ordinary differential equations involving discontinuous nonlinearities.

4. #### On ordinary differential inclusions with mixed boundary conditions

Bonanno, Gabriele; Iannizzotto, Antonio; Marras, Monica
By means of nonsmooth critical point theory, we prove existence of three weak solutions for an ordinary differential inclusion of Sturm-Liouville type involving a general set-valued reaction term depending on a parameter, and coupled with mixed boundary conditions. As an application, we give a multiplicity result for ordinary differential equations involving discontinuous nonlinearities.

5. #### On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Ciesielski, Jakub; Janczewska, Joanna; Waterstraat, Nils
We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb R$, $q\in\mathbb R^n$, $a\colon\mathbb R\to\mathbb R$ is a continuous positive bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb R\to\mathbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

6. #### On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Ciesielski, Jakub; Janczewska, Joanna; Waterstraat, Nils
We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb R$, $q\in\mathbb R^n$, $a\colon\mathbb R\to\mathbb R$ is a continuous positive bounded function, $G\colon\mathbb R^n\to\mathbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb R\to\mathbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

7. #### Positive semiclassical states for a fractional Schrödinger-Poisson system

Murcia, Edwin G.; Siciliano, Gaetano
We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e., in the semiclassical limit'') the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.

8. #### Positive semiclassical states for a fractional Schrödinger-Poisson system

Murcia, Edwin G.; Siciliano, Gaetano
We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e., in the semiclassical limit'') the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.

9. #### Three solutions for an elliptic system near resonance with the principal eigenvalue

Massa, Eugenio; Rossato, Rafael Antônio
We consider an elliptic system of Hamiltonian type with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least three solutions when the linear part is near resonance with the principal eigenvalue, either from above or from below. For two of these solutions, we also obtain information on the sign of its components. The system is associated to a strongly indefinite functional and the solutions are obtained trough saddle point theorem, after truncating the nonlinearity.

10. #### Three solutions for an elliptic system near resonance with the principal eigenvalue

Massa, Eugenio; Rossato, Rafael Antônio
We consider an elliptic system of Hamiltonian type with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least three solutions when the linear part is near resonance with the principal eigenvalue, either from above or from below. For two of these solutions, we also obtain information on the sign of its components. The system is associated to a strongly indefinite functional and the solutions are obtained trough saddle point theorem, after truncating the nonlinearity.

11. #### The focusing cubic NLS with inverse-square potential in three space dimensions

Killip, Rowan; Murphy, Jason; Visan, Monica; Zheng, Jiqiang
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS [7, 11]. We also prove failure of uniform space-time bounds at the

12. #### The focusing cubic NLS with inverse-square potential in three space dimensions

Killip, Rowan; Murphy, Jason; Visan, Monica; Zheng, Jiqiang
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS [7, 11]. We also prove failure of uniform space-time bounds at the

13. #### Multiple positive solutions for the $m$-Laplacian and a nonlinearity with many zeros

Iturriaga, Leonelo; Lorca, Sebastián; Massa, Eugenio
In this paper, we consider the quasilinear elliptic equation $-\Delta_m u=\lambda f(u)$, in a bounded, smooth and convex domain. When the nonnegative nonlinearity $f$ has multiple positive zeros, we prove the existence of at least two positive solutions for each of these zeros, for $\lambda$ large, without any hypothesis on the behavior at infinity of $f$. We also prove a result concerning the behavior of the solutions as $\lambda\to\infty$.

14. #### An infinite number of solutions for an elliptic problem with power nonlinearity

We consider the following nonlinear elliptic equation $$\tag{0.1} A_\frac{1}{2} u = K(x) |u|^{p-1}u \hbox{ in } \Omega, \;\; u=0 \hbox{ on } \partial\Omega,$$ where $\Omega$ is a bounded domain of $\mathbb{R}^n, n\geq 1, K(x)$ is a given function, $A_\frac{1}{2}$ represents the square root of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition and $1 < p < \frac{n+1}{n-1}$, $(p > 1$ if $n=1$). We apply the Brouwer's fixed point theorem to prove that (0.1) has infinitely many distinct solutions.

15. #### Ground states for a fractional scalar field problem with critical growth

Ambrosio, Vincenzo
We prove the existence of a ground state solution for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u= g(u) \mbox{ in } \mathbb R^{N} \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1, \beta}( \mathbb R, \mathbb R)$ is an odd function satisfying the critical growth assumption.

16. #### Regularity criteria for Navier-Stokes and related system

Fan, Jishan; Ozawa, Tohru
We show some regularity criteria for Navier-Stokes equations, the harmonic heat flow, two liquid crystals models, and a model for magneto-elastic materials. The method of proof depends on a systematic use of interpolation inequalities in Besov spaces and is independent on logarithmic inequalities.

17. #### The stochastic derivative nonlinear Schrödinger equation

Zhong, Sijia
In this paper, we will use a gauge transform to prove the local existence and uniqueness of the derivative nonlinear Schrödinger equation with additive noise, showing that for the initial data $u_0\in H^\frac{1}{2}(\mathbb{R})$, there is a local and unique solution almost surely.

18. #### A phase-field system with two temperatures and memory

Conti, Monica; Gatti, Stefania; Miranville, Alaine
Our aim, in this paper, is to study a generalization of the Caginalp phase-field system based on the Gurtin--Pipkin law with two temperatures for heat conduction with memory. In particular, we obtain well-posedness results and study the dissipativity, in terms of the global attractor with optimal regularity, of the associated solution operators. We also study the stability of the system as the memory kernel collapses to a Dirac mass.

19. #### On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations

Maryani, Sri; Saito, Hirokazu
The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dot\Omega =\Omega _+\cup\Omega _-$, where $\Omega _\pm$ are uniform $W_r^{2-1/r}$ domains of $N$-dimensional Euclidean space ${\mathbf{R}^N}$ ($N\geq 2$, $N < r < \infty$). More precisely, given a uniform $W_r^{2-1/r}$ domain $\Omega$ with two boundaries $\Gamma _\pm$ satisfying $\Gamma _+\cap \Gamma _-=\emptyset$, we suppose that some hypersurface $\Gamma$ divides $\Omega$ into two sub-domains, that is, there exist domains $\Omega _\pm\subset\Omega$ such that $\Omega _+\cap\Omega _-=\emptyset$ and $\Omega \setminus \Gamma =\Omega _+\cup\Omega _-,$ where $\Gamma \cap \Gamma _+=\emptyset$, $\Gamma \cap \Gamma _-=\emptyset$, and the boundaries of $\Omega _\pm$...

20. #### A time-splitting approach to quasilinear degenerate parabolic stochastic partial differential equations

Kobayasi, Kazuo; Noboriguchi, Dai
In this paper, we discuss the Cauchy problem for a degenerate parabolic-hyperbolic equation with a multiplicative noise. We focus on the existence of a solution. Using nondegenerate smooth approximations, Debussche, Hofmanová and Vovelle [8] proved the existence of a kinetic solution. On the other hand, we propose to construct a sequence of approximations by applying a time splitting method and prove that this converges strongly in $L^1$ to a kinetic solution. This method will somewhat give us not only a simpler and more direct argument but an improvement over the existence result.

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