Showing papers in "Differential and Integral Equations in 2012"

Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth

Abstract: This paper is concerned with the multiplicity of nontrivial solutions to the class of nonlocal boundary value problems of the Kirchhoff type, \begin{equation*} - \Big [M \Big (\int_{\Omega}| abla u|^{2} \ dx\ \Big ) \Big ]\Delta u = \lambda |u|^{q-2}u+|u|^{p-2}u \ \mbox{in} \ \Omega, \ \ \mbox{and} \ u=0 \ \mbox{on} \ \ \partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^{N}$, for $N=1,$ 2, and 3, is a bounded smooth domain, $1 < q < 2 < p \leq 2^{*}=6$ in the case $N=3$ and $2^{*}=\infty$ in the case $N=1$ or $N=2$. Our approach is based on the genus theory introduced by Krasnoselskii [22].

The elliptic Kirchhoff equation in $\mathbb R^N$ perturbed by a local nonlinearity

Abstract: In this paper we present a very simple proof of the existence of at least one nontrivial solution for a Kirchhoff-type equation on ${{\mathbb{R}^N}}$, for $N\ge 3$. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.

Energy decay estimates for wave equations with a fractional damping

Abstract: We consider the Cauchy problem in ${\bf R}^{n}$ for wave equations with a fractional damping. We generalize partially the previous results due to[12], and derive sharp decay estimates for the total energy and the $L^{2}$-norm of solutions based on the energy method in the Fourier space.

Nonlocal interaction equations: Stationary states and stability analysis

Abstract: This equation can be seen as a many particles limit of discret processes where particles (or individuals) can interact at a large distance, through an interaction potential W (see [27, 20]), and may be subjected to an external potential V . Such equations appear in various biological phenomenons like swarming (see [7, 15]), distribution of actin-filament networks (see [16, 19]), as well as in physical problems, for example in the field of granular media (see [2, 32, 14]). For some interaction potentials, this equation can lead to surprisingly complicated paterns, such as solutions converging to singular steady states, as shown in [4, 29, 17, 18], or more recently in [23, 8, 1]. Many of the above models couple the long-range interaction between particles with a diffusive term. Nevertheless, in this paper we shall not consider a diffusion term, and focus our study on the effect of a long-range interaction. Let us now describe typical interaction potentials W which appear in the models quoted above:

Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth

Abstract: This paper is concerned with a focusing nonlinear Schrodinger equation whose nonlinearity consists of the energy-critical local interaction term with a perturbation of the $L^2$-super and energy-subcritical term. We prove the existence of a ground state (= a standing-wave solution of minimal action) when the space dimension is four or higher and prove the nonexistence of any ground state when the space dimension is three and the perturbation is small. Once we have a ground state, a so-called potential-well scenario works well, so that we can give a sufficient condition for the nonexistence of global-in-time solutions.

Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics

Abstract: Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: {equation*} -\Delta v=\lambda\int_I V(\alpha,x,v)e^{\alpha v}\,\Pda -\frac{\lambda}{|\Omega|}\iint_{I\times\Om}V(\alpha,x,v)e^{\alpha v}\,\Pda dx, {equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered.

Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients

Abstract: . We derive local boundedness estimates for weak solutions of a large class of secondorder quasilinear equations. The structural assumptions imposed on an equation in the class allowvanishing of the quadratic form associated with its principal part and require no smoothness ofits coefficients. The class includes second order linear elliptic equations as studied in [GT] andsecond order subelliptic linear equations as in [SW1, 2]. Our results also extend ones obtained byJ. Serrin [S] concerning local boundedness of weak solutions of quasilinear elliptic equations. 1. IntroductionThe main purpose of this paper is to prove local boundedness of weak solutions uof roughsubelliptic quasilinear equations of the form(1.1) divA(x,u,∇u)= B(x,u,∇u)in an open set Ω ⊂ R n . Further regularity results will be studied in a sequel to this paper. Wewill assume that the vector-valued function Aand the scalar function Bsatisfy specific structuralrestrictions on their size, but not on their smoothness, relative to a symmetric nonnegative semi-definite matrix Q(x). Thus the quadratic form Q(x,ξ) = hQ(x)ξ,ξi,ξ∈ R

Simultaneous Blowup and Mass Separation During Collapse in an Interacting System of Chemotactic Species

Abstract: We study an interacting system of chemotactic species in two space dimensions. First, we show that there is a parameter region which ensures simultaneous blowup also for non-radially symmetric solutions. If the existence time of the solution is finite, there is a formation of collapse (possibly degenerate) for each component, total mass quantization, and formation of subcollapses. For radially symmetric solutions we can rigorously prove that the collapse concentrates mass on one component if the total masses of the other components are relatively small. Several related results are also shown.

Low regularity well-posedness for the periodic Kawahara equation

Abstract: In this paper, we consider the well-posedness for the Cauchy problem of the Kawahara equation with low regularity data in the periodic case. We obtain the local well-posedness for $s \geq -3/2$ by a variant of the Fourier restriction norm method introduced by Bourgain. Moreover, these local solutions can be extended globally in time for $s \geq -1$ by the I-method. On the other hand, we prove ill-posedness for $s < -3/2$ in some sense. This is a sharp contrast to the results in the case of $\mathbb{R}$, where the critical exponent is equal to $-2$.

Changing-sign solutions for the CR-Yamabe equation

Abstract: In this paper we prove that the CR-Yamabe equation on the Heisenberg group has innitely many changing-sign solutions. By means of the Cayley transform we will set the problem on the sphere S 2n+1 ; since the functional I associated with the equation does not satisfy the Palais-Smale compactness condition, we will nd a suitable closed subspace X on which we can apply the minmax argument for IjX. We generalize the result to any compact contact manifold of K-contact type.

Some new well-posedness results for the Klein-Gordon-Schrödinger system

Abstract: We consider the Cauchy problem for the 2D and 3D Klein-Gordon Schrodinger system. In 2D we show local well posedness for Schrodinger data in $H^s$ and wave data in $H^{\sigma} \times H^{\sigma -1}$ for $s=-1/4 \, +$ and $\sigma = -1/2$, whereas ill posedness holds for $s < - 1/4$ or $\sigma < -1/2$, and global well-posedness for $s\ge 0$ and $s-\frac{1}{2} \le \sigma < s+ \frac{3}{2}$. In 3D we show global well posedness for $s \ge 0$, $ s - \frac{1}{2} < \sigma \le s+1$. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru [2], and Bejenaru and Herr [3] for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrodinger system by Colliander, Holmer and Tzirakis [5].

The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions

Abstract: In this paper we prove the blow-up theorem in the critical case for weakly coupled systems of semilinear wave equations in high dimensions. The upper bound of the lifespan of the solution is precisely clarified.

Self-similarity, symmetries and asymptotic behavior in Morrey spaces for a fractional wave equation

Abstract: This paper is concerned with a fractional PDE that interpo- lates semilinear heat and wave equations We show results on global-in- time well-posedness for small initial data in the critical Morrey spaces and space dimension n 1 We also remark how to derive the local- in-time version of the results Qualitative properties of solutions like self-similarity, antisymmetry and positivity are also investigated More- over, we analyze the asymptotic stability of the solutions and obtain a class of asymptotically self-similar solutions

Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces

Abstract: We generalize the phase-plane method approach introduced in [9] to the case of higher dimensions. To this aim, the phase-space is assumed to be decomposable as a product of planes, where the corresponding components of the solutions can be controlled by means of suitable plane curves. We then apply our general result to the periodic problem associated to a system of coupled oscillators, with retraction forces having a linear growth, or with one-sided superlinear nonlinearities.

Existence of compact support solutions for a quasilinear and singular problem

Abstract: Let $\Omega$ be a $\mathcal{C}^{2}$ bounded domain of ${{\mathbb R}}^N$, $N\geq 2$. We consider the following quasilinear elliptic problem: $$ ({ P}_{\lambda})\left\lbrace \begin{array}{l} -\Delta_p u = K(x)(\lambda u^q-u^r),\quad \ \ \mbox{ in }\Omega, \\ \quad \;\;\;u= 0 \quad\mbox{ on }\partial\Omega, \quad u\geq 0\quad\mbox{ in }\Omega, \end{array}\right. $$ where $p>1$ and $\Delta_p u{\stackrel{{\rm {def}}}{=}} \mathrm{div} \left(\vert abla u\vert ^{p-2} abla u\right)$ denotes the $p$-Laplacian operator. In this paper, $\lambda>0$ is a real parameter, the exponents $q$ and $r$ satisfy $-1 0$ large enough, whereas it has only compact support solutions if $k\geq 1+r$.

Local and global well posedness for the chern-simons-dirac system in one dimension

Abstract: We consider the Cauchy problem for the Chern-Simons-Dirac system on $\mathbb{R}^{1+1}$ with initial data in $H^s$. Almost optimal local well-posedness is obtained. Moreover, we show that the solution is global in time, provided that initial data for the spinor component has finite charge, or $L^2$ norm.

A singular Sturm-Liouville equation under non-homogeneous boundary conditions

Abstract: Given $\alpha > 0$ and $f\in L^2(0,1)$, consider the following singular Sturm-Liouville equation: \[ \left\lbrace\begin{aligned} -(x^{2\alpha}u'(x))'+u(x) & =f(x) \ \hbox{ a.e. on } (0,1),\\ u(1) & =0. \end{aligned}\right. \] We prove existence of solutions under (weighted) non-homogeneous boundary conditions at the origin.

Infinite semipositone problems with asymptotically linear growth forcing terms

Abstract: We study the existence of positive solutions to the singular problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda f(u)-\frac{1}{u^{\alpha}} & \mbox{ in } \Omega, \\ u = 0 & \mbox{ on } \partial \Omega, \end{cases} \end{equation*} where $\lambda$ is a positive parameter, $\Delta_p u =\operatorname{div}(| abla{u}|^{p-2} abla{u})$, $p > 1$, $\Omega $ is a bounded domain in $\mathbb{R}^{n}, n \geq 1$ with smooth boundary $\partial\Omega$, $0 < \alpha < 1$, and $f:[0,\infty) \rightarrow \mathbb{R}$ is a continuous function which is asymptotically $p$-linear at $\infty$. We prove the existence of positive solutions for a certain range of $\lambda$ using the method of sub-supersolutions. We also extend our study to classes of systems which have forcing terms satisfying a combined asymptotically p-linear condition at $\infty$ and to corresponding problems on exterior domains.

Large-time behavior of solutions to the drift-diffusion equation with fractional dissipation

Abstract: We consider the Nernst-Planck-type drift-diffusion equation with fractional dissipation. For the initial-value problem of this equation, the well-posedness, the time-global existence, and the decay of solutions were already shown. When the dissipation operator is given by the Laplacian, the asymptotic expansion of the solution as $t\to\infty$ was obtained in a previous paper. We also derive the asymptotic expansion of the solution to the drift-diffusion equation with the fractional Laplacian.

Interior $ L^p$ estimates for degenerate elliptic equations in divergence form with $VMO$ coefficients

Abstract: We prove interior $L^p$ gradient weighted estimates for degenerate elliptic equations in divergence form with VMO coefficients.

On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain

Abstract: In this paper we prove a stability result for some classes of elliptic problems involving variable exponents More precisely, we consider the Dirichlet problem for an elliptic equation in a domain Ω, which is the p–Laplacian equation, −div(|∇u|p−2∇u) = f , in a subdomain Ω1 of Ω and the Laplace equation, −∆u = f , in its complementary (that is, our equation involves the so–called p(x)–Laplacian with a discontinuous exponent) We assume that the right-hand side f belongs to L∞(Ω) For this problem, we study the behaviour of the solutions as p goes to 1, showing that they converge to a function u, which is almost everywhere finite when the size of the datum f is small enough Moreover, we prove that this u is a solution of a limit problem involving the 1–Laplacian operator in Ω1 We also discuss uniqueness under a favorable geometry

The Davey Stewartson system in weak $L^p$ spaces

Abstract: We study the global Cauchy problem associated to the Davey-Stewartson system in ${\mathbb{R}}^n,\ n=2,3$ Existence and uniqueness of the solution are established for small data in some weak $L^p$ space We apply an interpolation theorem and the generalization of the Strichartz estimates for the Schrodinger equation derived in [9] As a consequence we obtain self-similar solutions

A Mathematical analysis of a model of structured population (I)

Abstract: This work deals with a mathematical analysis of an age-cycle structured population endowed with a transition rate and with a general biological rule. We show that the corresponding model is governed by a strongly continuous positive semigroup. Some of its spectral properties are proved.

Location of the blow-up set for a superlinear heat equation with small diffusion

Abstract: We consider the blow-up problem for a superlinear heat equation $$ \begin{cases} \partial_t u=\epsilon\Delta u+f(u), &x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, &x\in\partial\Omega, \,\,\, t>0 \quad\mbox{if}\quad \partial\Omega ot=\emptyset, \\ u(x,0)=\varphi_\epsilon(x)\ge 0\, ( ot\equiv 0), &x\in\Omega, \end{cases} $$ where $\epsilon>0$, $N\ge 1$, $\Omega$ is a domain in ${\bf R}^N$, $f=f(s)$ is a convex function in $s\in (0,\infty)$, and the initial function $\varphi_\epsilon$ is a nonnegative bounded continuous function in $\overline{\Omega}$. The typical examples of $f$ that we treat in this paper, are $f(u)=(u+\lambda)^p$ ($p>1$, $\lambda\ge 0$) and $f(u)=e^u$. In this paper, under suitable assumptions, we prove that the solution $u_\epsilon$ blows up only near the maximum points of the initial function $\varphi_\epsilon$ if $\epsilon>0$ is sufficiently small.

A characterization of the mountain pass geometry for functionals bounded from below

Abstract: In this paper it is proved that, when a regular functional is bounded from below, the mountain pass geometry and the existence of at least two distinct local minima are equivalent conditions. As a consequence, the classical mountain pass theorem, under the additional assumption of boundedness from below of the functional, ensures actually three distinct critical points. Moreover, as application, the existence of three solutions to Hamiltonian systems is established.

Global existence and blow up results for a heat equation with nonlinear nonlocal term

Abstract: We study the nonlocal parabolic equation $$ u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s) \, ds $$ with $p>1$. We assume that $k$ is continuous and there exists $\gamma \in {\mathbb{R}}$ such that $k(\lambda t, \lambda s)=\lambda^{-\gamma}k(t,s)$ for all $\lambda>0$, $0