Mostrando recursos 1 - 20 de 2.077

  1. The time derivative in a singular parabolic equation

    Lindqvist, Peter
    We study the Evolutionary $p$-Laplace Equation in the singular case $1 < p < 2$. We prove that a weak solution has a time derivative (in Sobolev's sense) which is a function belonging (locally) to a $L^q$-space.

  2. Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions

    Kato, Isao; Tsugawa, Kotaro
    We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $ (u(0), n(0), \partial_t n(0) )\in H^k(\mathbb R^d)\times \dot{H}^l(\mathbb R^d)\times \dot{H}^{l-1}(\mathbb R^d)$. According to Ginibre, Tsutsumi and Velo ([9]), the critical exponent of $(k,l)$ is $ ((d-3)/2,(d-4)/2 ). $ We prove the small data global well-posedness and the scattering at the critical space. It seems difficult to get the crucial bilinear estimate only by applying the $U^2,\ V^2$ type spaces introduced by Koch and Tataru ([23], [24]). To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.

  3. Logarithmic NLS equation on star graphs: Existence and stability of standing waves

    Ardila, Alex H.
    In this paper, we consider the logarithmic Schrödinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the existence of several families of standing waves. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we show that the ground states are orbitally stable via a variational approach.

  4. Well-posedness and flow invariance for semilinear functional differential equations governed by non-densely defined operators

    Sano, Hiroki; Tanaka, Naoki
    The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild solutions is introduced through a new type of variation of constants formula and the well-posedness is established under a semilinear stability condition with respect to a metric-like functional and a subtangential condition. The abstract result is applied to a size-structured model with birth delay.

  5. Almost automorphic solutions of Volterra equations on time scales

    Lizama, Carlos; Mesquitan, Jaqueline G.; Ponce, Rodrigo; Toon, Eduard
    The existence and uniqueness of almost automorphic solutions for linear and semilinear nonconvolution Volterra equations on time scales is studied. The existence of asymptotically almost automorphic solutions is proved. Examples that illustrate our results are given.

  6. On multiple solutions for nonlocal fractional problems via $\nabla$-theorems

    Molica Bisci, Giovanni; Mugnai, Dimitri; Servadei, Raffaella
    The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb R^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset \mathbb R^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper, we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. For this purpose, we employ a variational theorem of mixed type...

  7. Nonlocal variational constants of motion in dissipative dynamics

    Gorni, Gianluca; Zampieri, Gaetano
    We give a recipe to generate ``nonlocal'' constants of motion for ODE Lagrangian systems and we apply the method to find useful constants of motion which permit to prove global existence and estimates of solutions to dissipative mechanical systems, and to the Lane-Emden equation.

  8. Exact Evolution versus mean field with second-order correction for Bosons interacting via short-range two-body potential

    Kuz, Elif
    We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.

  9. A classification of solutions of a fourth order semi-linear elliptic equation in $\mathbb R^n$

    Chammakhi, Ridha; Harrabi, Abdellaziz; Selmi, Abdelbaki
    In this paper, we classify all regular sign changing~solutions~of $$ \Delta ^2 u=u_+^{p} \,\,\,\mbox {in}\, \, \mathbb R^n\ \ \,\,u_+^{p}\in L^1(\mathbb R^n), $$ where $\Delta ^2$ denotes the biharmonic operator in $\mathbb R^n$, $1 < p\leq \frac{n}{n-4}$ and $n\geq 5$. We prove by using the procedure of moving parallel planes that such solutions are radially symmetric about some point in $\mathbb R^n$. We also present a sup+inf type inequality for regular solutions of the following equation: $$ (-\Delta )^m u=u_+^{p}\,\,\,\mbox{in}\,\,\, \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^n$, $m\geq1$, $n\geq 2m+1$ and $p\in (1,(n+2m)/(n-2m) )$.

  10. A three critical points result in a bounded domain of a Banach space and applications

    Precup, Radu; Pucci, Patrizia; Varga, Csaba
    Using the bounded mountain pass lemma and the Ekeland variational principle, we prove a bounded version of the Pucci-Serrin three critical points result in the intersection of a ball with a wedge in a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems.

  11. The initial boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line

    Cavalcante, Márcio
    We prove local well-posedness for the initial-boundary value problem associated to some quadratic nonlinear Schrödinger equations on the half-line. The results are obtained in the low regularity setting by introducing an analytic family of boundary forcing operators, following the ideas developed in [14].

  12. Remarks on large time behavior of the $L^{2}$-norm of solutions to strongly damped wave equations

    Ikehata, Ryo; Onodera, Michiaki
    We reconsider the asymptotic behavior as $t \to +\infty$ of the $L^{2}$-norm of solutions to strongly damped wave equations in ${\bf R}^{n}$ with weighted initial data in the $n=1,2$ dimensional case. As an application, we shall apply those results to the analysis of the linearized compressible Navier-Stokes equations in ${\bf R}^{3}$.

  13. Heat equation with a nonlinear boundary condition and growing initial data

    Ishige, Kazuhiro; Sato, Ryuichi
    We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u, & x\in\Omega,\,t > 0, \\ \nabla u\cdot\nu(x)=u^p,\qquad &x\in\partial\Omega,\,\,t > 0, \\ u(x,0)=\varphi(x)\ge 0, & x\in\Omega, \end{array} \right. $$ where $N\ge 1$, $p > 1$, $\Omega$ is a smooth domain in ${\bf R}^N$ and $\varphi(x)=O(e^{\lambda d(x)^2})$ as $d(x)\to\infty$ for some $\lambda\ge 0$. Here, $d(x)=\mbox{dist}\,(x,\partial\Omega)$. Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary $\partial\Omega$.

  14. Nodal solutions to problem with mean curvature operator in Minkowski space

    Dai, Guowei; Wang, Jun
    This paper is devoted to investigate the existence and multiplicity of radial nodal solutions for the following Dirichlet problem with mean curvature operator in Minkowski space \begin{eqnarray} \begin{cases} -\text{div} \Big (\frac{\nabla v}{\sqrt{1-\vert \nabla v\vert^2}} \Big ) = \lambda f(\vert x\vert,v)\,\, &\text{in}\,\, B_R(0),\\ v=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial B_R(0). \end{cases} \nonumber \end{eqnarray} By bifurcation approach, we determine the interval of parameter $\lambda$ in which the above problem has two or four radial nodal solutions which have exactly $n-1$ simple zeros in $(0,R)$ according to linear/sublinear/ superlinear nonlinearity at zero. The asymptotic behaviors of radial nodal solutions as $\lambda \to +\infty$ and $n \to +\infty$ are also studied.

  15. Pullback dynamics of non-autonomous wave equations with acoustic boundary condition

    Ma, To Fu; Souza, Thales Maier
    This paper is concerned with a class of wave equations with acoustic boundary condition subject to non-autonomous external forces. Under some general assumptions, the problem generates a well-posed evolution process. Then, we establish the existence of a minimal pullback attractor within a universe of tempered sets defined by the forcing terms. We also, study the upper semicontinuity of attractors as the non-autonomous perturbation tends to zero. Our results allow unbounded external forces and nonlinearities with critical growth.

  16. Uniform bounds for solutions to volume-surface reaction diffusion systems

    Sharma, Vandana; Morgan, Jeff
    We consider reaction-diffusion systems where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish criteria guaranteeing that solutions are uniformly bounded in time.

  17. Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

    Bhakta, Mousomi; Mukherjee, Debangana
    In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda |u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a smooth bounded domain in $ \mathbb R^N $, $N > 2s$, $s\in(0, 1)$, $0 < q < 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to the fractional laplacian operator $-(-\Delta)^s $, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $ \lambda =1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign...

  18. Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

    Green, William R.; Toprak, Ebru
    We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions,...

  19. 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.

  20. 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.

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