Mostrando recursos 1 - 20 de 33

  1. Existence of entropy solutions to a doubly nonlinear integro-differential equation

    Scholtes, Martin; Wittbold, Petra
    We consider a class of doubly nonlinear history-dependent problems associated with the equation $$ \partial_{t}k\ast(b(v)- b(v_{0})) = \text{div}\, a(x,Dv) + f . $$ Our assumptions on the kernel $k$ include the case $k(t) = t^{-\alpha}/\Gamma(1-\alpha)$, in which case the left-hand side becomes the fractional derivative of order $\alpha\in (0,1)$ in the sense of Riemann-Liouville. Existence of entropy solutions is established for general $L^{1}-$data and Dirichlet boundary conditions. Uniqueness of entropy solutions has been shown in a previous work.

  2. Existence of entropy solutions to a doubly nonlinear integro-differential equation

    Scholtes, Martin; Wittbold, Petra
    We consider a class of doubly nonlinear history-dependent problems associated with the equation $$ \partial_{t}k\ast(b(v)- b(v_{0})) = \text{div}\, a(x,Dv) + f . $$ Our assumptions on the kernel $k$ include the case $k(t) = t^{-\alpha}/\Gamma(1-\alpha)$, in which case the left-hand side becomes the fractional derivative of order $\alpha\in (0,1)$ in the sense of Riemann-Liouville. Existence of entropy solutions is established for general $L^{1}-$data and Dirichlet boundary conditions. Uniqueness of entropy solutions has been shown in a previous work.

  3. Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition

    Kim, Jae-Myoung; Kim, Yun-Ho; Lee, Jongrak
    We are concerned with the following elliptic equations with variable exponents \begin{equation*} -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u) \quad \text{in} \quad \mathbb R^{N}, \end{equation*} where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \mathbb R^{N} \to (1,\infty)$, $V: \mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $f: \mathbb R^{N}\times \mathbb R \to \mathbb R$ satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass...

  4. Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition

    Kim, Jae-Myoung; Kim, Yun-Ho; Lee, Jongrak
    We are concerned with the following elliptic equations with variable exponents \begin{equation*} -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u) \quad \text{in} \quad \mathbb R^{N}, \end{equation*} where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \mathbb R^{N} \to (1,\infty)$, $V: \mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $f: \mathbb R^{N}\times \mathbb R \to \mathbb R$ satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass...

  5. Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

    do Ó, João Marcos; de Albuquerque, José Carlos
    In this paper, we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \begin{cases} (-\Delta)^{ \frac 12 } u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},\\ (-\Delta)^{ \frac 12 } v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{cases} $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is...

  6. Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

    do Ó, João Marcos; de Albuquerque, José Carlos
    In this paper, we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \begin{cases} (-\Delta)^{ \frac 12 } u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},\\ (-\Delta)^{ \frac 12 } v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{cases} $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is...

  7. A class of differential operators with complex coefficients and compact resolvent

    Behncke, Horst; Hinton, Don
    We consider the problem of the a second order singular differential operator with complex coefficients in the discrete spectrum case. The Titchmarsh-Weyl m-function is constructed without the use of nesting circles, and it is then used to give a representation of the resolvent operator. Under conditions on the growth of the coefficients, the resolvent operator is proved to be Hilbert-Schmidt and the root subspaces are shown to be complete in the associated Hilbert space. The operator is considered on both the half line and whole line cases.

  8. A class of differential operators with complex coefficients and compact resolvent

    Behncke, Horst; Hinton, Don
    We consider the problem of the a second order singular differential operator with complex coefficients in the discrete spectrum case. The Titchmarsh-Weyl m-function is constructed without the use of nesting circles, and it is then used to give a representation of the resolvent operator. Under conditions on the growth of the coefficients, the resolvent operator is proved to be Hilbert-Schmidt and the root subspaces are shown to be complete in the associated Hilbert space. The operator is considered on both the half line and whole line cases.

  9. On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$

    Strütt, David
    We study the system of linear partial differential equations given by \[ dw+a\wedge w=f, \] on open subsets of $\mathbb R^n$, together with the algebraic equation \[ da\wedge u=\beta, \] where $a$ is a given $1$-form, $f$ is a given $(k+1)$-form, $\beta$ is a given $k+2$-form, $w$ and $u$ are unknown $k$-forms. We show that if $\text{rank}[da]\geq 2(k+1)$ those equations have at most one solution, if $\text{rank}[da] \equiv 2m \geq 2(k+2)$ they are equivalent with $\beta=df+a\wedge f$ and if $\text{rank}[da]\equiv 2 m\geq2(n-k)$ the first equation always admits a solution. ¶ Moreover, the differential equation is closely linked to the Poincaré...

  10. On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$

    Strütt, David
    We study the system of linear partial differential equations given by \[ dw+a\wedge w=f, \] on open subsets of $\mathbb R^n$, together with the algebraic equation \[ da\wedge u=\beta, \] where $a$ is a given $1$-form, $f$ is a given $(k+1)$-form, $\beta$ is a given $k+2$-form, $w$ and $u$ are unknown $k$-forms. We show that if $\text{rank}[da]\geq 2(k+1)$ those equations have at most one solution, if $\text{rank}[da] \equiv 2m \geq 2(k+2)$ they are equivalent with $\beta=df+a\wedge f$ and if $\text{rank}[da]\equiv 2 m\geq2(n-k)$ the first equation always admits a solution. ¶ Moreover, the differential equation is closely linked to the Poincaré...

  11. On the global well-posedness of 3-d Navier-Stokes equations with vanishing horizontal viscosity

    Abidi, Hammadi; Paicu, Marius
    We study, in this paper, the axisymmetric $3$-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data in Lebesgue spaces.

  12. On the global well-posedness of 3-d Navier-Stokes equations with vanishing horizontal viscosity

    Abidi, Hammadi; Paicu, Marius
    We study, in this paper, the axisymmetric $3$-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data in Lebesgue spaces.

  13. Horizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamics

    Yamazaki, Kazuo
    We derive a Biot-Savart law type identity for the horizontal components of the solution to the fluid system of equations with incompressibility in general dimension. Along with another new decomposition of non-linear terms, we give its application to derive two regularity criteria for the four-dimensional magneto-hydrodynamics system, in particular a criteria in terms of two velocity field components, two magnetic field components and two partial derivatives of the other two magnetic field components in a scaling-invariant norm. It is an open problem to obtain a criterion in terms of just two velocity field components and two partial derivatives of two...

  14. Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain

    Bortot, César Augusto; Corrêa, Wellington José
    In this paper, we study the exponential stability for the semilinear defocusing Schrödinger equation with locally distributed damping on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$. The proofs are based on a result of unique continuation property due to Cavalcanti et al. [15] and on a forced smoothing effect due to Aloui [2] combined with ideas from Cavalcanti et. al. [15], [16] adapted to the present context.

  15. Nodal solutions for Lane-Emden problems in almost-annular domains

    Amadori, Anna Lisa; Gladiali, Francesca; Grossi, Massimo
    In this paper, we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{ in } \Omega , \\ u= 0 & \text{ on } \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in $\mathbb R^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1 < p_2 < \cdots$ with $\lim\limits_{k\rightarrow+\infty}p_k=+\infty$ such that for any real number $p > 1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide...

  16. Quasilinear elliptic systems convex-concave singular terms and $\Phi$-Laplacian operator

    Gonçalves, José V.; Carvalho, Marcos L.; Santos, Carlos Alberto
    This paper deals with the existence of positive solutions for a class of quasilinear elliptic systems involving the $\Phi$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbation techniques and comparison arguments in the setting of Orlicz-Sobolev spaces.

  17. A nondivergence parabolic problem with a fractional time derivative

    Allen, Mark
    We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Hölder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative $\alpha \to 1$.

  18. Global well posedness for a two-fluid model

    Giga, Yoshikazu; Ibrahim, Slim; Shen, Shengyi; Yoneda, Tsuyoshi
    We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently...

  19. On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise

    Kukavica, Igor; Uğurlu, Kerem; Ziane, Mohammed
    We investigate the convergence of the Galerkin approximations for the stochastic Navier-Stokes equations in an open bounded domain $\mathcal{O}$ with the non-slip boundary condition. We prove that \begin{equation*} \mathbb{E} \Big [ \sup_{t \in [0,T]} \phi_1(\lVert (u(t)-u^n(t)) \rVert^2_V) \Big ] \rightarrow 0, \end{equation*} as $n \rightarrow \infty$ for any deterministic time $T > 0$ and for a specified moment function $\phi_1$ where $u^n(t)$ denotes the Galerkin approximations of the solution $u(t)$. Also, we provide a result on uniform boundedness of the moment $\mathbb{E} [ \sup_{t \in [0,T]} \phi(\lVert u(t) \rVert^2_V) ] $ where $\phi$ grows as a single logarithm at infinity....

  20. Global stability of an SIS epidemic model with a finite infectious period

    Nakata, Yukihiko; Röst, Gergely
    Assuming a general distribution for the sojourn time in the infectious class, we consider an SIS type epidemic model formulated as a scalar integral equation. We prove that the endemic equilibrium of the model is globally asymptotically stable whenever it exists, solving the conjecture of Hethcote and van den Driessche (1995) for the case of nonfatal diseases.

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