Mostrando recursos 1 - 20 de 31

  1. Long-time behavior of solutions to the fifth-order modified KdV-type equation

    Okamoto, Mamoru
    We consider the long-time behavior of solutions to the fifth-order modified KdV-type equation. For the local-in-time well-posedness, we show that regularity conditions where by a tri-linear estimate holds in the Fourier restriction norm spaces, which is an extension of Kwon's result (2008). Using the method of testing by wave packets, we prove the small-data global existence and modified scattering. We derive the leading asymptotic in both the self-similar and oscillatory regions.

  2. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and higher dimensions

    Kato, Isao; Kinoshita, Shinya
    We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1} (\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.

  3. Global existence for the heat flow of symphonic maps into spheres

    Misawa, Masashi; Nakauchi, Nobumitsu
    In our previous papers, we introduce symphonic maps ([9]) and show a Hölder continuity of symphonic maps from domains of $\mathbb{R}^4$ into the spheres ([6], [7]). In this paper, we consider the heat flow of symphonic maps with values into spheres and prove a global existence of a weak solution to the Cauchy-Dirichlet problem for any given initial and boundary data.

  4. On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential for any given initial and boundary data.

    Cho, Yonggeun; Hwang, Gyeongha; Yang, Changhun
    In this paper, we study 3-d Hartree type fractional Schrödinger equations: $$ i\partial_{t}u-\vert\nabla\vert^{\alpha}u = \lambda \left ( |x|^{-\gamma} *\vert u\vert^{2} \right ) u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\; \lambda \in \mathbb R \setminus \{0\}. $$ In [7] it is known that no scattering occurs in $L^2$ for the long range ($0 < \gamma \le 1$). In [4, 10, 8] the short-range scattering ($1 < \gamma < 3$) was treated for the scattering in $H^s$. In this paper, we consider the critical case ($\gamma = 1$) and prove a modified scattering in $L^\infty$ on the frequency to the...

  5. Ground and bound state solutions for a Schrödinger system with linear and nonlinear couplings in $\mathbb{R}^N$

    Perera, Kanishka; Tintarev, Cyril; Wang, Jun; Zhang, Zhitao
    We study the existence of ground and bound state solutions for a system of coupled Schrödinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.

  6. Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity

    Sobajima, Motohiro; Wakasugi, Yuta
    In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation having a degenerate diffusion at spatial infinity and weighted energy estimates for the damped wave equation. To construct a suitable weight function for the energy estimates, we study a certain elliptic problem.

  7. Existence results for non-local elliptic systems with nonlinearities interacting with the spectrum

    Miyagaki, Olímpio H.; Pereira, Fábio
    In this work, we establish an existence result for a class of non-local variational elliptic systems with critical growth, but with nonlinearities interacting with the fractional laplacian spectrum. More specifically, we treat the situation when the interval defined by two eigenvalues of the real matrix coming from the linear part contains an eigenvalue of the spectrum of the fractional laplacian operator. In this case, there are situations where resonance or double resonance phenomena can occur. The novelty here is because, up to our knowledge, the results that have been appeared in the literature up to now, this interval does not...

  8. Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces

    Costea, Nicuşor; Moroşanu, Gheorghe; Varga, Csaba
    We study PDI's of the type $-\Delta_\Phi u\in \partial_C f(x,u)$ subject to Dirichlet boundary condition in a bounded domain $\Omega\subset\mathbb{R}^N$ with Lipschitz boundary $\partial\Omega$. Here, $\Phi:\mathbb{R}\rightarrow [0,\infty)$ is the $N$-function defined by $\Phi(t):=\int_0^t a(|s|)s\,ds$, with $a:(0,\infty)\rightarrow (0,\infty)$ a prescribed function, not necessarily differentiable, and $\Delta_\Phi u:={\rm div}(a(|\nabla u|)\nabla u)$ is the $\Phi$-Laplacian. In addition, $f:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ is a locally Lipschitz function with respect to the second variable and $\partial_C$ denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also...

  9. Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity

    Majdoub, Mohamed; Otsmane, Sarah; Tayachi, Slim
    In this paper, we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\; t > 0,\; x \in \mathbb R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^m$ as $u\to 0,$ $m\geq 2$, $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.

  10. Mountain pass solutions for the fractional Berestycki-Lions problem

    Ambrosio, Vincenzo
    We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation \begin{align*} (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb R^{N}, \end{align*} where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian and $g: \mathbb R \rightarrow \mathbb R $ is an odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in [33]. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above...

  11. Traveling waves in a simplified gas-solid combustion model in porous media

    Ozbag, Fatih; Schecter, Stephen; Chapiro, Grigori
    We study the combustion waves that occur when air is injected into a porous medium containing initially some solid fuel and prove the existence of traveling waves using phase plane analysis. We also identify all the possible ways that combustion waves and contact discontinuities can combine to produce wave sequences that solve boundary value problems on infinite intervals with generic constant boundary data.

  12. Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity

    Sasaki, Takiko
    We study a blow-up curve for the one dimensional wave equation $\partial_t^2 u- \partial_x^2 u = |\partial_t u|^p$ with $p>1$. The purpose of this paper is to show that the blow-up curve is a $C^1$ curve if the initial values are large and smooth enough. To prove the result, we convert the equation into a first order system, and then apply a modification of the method of Caffarelli and Friedman [2]. Moreover, we present some numerical investigations of the blow-up curves. From the numerical results, we were able to confirm that the blow-up curves are smooth if the initial values...

  13. Regularity and time behavior of the solutions of linear and quasilinear parabolic equations

    Porzio, Maria Michaela
    In this paper, we study the regularity, the uniqueness and the asymptotic behavior of the solutions to a class of nonlinear operators in dependence of the summability properties of the datum $f$ and of the initial datum $u_0$. The case of only summable data $f$ and $u_0$ is allowed. We prove that these equations satisfy surprising regularization phenomena. Moreover, we prove estimates (depending continuously from the data) that for zero datum $f$ become well known decay (or ultracontractive) estimates.

  14. The Friedrichs extension for elliptic wedge operators of second order

    Krainer, Thomas; Mendoza, Gerardo A.
    Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such...

  15. The Friedrichs extension for elliptic wedge operators of second order

    Krainer, Thomas; Mendoza, Gerardo A.
    Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such...

  16. Asymptotics for the modified Boussinesq equation in one space dimension

    Hayashi, Nakao; Naumkin, Pavel I.
    We consider the Cauchy problem for the modified Boussinesq equation in one space dimension \begin{equation*} \begin {cases} w_{tt}=a^{2}\partial _{x}^{2}w-\partial _{x}^{4}w+\partial _{x}^{2} ( w^{3} ) ,\text{ } ( t,x ) \in \mathbb{R}^{2}, \\ w ( 0,x ) =w_{0} ( x ) ,\text{ }w_{t} ( 0,x ) =w_{1} ( x ) ,\text{ }x\in \mathbb{R}\text{,} \end {cases} \end{equation*} where $a > 0.$ We study the large time asymptotics of solutions to the Cauchy problem for the modified Boussinesq equation. We apply the factorization technique developed recently in papers [5], [6], [7], [8].

  17. Asymptotics for the modified Boussinesq equation in one space dimension

    Hayashi, Nakao; Naumkin, Pavel I.
    We consider the Cauchy problem for the modified Boussinesq equation in one space dimension \begin{equation*} \begin {cases} w_{tt}=a^{2}\partial _{x}^{2}w-\partial _{x}^{4}w+\partial _{x}^{2} ( w^{3} ) ,\text{ } ( t,x ) \in \mathbb{R}^{2}, \\ w ( 0,x ) =w_{0} ( x ) ,\text{ }w_{t} ( 0,x ) =w_{1} ( x ) ,\text{ }x\in \mathbb{R}\text{,} \end {cases} \end{equation*} where $a > 0.$ We study the large time asymptotics of solutions to the Cauchy problem for the modified Boussinesq equation. We apply the factorization technique developed recently in papers [5], [6], [7], [8].

  18. Global regularity of the 2D magnetic Bénard system with partial dissipation

    Ye, Zhuan
    In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or...

  19. Global regularity of the 2D magnetic Bénard system with partial dissipation

    Ye, Zhuan
    In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or...

  20. On the focusing energy-critical fractional nonlinear Schrödinger equations

    Cho, Yonggeun; Hwang, Gyeongha; Ozawa, Tohru
    We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion $|\nabla|^{\alpha}$ and focusing energy-critical Hartree type nonlinearity $[-(|x|^{-2{\alpha}}*|u|^2)u]$. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].

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