On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

A two-point problem for first-order systems

Let f be a continuous function from [a, b] ×\mathbbRⁿ [a, b] \times \mathbb{R}^n into \mathbbRⁿ \mathbb{R}^n . In this paper we prove that the problem¶¶ { llu^￠ = f(t,u)+ lu(a)=u(b)=0  \left \{ \begin{array}{ll}u^{\prime}= f(t,u)+ \lambda \\[3pt]u(a)=u(b)=0\end{array}\right.\ ¶¶ has a (classical) solution for a wide class of functions f. Next we point out a particular case.     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

Inequalities of Hlawka type for matrices

A generalized Hlawka's inequality says that for any n (\geqq 2) (\geqq 2) complex numbers¶ x₁, x₂, ..., x_n,¶¶ ?_i=1ⁿ|x_i - ?_j=1ⁿx_j| \leqq ?_i=1ⁿ|x_i| + (n - 2)|?_j=1ⁿx_j|. \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. ¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶ ||A - Tr A ||₁ \leqq ||A||₁ + (n - 2)| Tr A|. ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. ¶¶ We consider also the related inequalities for p-norms (1 \leqq p \leqq ￥) (1 \leqq p \leqq \infty) on matrices.     

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient

We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR² é W? \mathbbR² u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      8. Harmonicity of functions satisfying a weak form of the mean value property      M. Hasson 《Archiv der Mathematik》2001,76(4):283-291 Let W ì \Bbb Rn\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C0(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-òSr([`(x)]) u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -òSr([`(x)]) u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere Sr([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-òSr([`(x)]) u d s- a = o(r2)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0.      9. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      10. Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity      Yongsheng Li  Wei Yan  Xingyu Yang 《Journal of Evolution Equations》2010,10(2):465-486 In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the H s (R) with ${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.} As a consequence of the conservation of the energy ||u||H1(R),{{||u||_{H^{1}(R)},}} we have the global well-posedness for the initial data in H 1(R).      11. Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces      K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573 Let W ì \mathbbRn \Omega \subset \mathbb{R}^n be an open set and l(x) | u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}      12. Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity      Jiabao Su 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,21(2):51-62 We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation {ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ￥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.      13. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      14. Banach Spaces of Solutions of the Helmholtz Equation in the Plane      Josefina Alvarez  Magali Folch-Gabayet  Salvador Pérez-Esteva 《Journal of Fourier Analysis and Applications》2001,7(1):49-62 The purpose of this article is to study the Hilbert space W2\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR2\Bbb{R}^2 that are the image under the Fourier transform of L2L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H2\mathcal{ H}^2 of all functions belonging to L2( | x | -3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR2\Bbb{R}^2. We calculate the reproducing kernel of W2\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces Hp\mathcal{H}^p, for $p>1$p>1.      15. Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid      S. Poborchi 《Journal of Mathematical Sciences》2011,175(3):363-374 We consider the Dirichlet problem for the equation - \textdiv( | ?u |p - 2?u ) + a| u |p - 2u = 0, - {\text{div}}\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + a{\left| u \right|^{p - 2}}u = 0,      16. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      17. Precise Asymptotic Formulas for Semilinear Eigenvalue Problems      T. Shibata 《Annales Henri Poincare》2001,2(4):713-732 . We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L2-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = ln (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L2 norm of the eigenfunction associated with l \lambda .      18. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      19. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      20. Asymptotisches Verhalten der Maxima der Lösungen gewisser Differentialgleichungen      Gerd Herzog  Raymond M. Redheffer 《Archiv der Mathematik》1999,72(2):132-144 Eine oszillatorische Lösung u der Gleichung u" + qu = 0u' + qu = 0 hat die Wintner-Eigenschaft, wenn die relativen Maxima von q1/4 |u|q^{1/4} |u| für t ? ￥t \to \infty gegen einen positiven endlichen Grenzwert streben. In Verallgemeinerung eines Satzes von Hartman werden neue Methoden zum Nachweis der Wintner-Eigenschaft beschrieben. Andererseits werden Funktionen q konstruiert, die belegen, daß i.a. das Verhalten von q1/4 uq^{1/4} u fast beliebig ist. Durch Methoden aus dem Bereich der universellen Funktionen wird die Residualität solcher Funktionen q bewiesen.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Harmonicity of functions satisfying a weak form of the mean value property

Let W ì \Bbb Rⁿ\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C⁰(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-ò_{Sr([`(x)])</sub> u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -ò<sub>Sr([`(x)])} u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere S_r([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W^2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-ò_Sr([`(x)]) u d s- a = o(r²)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0.     

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations

ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      10. Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity      Yongsheng Li  Wei Yan  Xingyu Yang 《Journal of Evolution Equations》2010,10(2):465-486 In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the H s (R) with ${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.} As a consequence of the conservation of the energy ||u||H1(R),{{||u||_{H^{1}(R)},}} we have the global well-posedness for the initial data in H 1(R).      11. Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces      K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573 Let W ì \mathbbRn \Omega \subset \mathbb{R}^n be an open set and l(x) | u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}      12. Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity      Jiabao Su 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,21(2):51-62 We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation {ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ￥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.      13. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      14. Banach Spaces of Solutions of the Helmholtz Equation in the Plane      Josefina Alvarez  Magali Folch-Gabayet  Salvador Pérez-Esteva 《Journal of Fourier Analysis and Applications》2001,7(1):49-62 The purpose of this article is to study the Hilbert space W2\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR2\Bbb{R}^2 that are the image under the Fourier transform of L2L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H2\mathcal{ H}^2 of all functions belonging to L2( | x | -3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR2\Bbb{R}^2. We calculate the reproducing kernel of W2\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces Hp\mathcal{H}^p, for $p>1$p>1.      15. Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid      S. Poborchi 《Journal of Mathematical Sciences》2011,175(3):363-374 We consider the Dirichlet problem for the equation - \textdiv( | ?u |p - 2?u ) + a| u |p - 2u = 0, - {\text{div}}\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + a{\left| u \right|^{p - 2}}u = 0,      16. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      17. Precise Asymptotic Formulas for Semilinear Eigenvalue Problems      T. Shibata 《Annales Henri Poincare》2001,2(4):713-732 . We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L2-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = ln (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L2 norm of the eigenfunction associated with l \lambda .      18. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      19. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      20. Asymptotisches Verhalten der Maxima der Lösungen gewisser Differentialgleichungen      Gerd Herzog  Raymond M. Redheffer 《Archiv der Mathematik》1999,72(2):132-144 Eine oszillatorische Lösung u der Gleichung u" + qu = 0u' + qu = 0 hat die Wintner-Eigenschaft, wenn die relativen Maxima von q1/4 |u|q^{1/4} |u| für t ? ￥t \to \infty gegen einen positiven endlichen Grenzwert streben. In Verallgemeinerung eines Satzes von Hartman werden neue Methoden zum Nachweis der Wintner-Eigenschaft beschrieben. Andererseits werden Funktionen q konstruiert, die belegen, daß i.a. das Verhalten von q1/4 uq^{1/4} u fast beliebig ist. Durch Methoden aus dem Bereich der universellen Funktionen wird die Residualität solcher Funktionen q bewiesen.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity

In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the H ^s (R) with ${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.} As a consequence of the conservation of the energy ||u||_H¹(R),{{||u||_{H^{1}(R)},}} we have the global well-posedness for the initial data in H ¹(R).     

Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Let W ì \mathbbRⁿ \Omega \subset \mathbb{R}^n be an open set and l(x) | u |_p,l = ( ò_W l^p (x)| u(x) |^p dx )^1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}     

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      14. Banach Spaces of Solutions of the Helmholtz Equation in the Plane      Josefina Alvarez  Magali Folch-Gabayet  Salvador Pérez-Esteva 《Journal of Fourier Analysis and Applications》2001,7(1):49-62 The purpose of this article is to study the Hilbert space W2\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR2\Bbb{R}^2 that are the image under the Fourier transform of L2L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H2\mathcal{ H}^2 of all functions belonging to L2( | x | -3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR2\Bbb{R}^2. We calculate the reproducing kernel of W2\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces Hp\mathcal{H}^p, for $p>1$p>1.      15. Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid      S. Poborchi 《Journal of Mathematical Sciences》2011,175(3):363-374 We consider the Dirichlet problem for the equation - \textdiv( | ?u |p - 2?u ) + a| u |p - 2u = 0, - {\text{div}}\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + a{\left| u \right|^{p - 2}}u = 0,      16. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      17. Precise Asymptotic Formulas for Semilinear Eigenvalue Problems      T. Shibata 《Annales Henri Poincare》2001,2(4):713-732 . We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L2-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = ln (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L2 norm of the eigenfunction associated with l \lambda .      18. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      19. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      20. Asymptotisches Verhalten der Maxima der Lösungen gewisser Differentialgleichungen      Gerd Herzog  Raymond M. Redheffer 《Archiv der Mathematik》1999,72(2):132-144 Eine oszillatorische Lösung u der Gleichung u" + qu = 0u' + qu = 0 hat die Wintner-Eigenschaft, wenn die relativen Maxima von q1/4 |u|q^{1/4} |u| für t ? ￥t \to \infty gegen einen positiven endlichen Grenzwert streben. In Verallgemeinerung eines Satzes von Hartman werden neue Methoden zum Nachweis der Wintner-Eigenschaft beschrieben. Andererseits werden Funktionen q konstruiert, die belegen, daß i.a. das Verhalten von q1/4 uq^{1/4} u fast beliebig ist. Durch Methoden aus dem Bereich der universellen Funktionen wird die Residualität solcher Funktionen q bewiesen.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid

We consider the Dirichlet problem for the equation

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let X ì \mathbb Rⁿ{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional

Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Asymptotisches Verhalten der Maxima der Lösungen gewisser Differentialgleichungen

Eine oszillatorische Lösung u der Gleichung u" + qu = 0u' + qu = 0 hat die Wintner-Eigenschaft, wenn die relativen Maxima von q^1/4 |u|q^{1/4} |u| für t ? ￥t \to \infty gegen einen positiven endlichen Grenzwert streben. In Verallgemeinerung eines Satzes von Hartman werden neue Methoden zum Nachweis der Wintner-Eigenschaft beschrieben. Andererseits werden Funktionen q konstruiert, die belegen, daß i.a. das Verhalten von q^1/4 uq^{1/4} u fast beliebig ist. Durch Methoden aus dem Bereich der universellen Funktionen wird die Residualität solcher Funktionen q bewiesen.