Positive solutions of a Schr?dinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation -Du+la(x)u=mu+u^2*-1,  u ? \mathbbR^N,-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N}, with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

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}      3. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      4. On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures      V. I. Bogachev  M. R?ckner  S. V. Shaposhnikov 《Journal of Mathematical Sciences》2011,179(1):7-47 We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ t μ = L * μ for bounded Borel measures on ℝ d  × (0, T), where L is a second order elliptic operator, for example, Lu = Dxu + ( b,?xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity ò( ?tu + Lu )dm = 0 \int \left( {{\partial_t}u + Lu} \right)d\mu = 0      5. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      6. Multiplicity of solutions for a fourth order equation with power-type nonlinearity      Juan Dávila  Isabel Flores  Ignacio Guerra 《Mathematische Annalen》2010,348(1):143-193 Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbRN{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      7. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      8. Growth rate for beta-expansions      De-Jun Feng  Nikita Sidorov 《Monatshefte für Mathematik》2011,50(11):41-60 Let β > 1 and let m > β be an integer. Each x ? Ib:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form x=?k=1￥ ekb-k,x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},      9. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      10. 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.      11. Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature      Manuel del Pino  Michal Kowalczyk  Juncheng Wei  Jun Yang 《Geometric And Functional Analysis》2010,20(4):918-957 Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation e2 D[(g)\tilde] u  +  (1 - u2 )u = 0     in  M,\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},      12. 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}      13. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      14. Diffuse measures and nonlinear parabolic equations      Francesco Petitta  Augusto C. Ponce  Alessio Porretta 《Journal of Evolution Equations》2011,11(4):861-905 Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb RN{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of ut-Dp u = m    \textin Qu_t-\Delta_p u = \mu \quad \text{in }Q      15. Sojourn time of almost semicontinuous integral-valued processes in a fixed state      D. V. Gusak 《Ukrainian Mathematical Journal》2012,63(8):1176-1186 Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps xk:E[ zxk / xk < 0 ] = \frac1 - bz - b,   0 \leqslant b < 1. {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1.      16. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      7. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      8. Growth rate for beta-expansions      De-Jun Feng  Nikita Sidorov 《Monatshefte für Mathematik》2011,50(11):41-60 Let β > 1 and let m > β be an integer. Each x ? Ib:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form x=?k=1￥ ekb-k,x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},      9. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      10. 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.      11. Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature      Manuel del Pino  Michal Kowalczyk  Juncheng Wei  Jun Yang 《Geometric And Functional Analysis》2010,20(4):918-957 Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation e2 D[(g)\tilde] u  +  (1 - u2 )u = 0     in  M,\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},      12. 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}      13. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      14. Diffuse measures and nonlinear parabolic equations      Francesco Petitta  Augusto C. Ponce  Alessio Porretta 《Journal of Evolution Equations》2011,11(4):861-905 Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb RN{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of ut-Dp u = m    \textin Qu_t-\Delta_p u = \mu \quad \text{in }Q      15. Sojourn time of almost semicontinuous integral-valued processes in a fixed state      D. V. Gusak 《Ukrainian Mathematical Journal》2012,63(8):1176-1186 Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps xk:E[ zxk / xk < 0 ] = \frac1 - bz - b,   0 \leqslant b < 1. {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1.      16. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

Highly time-oscillating solutions for very fast diffusion equations

This work is concerned with the fast diffusion equation

Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each x ? I_b:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form

Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion

WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      10. 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.      11. Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature      Manuel del Pino  Michal Kowalczyk  Juncheng Wei  Jun Yang 《Geometric And Functional Analysis》2010,20(4):918-957 Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation e2 D[(g)\tilde] u  +  (1 - u2 )u = 0     in  M,\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},      12. 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}      13. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      14. Diffuse measures and nonlinear parabolic equations      Francesco Petitta  Augusto C. Ponce  Alessio Porretta 《Journal of Evolution Equations》2011,11(4):861-905 Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb RN{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of ut-Dp u = m    \textin Qu_t-\Delta_p u = \mu \quad \text{in }Q      15. Sojourn time of almost semicontinuous integral-valued processes in a fixed state      D. V. Gusak 《Ukrainian Mathematical Journal》2012,63(8):1176-1186 Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps xk:E[ zxk / xk < 0 ] = \frac1 - bz - b,   0 \leqslant b < 1. {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1.      16. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

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

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation

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}      13. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      14. Diffuse measures and nonlinear parabolic equations      Francesco Petitta  Augusto C. Ponce  Alessio Porretta 《Journal of Evolution Equations》2011,11(4):861-905 Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb RN{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of ut-Dp u = m    \textin Qu_t-\Delta_p u = \mu \quad \text{in }Q      15. Sojourn time of almost semicontinuous integral-valued processes in a fixed state      D. V. Gusak 《Ukrainian Mathematical Journal》2012,63(8):1176-1186 Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps xk:E[ zxk / xk < 0 ] = \frac1 - bz - b,   0 \leqslant b < 1. {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1.      16. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      14. Diffuse measures and nonlinear parabolic equations      Francesco Petitta  Augusto C. Ponce  Alessio Porretta 《Journal of Evolution Equations》2011,11(4):861-905 Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb RN{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of ut-Dp u = m    \textin Qu_t-\Delta_p u = \mu \quad \text{in }Q      15. Sojourn time of almost semicontinuous integral-valued processes in a fixed state      D. V. Gusak 《Ukrainian Mathematical Journal》2012,63(8):1176-1186 Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps xk:E[ zxk / xk < 0 ] = \frac1 - bz - b,   0 \leqslant b < 1. {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1.      16. Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions      Sergei B. Kuksin 《Geometric And Functional Analysis》2010,20(6):1431-1463 For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I 1, I 2, . . . ), evaluated along a solution u ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation [(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

Diffuse measures and nonlinear parabolic equations

Given a parabolic cylinder Q = (0, T) × Ω, where W ì \mathbb R^N{\Omega\subset \mathbb {R}^N} is a bounded domain, we prove new properties of solutions of

Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps

Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions

For the damped-driven KdV equation $ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I ₁, I ₂, . . . ), evaluated along a solution u ^ν (t, x), as ν → 0. We prove that for ${0 \leq \tau := {\nu}t \lesssim 1}For the damped-driven KdV equation

[(u)\dot]-nuxx + uxxx - 6uux = ?{n} h(t, x), x ? S1, òudx o òhdx o 0, \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0,      17. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      18. On wave equations with supercritical nonlinearities      P. Brenner  P. Kumlin 《Archiv der Mathematik》2000,74(2):129-147 We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { ?2tu-Dxu+ m2u+|u|r-1u=0,  t > 0,  x ? \Bbb Rn, u|t=0(x)=f(x), ?tu|t=0(x)=y(x). \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case.      19. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian      Zhuoran Du  Zheng Zhou  Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114 We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian -Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,      20. Bound states to critical quasilinear Schr?dinger equations      Youjun Wang  Wenming Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):19-47 In this paper, we consider the critical quasilinear Schr?dinger equations of the form -e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u),    x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N,      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

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

On wave equations with supercritical nonlinearities

We prove that the solution operators e_t (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]¹ ?L_r+1) ×L₂(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]^s_q￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e_t(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of {

Saddle solutions of nonlinear elliptic equations involving the p-Laplacian

We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian

Bound states to critical quasilinear Schr?dinger equations

In this paper, we consider the critical quasilinear Schr?dinger equations of the form