Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

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

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 for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

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

Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems

We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rⁿ{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form

u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      7. 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.      8. 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}      9. Quenching phenomena for a non-local diffusion equation with a singular absorption      Raúl Ferreira 《Israel Journal of Mathematics》2011,184(1):387-402 In this paper we study the quenching problem for the non-local diffusion equation ut(x,t) = òW J(x - y)u(y,t)dy + ò\mathbbRN\W J(x - y)dy - u(x,t) - lu - p(x,t) {u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} }      10. Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition      Mikhail Perepelitsa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,114(1):267-276 We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR3+{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ 0 is bounded and the magnitude of the initial velocity u 0 is suitably restricted in the norm ||?{r0(·)}u0(·)||L2(W) + ||?u0(·)||L2(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.      11. Large solutions to an anisotropic quasilinear elliptic problem      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《Annali di Matematica Pura ed Applicata》2010,189(4):689-712 In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      12. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds      Xianfa Song  Lin Zhao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,278(1):655-662 Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote Dg=-divg?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation Dgu(x)+h(x)u(x)=A(x)up(x)+\fracB(x)uq(x)\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}      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. Elliptic equation with singular potential      B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999 We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3: - Du = V(x)u,     u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right.      15. Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means      Zhiting?XuEmail author  Hongyan?Xing 《Annali di Matematica Pura ed Applicata》2005,184(3):395-406 Oscillation criteria for PDE with p-Laplacian div(|Du|p-2A(x)Du)+c(x)|u|p-2u=0\mbox{div}(|Du|^{p-2}A(x)Du)+c(x)|u|^{p-2}u=0      16. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems      Zong Ming Guo  Yao Yong Yan 《数学学报(英文版)》2011,27(11):2177-2190 We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: - eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,      17. 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,      18. 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,      19. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      20. Gaussian upper bounds for fundamental solutions of a family of parabolic equations      Yury A. Alkhutov  Vitali Liskevich 《Journal of Evolution Equations》2012,12(1):165-179 We study the family of divergence-type second-order parabolic equations we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x n  < 0 and we=e{\omega_\varepsilon=\varepsilon} for x n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .      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 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 {

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}      9. Quenching phenomena for a non-local diffusion equation with a singular absorption      Raúl Ferreira 《Israel Journal of Mathematics》2011,184(1):387-402 In this paper we study the quenching problem for the non-local diffusion equation ut(x,t) = òW J(x - y)u(y,t)dy + ò\mathbbRN\W J(x - y)dy - u(x,t) - lu - p(x,t) {u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} }      10. Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition      Mikhail Perepelitsa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,114(1):267-276 We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR3+{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ 0 is bounded and the magnitude of the initial velocity u 0 is suitably restricted in the norm ||?{r0(·)}u0(·)||L2(W) + ||?u0(·)||L2(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.      11. Large solutions to an anisotropic quasilinear elliptic problem      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《Annali di Matematica Pura ed Applicata》2010,189(4):689-712 In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      12. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds      Xianfa Song  Lin Zhao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,278(1):655-662 Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote Dg=-divg?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation Dgu(x)+h(x)u(x)=A(x)up(x)+\fracB(x)uq(x)\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}      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. Elliptic equation with singular potential      B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999 We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3: - Du = V(x)u,     u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right.      15. Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means      Zhiting?XuEmail author  Hongyan?Xing 《Annali di Matematica Pura ed Applicata》2005,184(3):395-406 Oscillation criteria for PDE with p-Laplacian div(|Du|p-2A(x)Du)+c(x)|u|p-2u=0\mbox{div}(|Du|^{p-2}A(x)Du)+c(x)|u|^{p-2}u=0      16. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems      Zong Ming Guo  Yao Yong Yan 《数学学报(英文版)》2011,27(11):2177-2190 We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: - eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,      17. 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,      18. 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,      19. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      20. Gaussian upper bounds for fundamental solutions of a family of parabolic equations      Yury A. Alkhutov  Vitali Liskevich 《Journal of Evolution Equations》2012,12(1):165-179 We study the family of divergence-type second-order parabolic equations we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x n  < 0 and we=e{\omega_\varepsilon=\varepsilon} for x n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .      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

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      12. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds      Xianfa Song  Lin Zhao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,278(1):655-662 Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote Dg=-divg?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation Dgu(x)+h(x)u(x)=A(x)up(x)+\fracB(x)uq(x)\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}      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. Elliptic equation with singular potential      B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999 We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3: - Du = V(x)u,     u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right.      15. Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means      Zhiting?XuEmail author  Hongyan?Xing 《Annali di Matematica Pura ed Applicata》2005,184(3):395-406 Oscillation criteria for PDE with p-Laplacian div(|Du|p-2A(x)Du)+c(x)|u|p-2u=0\mbox{div}(|Du|^{p-2}A(x)Du)+c(x)|u|^{p-2}u=0      16. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems      Zong Ming Guo  Yao Yong Yan 《数学学报(英文版)》2011,27(11):2177-2190 We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: - eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,      17. 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,      18. 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,      19. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      20. Gaussian upper bounds for fundamental solutions of a family of parabolic equations      Yury A. Alkhutov  Vitali Liskevich 《Journal of Evolution Equations》2012,12(1):165-179 We study the family of divergence-type second-order parabolic equations we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x n  < 0 and we=e{\omega_\varepsilon=\varepsilon} for x n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .      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

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote D_g=-div_g?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation

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. Elliptic equation with singular potential      B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999 We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3: - Du = V(x)u,     u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right.      15. Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means      Zhiting?XuEmail author  Hongyan?Xing 《Annali di Matematica Pura ed Applicata》2005,184(3):395-406 Oscillation criteria for PDE with p-Laplacian div(|Du|p-2A(x)Du)+c(x)|u|p-2u=0\mbox{div}(|Du|^{p-2}A(x)Du)+c(x)|u|^{p-2}u=0      16. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems      Zong Ming Guo  Yao Yong Yan 《数学学报(英文版)》2011,27(11):2177-2190 We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: - eDp u = f(x,u)inW, - \varepsilon \Delta _p u = f(x,u)in\Omega ,      17. 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,      18. 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,      19. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      20. Gaussian upper bounds for fundamental solutions of a family of parabolic equations      Yury A. Alkhutov  Vitali Liskevich 《Journal of Evolution Equations》2012,12(1):165-179 We study the family of divergence-type second-order parabolic equations we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x n  < 0 and we=e{\omega_\varepsilon=\varepsilon} for x n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .      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

Elliptic equation with singular potential

We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R ⁿ , n ≥ 3:

Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means

Oscillation criteria for PDE with p-Laplacian

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Bound states to critical quasilinear Schr?dinger equations

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

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,      19. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      20. Gaussian upper bounds for fundamental solutions of a family of parabolic equations      Yury A. Alkhutov  Vitali Liskevich 《Journal of Evolution Equations》2012,12(1):165-179 We study the family of divergence-type second-order parabolic equations we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x n  < 0 and we=e{\omega_\varepsilon=\varepsilon} for x n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .

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Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}

In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbR^N{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .