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 {

Oscillation and Nonoscillation of Second Order Neutral Delay Difference Equations

Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

We consider the Neumann initial boundary-value problem for the equation

Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes

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,      6. 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}}      7. Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $      Saïd Benachour  Philippe Laurenccedil;ot 《Journal of Evolution Equations》2003,3(1):27-37 The large time behaviour of the Lq L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ ut - Du + ?i=1m |uxi|pi = 0      in   \mathbbR+×\mathbbRN,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=￥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and pi ? [1,+￥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L1 L^1 -norm is identified, and temporal decay estimates for the L￥ L^\infty -norm are obtained, according to the values of the pi p_i 's. The main tool in our approach is the derivation of L￥ L^\infty -decay estimates for ?(ua ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.      8. An L p -theory for stochastic integral equations      Wolfgang Desch  Stig-Olof Londen 《Journal of Evolution Equations》2011,11(2):287-317 We investigate the stochastic parabolic integral equation of convolution type u=k1*Apu +?k=1￥ k2*gk+u0,   t 3 0, u=k_1\ast A_pu +\sum\limits_{k=1}^{\infty} k_2\star g^k+u_0, \; t\geq 0,      9. 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}.      10. 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}      11. 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,      12. Sum of weighted Lebesgue spaces and nonlinear elliptic equations      Marino Badiale  Lorenzo Pisani  Sergio Rolando 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(4):369-405 We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space L( W) = {u1+u2:u1 ? Lq1 (W),u2 ? Lq2 ( W) }, Lqi ( W) :=Lqi ( W,dm),L\left( \Omega \right) =\left\{u_{1}+u_{2}:u_{1} \in L^{q_{1}} \left(\Omega \right),u_{2} \in L^{q_{2}} \left( \Omega \right) \right\}, L^{q_{i}} \left( \Omega \right) :=L^{q_{i}} \left( \Omega ,d\mu \right),      13. Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term      Sami Baraket  Ines Ben Omrane  Taieb Ouni 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):59-78 Given a bounded open regular set W ì \mathbbR2{\Omega \subset \mathbb{R}^2} and x1, x2, ?, xm ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem -div(a(u)?u) = r2 f(u) -{\rm div}(a(u)\nabla u)= \rho^{2} f(u)      14. The evaluation of a quartic integral via Schwinger, Schur and Bessel      Tewodros Amdeberhan  Victor H. Moll  Christophe Vignat 《The Ramanujan Journal》2012,28(1):1-14 We provide additional methods for the evaluation of the integral N0,4(a;m) : = ò0￥ \fracdx( x4 + 2ax2 + 1 )m+1,N_{0,4}(a;m) := \int_{0}^{\infty} \frac{dx}{( x^{4} + 2ax^{2} + 1 )^{m+1}},      15. 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)      16. 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      17. 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}      18. 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      19. 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)      20. A modular transformation for a generalized theta function with multiple parameters      S. Bhargava  M. S. Mahadeva Naika  M. C. Maheshkumar 《Ukrainian Mathematical Journal》2009,61(8):1233-1249 We obtain a modular transformation for the theta function ? - ￥￥ ? - ￥￥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, }      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

Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $

The large time behaviour of the L^q L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ u_t - Du + ?_i=1^m |u_xi|^p_i = 0      in   \mathbbR₊×\mathbbR^N,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=￥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and p_i ? [1,+￥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L¹ L^1 -norm is identified, and temporal decay estimates for the L^￥ L^\infty -norm are obtained, according to the values of the p_i p_i 's. The main tool in our approach is the derivation of L^￥ L^\infty -decay estimates for ?(u^a ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.     

An L p -theory for stochastic integral equations

We investigate the stochastic parabolic integral equation of convolution type

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}.     

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}      11. 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,      12. Sum of weighted Lebesgue spaces and nonlinear elliptic equations      Marino Badiale  Lorenzo Pisani  Sergio Rolando 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(4):369-405 We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space L( W) = {u1+u2:u1 ? Lq1 (W),u2 ? Lq2 ( W) }, Lqi ( W) :=Lqi ( W,dm),L\left( \Omega \right) =\left\{u_{1}+u_{2}:u_{1} \in L^{q_{1}} \left(\Omega \right),u_{2} \in L^{q_{2}} \left( \Omega \right) \right\}, L^{q_{i}} \left( \Omega \right) :=L^{q_{i}} \left( \Omega ,d\mu \right),      13. Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term      Sami Baraket  Ines Ben Omrane  Taieb Ouni 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):59-78 Given a bounded open regular set W ì \mathbbR2{\Omega \subset \mathbb{R}^2} and x1, x2, ?, xm ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem -div(a(u)?u) = r2 f(u) -{\rm div}(a(u)\nabla u)= \rho^{2} f(u)      14. The evaluation of a quartic integral via Schwinger, Schur and Bessel      Tewodros Amdeberhan  Victor H. Moll  Christophe Vignat 《The Ramanujan Journal》2012,28(1):1-14 We provide additional methods for the evaluation of the integral N0,4(a;m) : = ò0￥ \fracdx( x4 + 2ax2 + 1 )m+1,N_{0,4}(a;m) := \int_{0}^{\infty} \frac{dx}{( x^{4} + 2ax^{2} + 1 )^{m+1}},      15. 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)      16. 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      17. 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}      18. 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      19. 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)      20. A modular transformation for a generalized theta function with multiple parameters      S. Bhargava  M. S. Mahadeva Naika  M. C. Maheshkumar 《Ukrainian Mathematical Journal》2009,61(8):1233-1249 We obtain a modular transformation for the theta function ? - ￥￥ ? - ￥￥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, }      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

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

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space

Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term

Given a bounded open regular set W ì \mathbbR²{\Omega \subset \mathbb{R}^2} and x₁, x₂, ?, x_m ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem

The evaluation of a quartic integral via Schwinger, Schur and Bessel

We provide additional methods for the evaluation of the integral

Highly time-oscillating solutions for very fast diffusion equations

This work is concerned with the fast diffusion equation

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

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}      18. 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      19. 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)      20. A modular transformation for a generalized theta function with multiple parameters      S. Bhargava  M. S. Mahadeva Naika  M. C. Maheshkumar 《Ukrainian Mathematical Journal》2009,61(8):1233-1249 We obtain a modular transformation for the theta function ? - ￥￥ ? - ￥￥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, }      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

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      19. 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)      20. A modular transformation for a generalized theta function with multiple parameters      S. Bhargava  M. S. Mahadeva Naika  M. C. Maheshkumar 《Ukrainian Mathematical Journal》2009,61(8):1233-1249 We obtain a modular transformation for the theta function ? - ￥￥ ? - ￥￥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, }      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

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)      20. A modular transformation for a generalized theta function with multiple parameters      S. Bhargava  M. S. Mahadeva Naika  M. C. Maheshkumar 《Ukrainian Mathematical Journal》2009,61(8):1233-1249 We obtain a modular transformation for the theta function ? - ￥￥ ? - ￥￥ qa( m2 + mn ) + cn2 + lm + mn + nV Am + BnZCm + Dn, \sum\limits_{ - \infty }^\infty {\sum\limits_{ - \infty }^\infty {{q^{a\left( {{m^2} + mn} \right) + c{n^2} + \lambda m + \mu n + {\nu_\varsigma }Am + B{n_Z}Cm + Dn}}}, }

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

A modular transformation for a generalized theta function with multiple parameters

We obtain a modular transformation for the theta function