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:

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}      3. Forward dynamical problem for the Timoshenko beam      M. I. Belishev  A. L. Pestov 《Journal of Mathematical Sciences》2010,167(5):603-621 The initial boundary value problem $ {*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ $ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array}      4. 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}}      5. Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm      Uğur Yüksel 《Advances in Applied Clifford Algebras》2010,20(1):201-209 This paper deals with the initial value problem of the type \frac?u(t,x) ?t = Lu(t,x),     u(0,x) = u0(x)\frac{\partial u(t,x)} {\partial t} = {\mathcal{L}}u(t,x), \quad u(0,x) = u_{0}(x)      6. 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}      7. 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)      8. On a type of evolution of self-referred and hereditary phenomena      Michele Miranda Jr.  Eduardo Pascali 《Aequationes Mathematicae》2006,71(3):253-268       9. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      10. A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order      M. N. Yakovlev 《Journal of Mathematical Sciences》2010,165(5):601-605 For the Lidstone boundary-value problem *20c u(4) + q(t)u = f(t),   0 < t < 1, u(0) = u"(0) = u(1) = u"(1) = 0 \begin{array}{*{20}{c}} {{u^{(4)}} + q(t)u = f(t),\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = u(1) = u'(1) = 0} \\ \end{array}      11. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    12. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      13. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      14. 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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

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

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

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}      7. 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)      8. On a type of evolution of self-referred and hereditary phenomena      Michele Miranda Jr.  Eduardo Pascali 《Aequationes Mathematicae》2006,71(3):253-268       9. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      10. A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order      M. N. Yakovlev 《Journal of Mathematical Sciences》2010,165(5):601-605 For the Lidstone boundary-value problem *20c u(4) + q(t)u = f(t),   0 < t < 1, u(0) = u"(0) = u(1) = u"(1) = 0 \begin{array}{*{20}{c}} {{u^{(4)}} + q(t)u = f(t),\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = u(1) = u'(1) = 0} \\ \end{array}      11. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    12. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      13. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      14. 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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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)      8. On a type of evolution of self-referred and hereditary phenomena      Michele Miranda Jr.  Eduardo Pascali 《Aequationes Mathematicae》2006,71(3):253-268       9. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      10. A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order      M. N. Yakovlev 《Journal of Mathematical Sciences》2010,165(5):601-605 For the Lidstone boundary-value problem *20c u(4) + q(t)u = f(t),   0 < t < 1, u(0) = u"(0) = u(1) = u"(1) = 0 \begin{array}{*{20}{c}} {{u^{(4)}} + q(t)u = f(t),\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = u(1) = u'(1) = 0} \\ \end{array}      11. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    12. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      13. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      14. 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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      10. A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order      M. N. Yakovlev 《Journal of Mathematical Sciences》2010,165(5):601-605 For the Lidstone boundary-value problem *20c u(4) + q(t)u = f(t),   0 < t < 1, u(0) = u"(0) = u(1) = u"(1) = 0 \begin{array}{*{20}{c}} {{u^{(4)}} + q(t)u = f(t),\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = u(1) = u'(1) = 0} \\ \end{array}      11. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    12. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      13. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      14. 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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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

A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order

For the Lidstone boundary-value problem

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:

(SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      13. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      14. 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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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 problem of determining the parameter of a parabolic equation

We study the boundary-value problem of determining the parameter p of a parabolic equation

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      15. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      16. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations      Tetsutaro Shibata 《Results in Mathematics》2010,58(1-2):105-118 We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.      17. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      18. On global transformations of ordinary differential equations of the second order      Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):499-508 The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz      19. 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},      20. Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres      Thomas Bartsch  Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804 We study the radially symmetric Schr?dinger equation $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\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

Best constant in a three-parameter Poincaré inequality

We study the problem of finding the best constant in the generalized Poincaré inequality

Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations

We consider the simple pendulum equation with friction: $${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$ where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u _λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.     

A remark on global existence of solutions of shadow systems

Let Ω be an open, bounded domain in \mathbbRⁿ  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d ₁, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:

On global transformations of ordinary differential equations of the second order

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form

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

Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres

We study the radially symmetric Schr?dinger equation