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:

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.      3. Packing circuits in matroids      Guoli Ding  Wenan Zang 《Mathematical Programming》2009,119(1):137-168 The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M), Our characterization contains a complete solution to a research problem on 2-edge-connected subgraph polyhedra posed by Cornuéjols, Fonlupt, and Naddef in 1985, which was independently solved by Vandenbussche and Nemhauser in Vandenbussche and Nemhauser (J. Comb. Optim. 9:357–379, 2005). W. Zang’s research partially supported by the Research Grants Council of Hong Kong.      4. A relaxation result for micromagnetics in SBV      Pedro M. Santos 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(3):327-353 An integral representation for the functional is obtained. This problem is motivated by equilibria issues in micromagnetics.       5. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(2):395-400 Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d 1, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].      6. A note on a fourth order degenerate parabolic equation in higher space dimensions      Junjie Li 《Mathematische Annalen》2007,339(2):251-285 We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions       7. On a type of evolution of self-referred and hereditary phenomena      Michele Miranda Jr.  Eduardo Pascali 《Aequationes Mathematicae》2006,71(3):253-268       8. On ground state solutions for superlinear Hamiltonian elliptic systems      Leiga Zhao  Fukun Zhao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(3):403-418 In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where ${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.      9. A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces      Sonja Gisela Cox  Erika Hausenblas 《Journal of Evolution Equations》2013,13(4):795-827 We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C 0-semigroup on a UMD Banach space X, H is a Hilbert space, W H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A 0 denote the perturbed operator and U 0 the solution to (SDE) with A substituted by A 0. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.      10. On the measurable solutions of a functional equation      Antal Járai 《Aequationes Mathematicae》2010,80(1-2):131-139 The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.      11. Vector-valued stochastic delay equations—a semigroup approach      Sonja Cox  Mariusz Górajski 《Semigroup Forum》2011,82(3):389-411 Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation in E: $ \left\{{l}dX(t)= AX(t) + C X_t + B(X(t),X_t)dW_H(t),\quad t>0;\\[5pt]X(0)=x_0;\\[5pt]X_0=f_0,\right. $ \left\{\begin{array}{l}dX(t)= AX(t) + C X_t + B(X(t),X_t)dW_H(t),\quad t>0;\\[5pt]X(0)=x_0;\\[5pt]X_0=f_0,\end{array}\right.      12. The Existence of Infinitely Many Solutions for the Nonlinear Schrödinger–Maxwell Equations      Wen-nian Huang  X. H. Tang 《Results in Mathematics》2014,65(1-2):223-234 In this paper, by using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of infinitely many solutions for the following nonlinear Schrödinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l}-\triangle u + V(x)u + \phi u = f(x, u), \quad \; \, {\rm in} \, \mathbb{R}^{3},\\ -\triangle \phi = u^{2}, \quad \quad \qquad \quad \quad \quad \quad {\rm in} \, \mathbb{R}^{3},\end{array}\right.$$ where the potential V is allowed to be sign-changing, under some more assumptions on f, we get infinitely many solutions for the system.      13. Local estimates for parabolic equations with nonlinear gradient terms      Tommaso Leonori  Francesco Petitta 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):153-187 In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .      14. Logarithmically Improved Regularity Conditions for the 3D Navier–Stokes Equations      Hui Zhang 《Integral Equations and Operator Theory》2013,75(3):433-441 In this paper, we consider two new regularity criteria for the 3D Navier–Stokes equations involving partial components of the velocity in multiplier spaces. It is proved that if the horizontal velocity ? = (u 1,u 2,0) satisfies $$\int_{0}^{T} \frac{\|\tilde{u}\|_{\dot{X}_{r}}^{\frac{2}{1-r}}}{1+ln(e + \|u(t,.)\|_{\infty})}{\rm d}t < \infty, \quad r \in[0, 1),$$ or the horizontal gradient field satisfies $$\int_{0}^{T}\frac{\|\nabla_{h}\tilde{u}\|_{\dot{X}_{r}}^{\frac{2}{2-r}}}{1 + ln(e + \|u(t,.)\|_{\infty})}{\rm d}t < \infty, \quad r \in[0, 1],$$ then the local strong solution remains smooth on [0, T].      15. On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian      Xian Wu  Jilin Chen 《NoDEA : Nonlinear Differential Equations and Applications》2008,15(6):745-755 In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian ((1.λ)) and ((2.λ)) are studied and some new multiplicity results of solutions for systems (1.λ) and (2.λ) are obtained. Moreover, by using the KKM principle we give also two new existence results of solutions for systems (1.1) and (2.1). This Work is supported in part by the National Natural Science Foundation of China (10561011).      16. Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity      Bitao Cheng  Xian Wu  Jun Liu 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(5):521-537 In the present paper, by applying variant mountain pass theorem and Ekeland variational principle we study the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity $$ \left\{\begin{array}{ll} -(a + b \int\nolimits_{\Omega} |\nabla{u}|^{2})\triangle{u} = \alpha(x)|u|^{q-2}u + f(x, u),\quad{\rm in}\;\Omega,\\ u = 0,\;\quad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm on}\;\partial\Omega, \end{array} \right. $$ A new existence theorem and an interesting corollary of four nontrivial solutions are obtained.      17. Properties of global decaying solution to the Cauchy problem of nonlinear evolution equations      Walter Allegretto  Yanping Lin  Zhiyong Zhang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,22(5):848-868 We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations, (1) . for initial data with different end states, (2) which displays the complexity in between ellipticity and dissipation. Due to smoothing effect of the parabolic operator, we detail the regularity property and estimates when t > 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data. Also we discuss the decay estimates without the restriction of L 1 bound as in Tang and Zhao [17], Wang [20]. Related to recent work by [15], our derivation may also establish the same estimates directly if under the same condition. Work supported by NSERC (Canada).      18. The geometric Neumann problem for the Liouville equation      José A. Gálvez  Asun Jiménez  Pablo Mira 《Calculus of Variations and Partial Differential Equations》2012,44(3-4):577-599 Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.      19. Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids      Hongzhi Liu  Hongjun Yuan  Jiezeng Qiao  Fanpei Li 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(5):865-878 We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: $$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$ Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.      20. THE GLOBAL ASYMPTOTICAL STABILITY OF SOLUTION OF ITO RANDOM DIFFERENTIAL EQUATION      Yu Zhongming 《数学年刊B辑(英文版)》1980,1(34):459-468 In this paper we discuss the stability and global asymptotical stability of the solu- tion of It? random differeirtial equation \[\left\{ \begin{array}{l} d\xi (t) = b(\xi (t),t)dt + \sigma (\xi (t),t)dw(t)\\xi ({t_0}) = {\xi _0} \end{array} \right.\] here \({\xi _0}\) is a bounded random vector. Suffloient conditions for the existence of the two typical stability are given. These conditions are natural extension of Lyapunov function in deterministic system. Our results extend some results due to Friedman, and pinsky (see[l]). We suggest an opinion about definition of asymptotical stability of solution of the following It? random differential equation \[\left\{ \begin{array}{l} d\xi (t) = b(\xi (t)dt + \sigma (\xi (t))dw(t)\\xi ({t_0}) = {x_0} \end{array} \right.\] where \({x_0}\) is a point of n-dimensional Eiiolidian space.

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Packing circuits in matroids

The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M),

A relaxation result for micromagnetics in SBV

An integral representation for the functional

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

On ground state solutions for superlinear Hamiltonian elliptic systems

In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where ${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.     

A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C ₀-semigroup on a UMD Banach space X, H is a Hilbert space, W _H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A ₀ denote the perturbed operator and U ₀ the solution to (SDE) with A substituted by A ₀. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.     

On the measurable solutions of a functional equation

The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.     

Vector-valued stochastic delay equations—a semigroup approach

Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation in E:

The Existence of Infinitely Many Solutions for the Nonlinear Schrödinger–Maxwell Equations

In this paper, by using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of infinitely many solutions for the following nonlinear Schrödinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l}-\triangle u + V(x)u + \phi u = f(x, u), \quad \; \, {\rm in} \, \mathbb{R}^{3},\\ -\triangle \phi = u^{2}, \quad \quad \qquad \quad \quad \quad \quad {\rm in} \, \mathbb{R}^{3},\end{array}\right.$$ where the potential V is allowed to be sign-changing, under some more assumptions on f, we get infinitely many solutions for the system.     

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .     

Logarithmically Improved Regularity Conditions for the 3D Navier–Stokes Equations

In this paper, we consider two new regularity criteria for the 3D Navier–Stokes equations involving partial components of the velocity in multiplier spaces. It is proved that if the horizontal velocity ? = (u ₁,u ₂,0) satisfies $$\int_{0}^{T} \frac{\|\tilde{u}\|_{\dot{X}_{r}}^{\frac{2}{1-r}}}{1+ln(e + \|u(t,.)\|_{\infty})}{\rm d}t < \infty, \quad r \in[0, 1),$$ or the horizontal gradient field satisfies $$\int_{0}^{T}\frac{\|\nabla_{h}\tilde{u}\|_{\dot{X}_{r}}^{\frac{2}{2-r}}}{1 + ln(e + \|u(t,.)\|_{\infty})}{\rm d}t < \infty, \quad r \in[0, 1],$$ then the local strong solution remains smooth on [0, T].     

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity

In the present paper, by applying variant mountain pass theorem and Ekeland variational principle we study the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity $$ \left\{\begin{array}{ll} -(a + b \int\nolimits_{\Omega} |\nabla{u}|^{2})\triangle{u} = \alpha(x)|u|^{q-2}u + f(x, u),\quad{\rm in}\;\Omega,\\ u = 0,\;\quad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm on}\;\partial\Omega, \end{array} \right. $$ A new existence theorem and an interesting corollary of four nontrivial solutions are obtained.     

Properties of global decaying solution to the Cauchy problem of nonlinear evolution equations

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations,

The geometric Neumann problem for the Liouville equation

Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.     

Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: $$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$ Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.     

THE GLOBAL ASYMPTOTICAL STABILITY OF SOLUTION OF ITO RANDOM DIFFERENTIAL EQUATION

In this paper we discuss the stability and global asymptotical stability of the solu- tion of It? random differeirtial equation \[\left\{ \begin{array}{l} d\xi (t) = b(\xi (t),t)dt + \sigma (\xi (t),t)dw(t)\\xi ({t_0}) = {\xi _0} \end{array} \right.\] here \({\xi _0}\) is a bounded random vector. Suffloient conditions for the existence of the two typical stability are given. These conditions are natural extension of Lyapunov function in deterministic system. Our results extend some results due to Friedman, and pinsky (see[l]). We suggest an opinion about definition of asymptotical stability of solution of the following It? random differential equation \[\left\{ \begin{array}{l} d\xi (t) = b(\xi (t)dt + \sigma (\xi (t))dw(t)\\xi ({t_0}) = {x_0} \end{array} \right.\] where \({x_0}\) is a point of n-dimensional Eiiolidian space.