On a Class of Infinite-Dimensional Hamiltonian Systems with Asymptotically Periodic Nonlinearities

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on R × RN:{■tu-xu + b ·▽xu + au + V(t,x)v = Hv(t,x,u,v),-■tv-xv-b·▽xv + av + V(t,x)u = Hu(t,x,u,v),where z =(u,v):R × RN → Rm × Rm,a > 0,b =(b1,···,bN) is a constant vector and V ∈ C(R × RN,R),H ∈ C1(R × RN × R2m,R).Under suitable conditions on V(t,x) and the nonlinearity for H(t,x,z),at least one non-stationary homoclinic solution with least energy is obtained.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      3. Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line      Ru Ying  XUE 《数学学报(英文版)》2010,26(12):2421-2442 we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line {δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0. u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）, u（x,0）=f（x）,δtu（x,0）=δxh（x）. The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋） 1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.      4. On solution properties of some types of <Emphasis Type="Italic">p</Emphasis>-adic kinetic equations of the form reaction-diffusion      Albert Kh. Bikulov 《P-Adic Numbers, Ultrametric Analysis, and Applications》2010,2(3):187-206 In this paper, solutions for two types of ultrametric kinetic equations of the form reaction-diffusion are obtained and properties of these solutions are investigated. General method to find the solution of equation of the form $ \tfrac{\partial } {{\partial t}}f(x,t) = \int_{\mathbb{Q}_p } {W(|x - y|_p )(f(y,t) - f(x,t))dy + V(|x|_p )f(x,t),f(x,0) = \phi (|x|_p ),} $ \tfrac{\partial } {{\partial t}}f(x,t) = \int_{\mathbb{Q}_p } {W(|x - y|_p )(f(y,t) - f(x,t))dy + V(|x|_p )f(x,t),f(x,0) = \phi (|x|_p ),}       5. Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions      Xiu Hui YANG Fu Cai LI Chun Hong XIE 《数学学报(英文版)》2005,21(4):923-928 Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.      6. Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations      Chen  Cui  Wang  Ya-Nan  Yan  Jun 《中国科学 数学(英文版)》2021,64(8):1789-1800 In this paper,we investigate the non-autonomous Hamilton-Jacobi equation ■ where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary.We obtain the viscosity solution denoted by T_(t_0)~tφ(x) and show T_(t_0)~tφ(x) converges uniformly to a time-periodic viscosity solution u~*(x,t) of ?_tu+H(t,x,?_xu,u)=0.      7. On Global Solvability of Nonlinear Kirchhoff Model in Infinitely Increasing Moving Domains      Rachid Benabidallah  FranÇois Ebobisse 《Applicable analysis》2013,92(3):705-723 The purpose of this article is to study the existence and uniqueness of global solution for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier type: $$ u_{tt} + \mu u_t - M\left (\int _{\Omega _t}|\nabla u|^2dx\right )\Delta u = 0\quad \hbox {in}\ \Omega _t\quad \hbox {and}\quad u|_{\Gamma _t} = \dot \gamma $$ where $ \Omega _t = \{x\in {\shadR}^2 | \ x = y\gamma (t), \ y\in \Omega \} $ with boundary o t , w is a positive constant and n ( t ) is a positive function such that lim t M X n ( t ) = + X . The real function M is such that $ M(r) \geq m_0 \gt 0 \forall r\in [0,\infty [ $ .      8. Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity      Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,41(6):604-614 We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space $ \left\{{{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} } \right. $ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right.      9. Chemotaxis-fluild 系统的全局弱解      刘梅  余孟玲  罗宏 《数学研究及应用》2019,39(2):181-195 We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.      10. Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations      Quan-sen Jiu  Huan Yu 《应用数学学报(英文版)》2016,32(1):1-16 In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion: $$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$ With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.       11. Self-similar solutions for a convection-diffusion equation with absorption inR N      M. Escobedo  E. Zuazua 《Israel Journal of Mathematics》1991,74(1):47-64 We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.      12. A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals      Lucio Boccardo 《Milan Journal of Mathematics》2011,79(1):193-206 The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type $\left\{{ll}-{\rm div}([a(x) + |u|^q] \nabla u) + b(x)u|u|^{p-1}|\nabla u|^2 = f(x), & {\rm in}\,\Omega;\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \; u = 0, & \,{\rm on}\,\partial\Omega. \right.$\left\{\begin{array}{ll}-{\rm div}([a(x) + |u|^q] \nabla u) + b(x)u|u|^{p-1}|\nabla u|^2 = f(x), & {\rm in}\,\Omega;\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \; u = 0, & \,{\rm on}\,\partial\Omega. \end{array}\right.      13. Local bifurcation-branching analysis of global and ��blow-up�� patterns for a fourth-order thin film equation      P. ��lvarez-Caudevilla  V. A. Galaktionov 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(5):483-537 Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4) $u_t=-\nabla \cdot \left(|u|^n \nabla \Delta u\right) \quad {\rm in} \quad \mathbb{R}^{N}\times\mathbb{R}_{+} \quad{\rm where}\quad n >0 ,$u_t=-\nabla \cdot \left(|u|^n \nabla \Delta u\right) \quad {\rm in} \quad \mathbb{R}^{N}\times\mathbb{R}_{+} \quad{\rm where}\quad n >0 ,      14. 一类具有奇异灵敏度的logistic趋化系统解的渐近行为      杜宛娟 《数学研究及应用》2021,41(5):473-480 本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.      15. Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions          下载免费PDF全文 Ahmed Ahme  Mohamed Saad Bouh ELEMINE VALL 《Journal of Applied Analysis & Computation》2024,14(4) The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.      16. Homoclinic orbits for an unbounded superquadratic      Jun Wang  Junxiang Xu  Fubao Zhang  Lei Wang 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):411-435 We consider the following nonperiodic diffusion systems $ \left\{{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N}, $ \left\{\begin{array}{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \end{array}\right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N},      17. Existence and Comparison Results for Non-Uniformly Parabolic Problems      Francesco Della Pietra  Giuseppina di Blasio 《Mediterranean Journal of Mathematics》2010,7(3):323-340 In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem $\left\{{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\right.$\left\{\begin{array}{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\end{array}\right.      18. A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier—Stokes System involving Dirichlet Boundary Conditions for the Signal      Wang  Yu Lan  Winkler  Michael  Xiang  Zhao Yin 《数学学报(英文版)》2022,38(6):985-1001 Acta Mathematica Sinica, English Series - The chemotaxis-Navier—Stokes system $$\left\{{\matrix{{{n_t} + u \cdot \nabla n = \Delta n - \nabla \cdot \left({n\nabla c} \right),} \hfill \cr...      19. On the convergence of double Fourier series of functions from Lp,p > 1      I. L. Bloshanskii 《Mathematical Notes》1977,21(6):438-444 It is proved that if a function from Lp, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.      20. Stability of sets for impulsive functional differential equations via Razumikhin method      Shenglan Xie  Jianhua Shen 《Journal of Mathematical Sciences》2011,177(3):474-486 In this paper, we consider the functional differential equation with impulsive perturbations $ \left\{ {{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ } \right. $ \left\{ {\begin{array}{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ \end{array} } \right.      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

Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line

we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line
{δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0.
u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）,
u（x,0）=f（x）,δtu（x,0）=δxh（x）.
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space
H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋）
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.     

On solution properties of some types of <Emphasis Type="Italic">p</Emphasis>-adic kinetic equations of the form reaction-diffusion

In this paper, solutions for two types of ultrametric kinetic equations of the form reaction-diffusion are obtained and properties of these solutions are investigated. General method to find the solution of equation of the form

Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.     

Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations

In this paper,we investigate the non-autonomous Hamilton-Jacobi equation ■ where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary.We obtain the viscosity solution denoted by T_(t_0)~tφ(x) and show T_(t_0)~tφ(x) converges uniformly to a time-periodic viscosity solution u~*(x,t) of ?_tu+H(t,x,?_xu,u)=0.     

On Global Solvability of Nonlinear Kirchhoff Model in Infinitely Increasing Moving Domains

The purpose of this article is to study the existence and uniqueness of global solution for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier type: $$ u_{tt} + \mu u_t - M\left (\int _{\Omega _t}|\nabla u|^2dx\right )\Delta u = 0\quad \hbox {in}\ \Omega _t\quad \hbox {and}\quad u|_{\Gamma _t} = \dot \gamma $$ where $ \Omega _t = \{x\in {\shadR}^2 | \ x = y\gamma (t), \ y\in \Omega \} $ with boundary o t , w is a positive constant and n ( t ) is a positive function such that lim t M X n ( t ) = + X . The real function M is such that $ M(r) \geq m_0 \gt 0 \forall r\in [0,\infty [ $ .     

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion:

$$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$

With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.

     

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

Local bifurcation-branching analysis of global and ��blow-up�� patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

Homoclinic orbits for an unbounded superquadratic

We consider the following nonperiodic diffusion systems

Existence and Comparison Results for Non-Uniformly Parabolic Problems

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier—Stokes System involving Dirichlet Boundary Conditions for the Signal

Acta Mathematica Sinica, English Series - The chemotaxis-Navier—Stokes system $$\left\{{\matrix{{{n_t} + u \cdot \nabla n = \Delta n - \nabla \cdot \left({n\nabla c} \right),} \hfill \cr...     

On the convergence of double Fourier series of functions from Lp,p > 1

It is proved that if a function from L_p, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.     

Stability of sets for impulsive functional differential equations via Razumikhin method

In this paper, we consider the functional differential equation with impulsive perturbations