共查询到20条相似文献,搜索用时 15 毫秒
1.
Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions
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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. 相似文献
2.
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$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$. 相似文献
3.
Dong Guangchang 《数学年刊B辑(英文版)》1986,7(3):277-302
In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation
$$\[\left\{ {\begin{array}{*{20}{c}}
{\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}}
{}&{(x,t) \in [0,T]}
\end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T}
\end{array}} \right.\]$$
$$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}}
{and}&{v(u) \to 0\begin{array}{*{20}{c}}
{as}&{u \to 0}
\end{array}}
\end{array}} \right)\]$$
under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$ 相似文献
4.
We consider integral functionals in which the density has growth p i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable. 相似文献
5.
Gong Guanglu 《数学年刊B辑(英文版)》1982,3(6):803-812
For a differential operator
$\Omega u=\sum\limits_i,j=1^n \frac{\partial}{\partial x_i}(a_ij(x)\frac{\partial u}{\partial x_j})+\sum\limits_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}+c(x)u$
with unbounded coefficients in R^n, a standard continuous paths process with infinitesimal operator \Omega has been constructed in this paper, and the invariance of such process under a transformation group of phase space has been discussed. 相似文献
6.
Yan Ziqian 《数学年刊B辑(英文版)》1982,3(1):67-78
In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below
$\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$
are discussed.The boundary value conditions are
$\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$
$\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$
Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved. 相似文献
7.
Ismail Kombe 《Proceedings of the American Mathematical Society》2004,132(9):2683-2691
In this paper we consider the following initial value problem:
where and . Nonexistence of positive solutions is analyzed.
where and . Nonexistence of positive solutions is analyzed.
8.
Li XUNJING 《数学年刊B辑(英文版)》1980,1(34):453-458
In this paper we consider the systems governed, by parabolioc equations
\[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\]
subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\]
We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control
\[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time.
The following maximum principle is proved:
Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a
solution to the equation
\[\left\{ {\begin{array}{*{20}{c}}
{ - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0}
\end{array}} \right.\]
with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that
\[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \] 相似文献
9.
Chen Yazhe 《数学年刊B辑(英文版)》1984,5(4):661-678
In this paper the author discusses the quasilinear parabolic equation
$$\[\frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial {x_i}}}[{a_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}] + {b_i}(x,t,u)\frac{{\partial u}}{{\partial {x_i}}} + c(x,t,u)\]$$
Which is uniformly degenerate at $\[u = 0\]$. Let $\[u(x,t)\]$ be a classical solution of the equation satisfying $\[0 < u(x,t) \le M\]$. Under some assumptions the author establishes the interior estimations of Holder
coefficient of the solution for the equation and the global estimations for Cauchy problems and the first boundary value problems, where Holder ooeffioients and exponents are independent of the lower positive bound of $\[u(x,t)\]$. 相似文献
10.
In this paper, we consider the complex Swift-Hohenberg(CSH) equation $\frac{\partial u}{\partial t}=\lambda u-(\alpha+\mathrm{i}\beta)\l(1+\frac{\partial^2}{\partial x^2}\r)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for
the above equation. 相似文献
11.
Some Coneral Results on the First Boundry Value Problem for
Quasiliear Degenerate Parabolic Equation
In this paper, the authors investigate the first boundary value problem for equations of the form
$\[Lu = \frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial {x_i}}}({a^{ij}}(u,x,t)\frac{{\partial u}}{{\partial {x_j}}}) - \frac{{\partial {f^i}(u,x,t)}}{{\partial {x_i}}} = g(u,x,t)\]$
with $a^ij(u,x,t)\xi_i\xi_j\geq 0$
An existence theorem of solution in BV_1,1/2(Q_T) is proved. The principal condition is that there exists \delta>0 such that for any (x, t)\in Q_T,|u|\geq M
$a^ij(u,x,t)\xi_i\xi_j-\delta\sum\limits_i,j=1^m(a_x^ij(u,x,t)\xi_i)^2\geq 0$ 相似文献
12.
We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C 2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$ 相似文献
13.
In this paper we study the first and tiie third boundary value problems for the elliptic equation
\[\begin{array}{l}
\varepsilon \left( {\sum\limits_{i,j = 1}^m {{d_{i,j}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{i = 1}^m {{d_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + d(x)u} } } \right) + \sum\limits_{i = 1}^m {{a_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + b(x) + c} \ = f(x),x \in G(0 < \varepsilon \le 1),
\end{array}\]
as the degenerated operator bas singular points, where
\[\sum\limits_{i,j = 1}^m {{d_{i,j}}(x){\xi _i}{\xi _j}} \ge {\delta _0}\sum\limits_{i = 1}^m {\xi _i^2} ,({\delta _0} > 0,x \in G).\]
The uniformly valid asymptotic solutions of boundary value problems have been
obtained under the condition of
\[\sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} > 0,or} \sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} < 0} ,\]
where \(n = ({n_1}(x),{n_2}(x), \cdots ,{n_m}(x))\) is the interior normal to \({\partial G}\). 相似文献
14.
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. 相似文献
15.
朱超娜 《数学年刊A辑(中文版)》2018,(4):349-366
设(M,g,e~(-f)dv_g)是n维完备光滑的度量测度空间.考虑以下非线性椭圆方程△_f~u+hu~α=0,1α(n+m)/(n+m-2)(n+m≥4)和非线性抛物方程(△_f-?/?t)u+hu~α=0,α0正解的梯度估计.对于经典的Laplace情形,Li (Li J. Gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds [J]. J Funct Anal,1991, 100:233-256.)证明了正解的梯度估计和Liouville定理.在本文中,对于上述的f-Laplace方程,作者将推导出相应的结果. 相似文献
16.
Giuseppe Maria Coclite Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli 《Semigroup Forum》2008,77(1):101-108
The solution u of the well-posed problem
depends continuously on (a
ij
,β,γ,q).
Dedicated to Karl H. Hofmann on his 75th birthday. 相似文献
17.
Chen Yazhe 《数学年刊B辑(英文版)》1987,8(3):343-356
This paper is concerned with the parabolic equation
$\frac{\partial u}{\partial t}-div(|\bigtriangledown u|^{p-2}\bigtriangledown u)=0,x\in \Omega \subseteqq R^n,t>0$
with $p>max{3/2,2N/(N+2)}$ which is degenerate if p<2 er singular if 3/2
相似文献
18.
Liselott Flodén Anders Holmbom Marianne Olsson Nils Svanstedt 《Annali dell'Universita di Ferrara》2007,53(2):217-232
Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y
1,y
2,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .
相似文献
19.
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. 相似文献
20.
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 相似文献
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