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Chang-Jun Li and Xiang Gao

Geometry and Topology, I. International Press, Cambridge, MA, 1994. [5] J. Q. Zhong and H. C. Yang, On the estimate of first eigenvalue of a compact Riemannian manifold , Sci. Sinica Ser. A 27 (1984), 1265-1273.

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Luis Maire

¯ , D ] , $$\begin{array}{} \displaystyle \sigma [\mathfrak{L},D] \gt \sigma [\bar{\mathfrak{L}} ,D], \end{array}$$ (see [ 16 , Theorem 3.2] if necessary). Moreover, by the uniqueness of the principal eigenvalue, we have σ [ L ¯ , D ] = λ 1 [ − Δ − n a , D ] = λ 1 [ − Δ , D ] − n a , $$\begin{array}{} \displaystyle \sigma [\bar{\mathfrak{L}} ,D] = {\lambda _1}[ - {\rm{\Delta }} - na,D] = {\lambda _1}[ - {\rm{\Delta }},D] - na, \end{array}$$ (3) where λ 1 [−Δ, D ] stands for the classical first eigenvalue of −Δ in D under Dirichlet homogeneous

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H. Chen, Jingfei Jiang, Dengqing Cao and Xiaoming Fan

of which describe the model nonlinear stochastic heat conduction. Some theoretical results which are invoked in numerical study are listed in the other part. 2.1 Model for nonlinear stochastic heat conduction Let D be a bounded domain with regular boundary in ℝ n , n =1,2,3, the govern equation for a heat conduction with nonlinear stochastic heat conduction is as follows ∂ u ∂ t = Δ u + a u − u 3 + g ( t ) u ξ $$\begin{array}{} \displaystyle \frac{\partial{u}}{\partial t}=\Delta u+au-u^3+g(t)u\xi \end{array}$$ (1) with Dirichlet boundary value

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J. Apraiz

\lambda_1\le\lambda_2\le\dots\leq\lambda_j\le\cdots $$ for the eigenvalues of −Δ with the zero Dirichlet boundary condition over ∂Ω, and { e j : j ≥1 } for the set of L 2 (Ω)-normalized eigenfunctions, i.e., { Δ e j + λ j e j = 0 ,   in   Ω , e j = 0 ,   on   ∂ Ω . $$ \begin{equation} \left\{\begin{array}{l} \Delta e_j+\lambda_je_j=0,\quad\mbox{in $\Omega$,} \\[1em] \displaystyle e_j=0, \quad\mbox{on $\partial\Omega$.} \end{array}\right. \end{equation} $$ (4) For λ > 0

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Farhad Khellat and Mahmud Beyk Khormizi

) ∥ h ( t ) ∥ L 2 ( Ω ) ≤ c 6 e α | t | , $$\begin{array}{} \displaystyle \|h(t)\|_{L^2(\Omega)}\leq c_6 e^{\alpha |t|}, \end{array}$$ (5) and 0 ≤ α < λ 1 where λ 1 > 0 is the first eigenvalue of the operator A = −Δ where Δ is the Laplace operator in L 2 (Ω) with zero Dirichlet boundary condition [ 7 ]. When the phase space is L 2 (Ω), in [ 1 ], it was proved that the problem (1) has a pullback global attractor in L 2 (Ω) with f satisfying (2) , (3) , (4) and under the condition of a polynomial bound on the forcing term instead of (5

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Matheus C. Bortolan and Felipe Rivero

) and sup t ∈ ℝ | ∂ s g ϵ ( t , s ) − f ′ ( s ) | ⩽ β ( ϵ ) ( 1 + | s | ρ − 1 ) ,  for all  s ∈ ℝ  and  ϵ ∈ [ 0 , 1 ] , $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|\partial_sg_\epsilon(t,s)-f'(s)|\leqslant \beta(\epsilon)(1+|s|^{\rho-1}), \hbox{ for all }s\in \mathbb{R} \hbox{ and } \epsilon\in [0,1], \end{array}$$ (H5) where λ 1 > 0 is the first eigenvalue of the negative Laplacian A = —Δ with Dirichlet boundary condition, for some α > 0 and 1 ⩽ ρ < n + 2 n − 2 $\begin{array}{} \displaystyle 1\leqslant \rho< \frac{n+2}{n-2} \end

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Jiahua Jin

| → + ∞ ( f 5 ) u ↦ f ( x , u ) | u | 3 $\begin{array}{} \displaystyle u \mapsto \frac{{f\left( {x,u} \right)}}{{{{\left| u \right|}^3}}} \end{array}$ is strictly increasing on (−∞,0) ∪ (0,+∞). Set V ( x ) ≡ 0 and replace R N by a bounded smooth domain Ω ⊆ R N respectively, ( 1 ) reduces to the following Dirichlet problem of Kirchhoff type { − ( a + b ∫ Ω | ▿ u | 2 d x ) ▵ u = f ( x , u ) , i n   Ω ; u = 0 , o n   ∂ Ω $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{c}} { - (a + b{\smallint _\Omega }|\nabla u{|^2}dx)\Delta u = f(x,u),} \hfill