共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper we study the asymptotic behavior of solutions of the following nonautonomous wave equation with nonlinear dissipation.where f is an analytic function, α is a small positive real and g(t, ·) tends to 0 sufficiently fast in L 2(Ω) as t tends to ∞.
$\left\{\begin{array}{ll} u_{tt}+\vert u_{t}\vert^{\alpha}u_{t}-\Delta u +f(u)=g(t,x),\quad{\rm in}\,\mathbb{R}_{+}\times\Omega,\\ \qquad\qquad u(t,x)=0,\quad\, {\rm on}\,\mathbb{R}_{+}\times\partial\Omega,\end{array}\right.$
We also obtain a general convergence result and the rate of decay of solutions for a class of second order ODE containing as a special case
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$\left\{\begin{array}{ll} \ddot{U}(t)+\Vert\dot{U}(t)\Vert^{\alpha}\dot{U}(t)+\nabla F(U(t))=g(t),\quad t \in \mathbb{R}_+,\\ \qquad U(0)=U_{0}\,\in \mathbb{R}^{N},\quad\dot{U}(0)=U_{1}\in \mathbb{R}^{N}. \end{array}\right.$
2.
Let Ω be a bounded smooth domain in \({{R}^N, N \geqq 2}\), and let us denote by d(x) the distance function d(x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model iswhere f belongs to \({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at \({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field \({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies \({B\cdot \nu\geqq \alpha}\) at ?Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.
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$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $
3.
In this paper we study the Dirichlet problem, where σ and ω are nonnegative Borel measures, and \({\Delta_p{u} = \nabla \cdot (\nabla{u} \, |\nabla{u}|^{p-2})}\) is the p-Laplacian. Here \({\Omega \subseteq \mathbf{R}^n}\) is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff’s potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.
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$\left\{\begin{array}{lll}-\Delta_p{u} = \sigma |u|^{p-2}u + \omega \quad {\rm in}\;\Omega,\\ u = 0 \qquad\quad\qquad\quad\;\qquad{\rm on}\;\partial\Omega,\end{array}\right.$
4.
5.
Stefano Bianchini Camillo De Lellis Roger Robyr 《Archive for Rational Mechanics and Analysis》2011,200(3):1003-1021
In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D x u and ? t u belong to the class SBV loc (Ω).
相似文献
$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$
6.
We focus here on the analysis of the regularity or singularity of solutions Ω 0 to shape optimization problems among convex planar sets, namely:where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:
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$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$
- i)under a suitable convexity property of the functional J, we prove that Ω 0 is a W 2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E f (Ω), λ1(Ω)) involves the area |Ω|, the Dirichlet energy E f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ1(Ω), and P(Ω) is the perimeter of Ω;
- ii)under a suitable concavity assumption on the functional J, we prove that Ω 0 is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
7.
We consider a family of non-convex integral functionals where W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the Γ-limits corresponding to linearization (h → 0) and homogenization (\({\varepsilon\rightarrow 0}\)) commute, provided W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of W.
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$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$
8.
Govind Menon 《Journal of Dynamics and Differential Equations》2016,28(3-4):1031-1038
Let G be the Green’s function for the Airy operator We show that the integral operator defined by G is Hilbert–Schmidt and that the 2-modified Fredholm determinant
相似文献
$$\begin{aligned} L\varphi := -\varphi ''+ x \varphi , \quad 0< x < \infty , \quad \varphi (0)=0. \end{aligned}$$
$$\begin{aligned} {\mathrm {det}}_2(1+zG) = \frac{{\mathrm {Ai}}(z)}{{\mathrm {Ai}}(0)} , \quad z \in {\mathbb {C}}. \end{aligned}$$
9.
In this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents,
$ \label{0.1}\left\{\begin{aligned}& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \quad \rm {in}\,\,\Omega,\quad\quad\quad(0.1)\\ & u=0 \quad {\rm on }\,\,\Omega, \end{aligned} \right.$ \label{0.1}\left\{\begin{aligned}& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \quad \rm {in}\,\,\Omega,\quad\quad\quad(0.1)\\ & u=0 \quad {\rm on }\,\,\Omega, \end{aligned} \right. 相似文献
10.
Daomin Cao Zhongyuan Liu Juncheng Wei 《Archive for Rational Mechanics and Analysis》2014,212(1):179-217
In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C 1-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ . 相似文献
11.
We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008). 相似文献
12.
Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation. 相似文献
13.
Eurica Henriques Rojbin Laleoglu 《Journal of Dynamics and Differential Equations》2018,30(3):1029-1051
In the context of measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality, we derive local and global sup bounds of the nonnegative weak subsolutions of 相似文献
$$\begin{aligned} (u^{q})_t-\nabla \cdot {(|\nabla u|^{p-2}\nabla u)}=0, \quad \mathrm {in} \ U_\tau = U \times (\tau _1, \tau _2] , \quad p>1,\quad q>1 \end{aligned}$$ 14.
We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator: 相似文献
$\partial_t u-\nabla\cdot(u\nabla p)=0, \quad p=(-\Delta)^{-s}u,\quad 0 < s < 1.$ 15.
16.
We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}
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