共查询到20条相似文献,搜索用时 15 毫秒
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Kazuhiro Ishige Tatsuki Kawakami 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):429-457
We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P). 相似文献
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Tatsuki Kawakami 《Journal of Mathematical Analysis and Applications》2010,368(1):320-329
We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:
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We consider the Laplace equation in ? d?1 × ?+ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order. 相似文献
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Lucas C. F. Ferreira Everaldo S. Medeiros Marcelo Montenegro 《Calculus of Variations and Partial Differential Equations》2013,47(3-4):667-682
We study the Laplace equation in the half-space ${\mathbb{R}_{+}^{n}}$ with a nonlinear supercritical Robin boundary condition ${\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}$ on ${\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}$ , where n ≥ 3 and λ ≥ 0. Existence of solutions ${u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}$ is obtained by means of a fixed point argument for a small data $f \in {L^{d}(\mathbb{R}^{n-1})}$ . The indexes p, q are chosen for the norm ${\Vert\cdot\Vert_{E_{pq}}}$ to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. ${x\in\mathbb{R}^{n-1}}$ . When f is radially symmetric, u is invariant under rotations around the axis {x n = 0}. Moreover, in a certain L q -norm, we show that solutions depend continuously on the parameter λ ≥ 0. 相似文献
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Jürgen Sprekels 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3028-3048
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate. 相似文献
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This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained. 相似文献
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Junichi Harada 《Calculus of Variations and Partial Differential Equations》2014,50(1-2):399-435
We consider positive solutions of $\varDelta u=0$ in $\mathbf{R}_+^n$ , $\partial _{\nu }u=u^q$ on $\partial \mathbf{R}_+^n$ , where $n\ge 3$ and $q>n/(n-2)$ . We investigate the qualitative property of positive $x_n$ -axial symmetric solutions. In particular, we are concerned with the asymptotic expansion and the intersection property of positive $x_n$ -axial symmetric solutions. 相似文献
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V. A. Kondrat’ev 《Differential Equations》2010,46(8):1114-1122
We consider a boundary value problem for the heat equation in the exterior of a bounded domain of space variables. On the
boundary of the domain, we pose a nonlinear boundary condition. We find sharp nonlinearity exponents for which there exists
no global solution. 相似文献
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This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses. 相似文献
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Juan Luis Vázquez 《Journal of Mathematical Analysis and Applications》2009,354(2):674-2161
This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem
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In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively. 相似文献
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Alexander Gladkov Kwang Ik Kim 《Journal of Mathematical Analysis and Applications》2008,338(1):264-273
In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(13):5211-5229
We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution. 相似文献
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We consider global solutions of a dynamical equation in ferrimagnet. We show that it admits a global weak solution by using the penalty method. By the energy estimates method we show there exists a unique global smooth solution. Finally we establish the relationship between this equation and wave maps. 相似文献
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Miccal T. Matthews James M. Hill 《Journal of Mathematical Analysis and Applications》2007,333(1):381-400
At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased. 相似文献
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Mads Kyed 《Mathematische Nachrichten》2008,281(2):253-271
The existence of travelling wave solutions for the heat equation ∂t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献