共查询到20条相似文献,搜索用时 516 毫秒
1.
Nathaniel Chafee 《Journal of Differential Equations》1974,15(3):522-540
We consider a parabolic partial differential equation ut = uxx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis. 相似文献
2.
Frederick Bloom 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1976,27(6):853-862
For the nonlinear wave equationu tt -Nu +G(t,u, u t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ 0 T ∥G(τ,u, u τ)∥2 dτ ≤ κ ∫ 0 T ∥?(τ)∥2 dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented. 相似文献
3.
Svatoslav Staněk 《Central European Journal of Mathematics》2014,12(11):1638-1655
In the first part, we investigate the singular BVP \(\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u\) , u(0) = A, u(1) = B, c D α u(t)| t=0 = 0, where \(\mathcal{H}\) is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems \(\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\) , u(0) = A, u(1) = B, \(\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\) where a < 0, 0 < β n ≤ α n < 1, lim n→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0. 相似文献
4.
Dinh Nho Ho 《Mathematical Methods in the Applied Sciences》1992,15(8):537-545
The non-characteristic Cauchy problem for the heat equation uxx(x,t) = u1(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), ux(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate. 相似文献
5.
Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u 1,…,u n ), $f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}$ , $f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}$ , i = l,…,n, F = (f 1…,f n ), ${\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}$ and ${\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}$ . Under some quite general conditions, we prove that either F0 = 0 and F∞ = ∞, or F0 = ∞ and F∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F0 = F∞ = 0, or F∞ = F∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F0 and F∞ > 0, or F0 and F∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone. 相似文献
6.
Shu-Yu Hsu 《manuscripta mathematica》2013,140(3-4):441-460
Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u t = Δu m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u 0(x) ≈ A|x|?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g ij /?t = ?Rg ij with u(x, 0) = u 0(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation. 相似文献
7.
Consider the renewal equation in the form (1) , where is a probability density on [0, ∞) and limt → ∞g(t) = g0. Asymptotic solutions of (1) are given in the case when f(t) has no expectation, i.e., . These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation. 相似文献
8.
Y. Guo 《Ukrainian Mathematical Journal》2011,62(9):1409-1419
We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations
*20c Da u(t) + l[ f( t,u(t) ) + q(t) ] = 0, 0 < t < 1, u(0) = 0, u(1) = bu(h), \begin{array}{*{20}{c}} {{{\mathbf{D}}^\alpha }u(t) + {{\lambda }}\left[ {f\left( {t,u(t)} \right) + q(t)} \right] = 0,\quad 0 < t < 1,} \\ {u(0) = 0,\quad u(1) = \beta u(\eta ),} \\ \end{array} 相似文献
9.
V. F. Butuzov 《Computational Mathematics and Mathematical Physics》2007,47(4):620-628
The singularly perturbed parabolic equation ?u t + ε2Δu ? f(u, x, ε) = 0, x ∈ D ? ?2, t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation. 相似文献
10.
Richard E. Ewing 《Journal of Mathematical Analysis and Applications》1979,71(1):167-186
Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)ux)x ? q(x)u = p(x)ut, 0 < x < 1,0 < t? T; ; ; p(0) ux(0, t) = g(t), 0 < t0 ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?1, ?2 and g are known only approximately. 相似文献
11.
In this paper, we consider the following nonlinear fractional three-point boundary-value problem:
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