We study the non-negative solutions of the boundary value problem is bounded with smooth boundary ?Ω.This problem arises in the theory of combustion. We study the estimates on the supremum norm of the solutions and estimates on the critical values of λ. 相似文献
In this paper we study the asymptotic behavior of solutions to the mixed initial boundary value problem for the system of nonlinear parabolic equations We show, under suitable technical assumptions, that these solutions converge to solutions of the Dirichlet problem for the corresponding limiting elliptic system, provided that the solution of the Dirichlet problem is unique. 相似文献
We perform analysis for a finite elements method applied to the singular self-adjoint problem. This method uses continuous
piecewise polynomial spaces for the trial and the test spaces. We fit the trial polynomial space by piecewise exponentials
and we apply so exponentially fitted Galerkin method to singular self-adjoint problem by approximating driving terms by Lagrange
piecewise polynomials, linear, quadratic and cubic. We measure the erroe in max norm. We show that method is optimal of the
first order in the error estimate. We also give numerical results for the Galerkin approximation. 相似文献
It is well known that every weak solution (with boundary values 0) of a semilinear equation is a regular solution if ? fulfils the growth condition . Here 2m is the order of A. In this paper we weaken this condition to . This requires a technique completely different from that which may be applied in case (1). 相似文献
Consider a smooth solution of and is C1, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, , and for simplicity, qr ? 0. Then the local energy decays faster than exponentially. 相似文献
This paper treats the quasilinear, parabolic boundary value problem u(0, t) = ?1(t); u(l, t) = ?2(t) on an infinite strip with the functions being periodic in t. The major theorem of the paper gives sufficient conditions on for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on and indicate a method for determining the initial estimate at which the iteration may begin. 相似文献
The boundary value problem where u(? ∞, t′) = v(∞, t′) = 0 andv(? ∞, t′) = u(∞, t′) = 1 for each t′ > 0 has been proposed by Murray as a model for the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We prove that there is a range of values for b and r over which the boundary value problem has traveling wave front solutions. 相似文献
Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, utt?uxx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in . 相似文献
In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation does not possess global solutions for arbitrary choices of ) when 0 < k ? 1 for a wide class of nonlinearities , which includes all the even powers of u and the functions u2n + 1, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy . The result is independent of the space dimension. 相似文献
This paper deals with some nonlinear elliptic problems arising from plasma physics. These problems involve an operator which is neither local, nor monotone, nor continuous, such as . Some of the problems considered here are free boundary problems. We apply techniques of multivalued analysis in order to prove the existence of solutions, or, in particular cases, the existence of one minimal and one maximal solution. 相似文献
We investigate the boundary value problem , , u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0. 相似文献
We consider the first initial-boundary value problem for and L1 are linear elliptic partial differential operators) and investigate the properties of u(x, t, ?) as ? ↓ 0 in the maximum norm. Special attention is paid to approximations obtained by the boundary layer method. We use a priori estimates. 相似文献
In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C1, and g(x, t)?L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (1) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method. 相似文献
We consider the mixed boundary value problem , where Ω is a bounded open subset of n whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on . The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset of the reals such that, if then for is a Fredholm operator if and only if s ∈ . Moreover, = ?xewx, where the sets x are determined algebraically by the coefficients of the operators at x. If n = 2, x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, x is either an open interval of length 1 or is empty; and finally, if n ? 4, x is an open interval of length 1. 相似文献
In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponentwhere is a bounded domain of with a smooth boundary . 相似文献
Several existence and uniqueness results are obtained for third order problems of the form specificied, plus one additional condition at x = 0. Using these results, we solve a nontrivial class of Riemann problems for a single conservation law, in the class of rarefraction waves plus discontinuities which are limits of fourth-order viscous profiles. For a nonconvex flux function, the solution so obtained differs, in many cases, from the classically “admissible” solution. 相似文献
In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation , where t∈J?, x∈. We interpret the equation in functional form as an ordinary differential equation for the mapping u:J→L2(R,μ), where L2(R,μ) is a weighted L2-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms. 相似文献