Applicable Analysis: An International Journal

Esser, P.; Laubin, P.

doi: 10.1080/00036818708839672pmid: N/A

This paper is devoted to the study of the second analytic wave front set of boundary values of holomorphic functions. After establishing an expression of hyperfunctions without second analytic support, we give an upper bound of the second analytic wave front set of boundary values. Finally, we construct a temperate distribution with only one point in the second analytic wave front set and with a minimal analytic wave front set.

Bank, Steven B.

doi: 10.1080/00036818708839673pmid: N/A

For the class of equations described in the title, there is a classical result which states that there are (deg P) + 2 critical rays, arg z = θj, such that for any ε > 0, all but finitely many zeros of any solution f(z) ≢ 0 must lie in the union of the sectors |arg z - θj| < ε. We prove that any infinite set of zeros in such a sector (for sufficiently small ε) must actually approach a definite ray, which is a translate ofthe critical ray (and which can be explicitely calculated). In addition, we estimate the rate at which the zeros approach the ray, thus obtaining new information on the exact location of the zeros. The class of equations treated here contains equations which arise in applications in other areas, such as Airy's equation and Titchmarsh's equation

Hinton, Don; Krall, Allan M.; Shaw, Ken

doi: 10.1080/00036818708839674pmid: N/A

The problem of assigning boundary conditions at singular endpoints is considered for differential operators which are neither limit-point or limit-circle at the singular endpoint. A solution is obtained by using Niessen subspaces. Additional asymptotic information yields a concrete description of the boundary conditions including a characterization of the Friedrich's extension.

McGhee, D.F.

doi: 10.1080/00036818708839675pmid: N/A

Let X and Y be Banach spaces and let . Various definitions of the notion of a simple eigenvalue of the multiparameter operator have appeared in the literature. Here, a generalised definition is proposed which gives rise to eigensurfaces in the parameter space. These eigensurfaces provide points of bifurcation for solutions of the non-linear problem where N is a nonlinear mapping satisfying some standard conditions.

Moller-Rettkowski, Andreas

doi: 10.1080/00036818708839676pmid: N/A

In bounded simply connected regions G with G n {x3 ≡ 0} ≠ Φ boundary value problems are studied for nonlinear equations of the form with , where T is the Tricomi operator and a,b,c,d are given functions of (x1,x2,x3)∈G. We prove the existence of generalized solutions by using apriori estimates for certain boundary value problems for the corresponding linear equation Lu ≡ g(x1,x2,x3). These estimates ensure the solvability of finite dimensional nonlinear equations related to (0); their solutions approximate in a specified sense the generalized solution of (0)

Vein, P. R.; Dale, P.

doi: 10.1080/00036818708839677pmid: N/A

It is well known that the Korteweg-de Vries equation is satisfied by a function of the form and that the Kadomtsev-Petviashvili equation is satisfied by a function of the form where H is a certain determinant of arbitrary order whose elements are functions of x, y, t, and H1 is a special case of H. These equations have been solved in the context of the theory of solitons but no satisfactory verifications of the solutions by differentiation and substitution have been published. It is the object of this paper to fill these gaps in the literature. Each solution has been verified by two distinct methods each of which apply recently developed techniques in the theory of determinants.

Marion, Martine

doi: 10.1080/00036818708839678pmid: N/A

In this paper, we study some questions related to attractors for two types of reaction-diffusion equations : an equation with a polynomial growth nonlinearity and systems admitting a positively invariant region. For these problems, we prove the existence of a maximal attractor which describes the long-time behaviour of the solutions and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our results are applied to several classical equations.

Yoshida, Norio

doi: 10.1080/00036818708839679pmid: N/A

We consider the semllinear elliptic difrerential equation Δu + c(x,u) = f(x) and we show that, under suitable conditions, there exists an infinite sequence of annular domains in which every solution has a zero.

Payne, L.E.; Philippin, G.A.

doi: 10.1080/00036818708839680pmid: N/A