A MIXED LOCAL-NONLOCAL PROBLEM FOR A DEGENERATE HIGH EVEN-ORDER EQUATION

Abstract

Within a rectangular region we formulate and study an initial-boundary value problem,
furnished with a combination of local and paired (linked) boundary conditions, for a high even-order partial
differential equation whose type is lost at once in the time variable - at each end of the time interval - and in
the spatial variable, and which additionally carries a subordinate term. Resting on an energy-integral
identity attuned to the temporal degeneracy and rounded out by a Gronwall-type bound, we settle the
uniqueness of the solution. Merging spectral analysis with the Green’s-function apparatus, we express the
solution as a Fourier series in the eigenfunctions of the attendant spectral problem, whose Green’s function
is written out explicitly and shown to be symmetric. Weighted 2-type bouns for the Fourier coefficients are
supplied, and an estimate of the solution is deduced, from which its continuous dependence on the prescribed
data follows.