New instability, blow-up and break-down effects for Sobolev-type evolution PDE: asymptotic analysis for a celebrated pseudo-parabolic model on the quarter-plane

We present previously unknown phenomena which emerge in the large-times asymptotic behavior of solutions to initial-boundary-value problems (IBVP) for mixed-derivative PDE of Sobolev type formulated in domains with semi-infinite boundaries. Our discovery is facilitated and elucidated by means of meticulous study of a ground-breaking equation with rational dispersion relation, widely known as Barenblatt’s pseudo-parabolic PDE, which effectively serves as an illustrative example. This PDE arises in miscellaneous important applications—ranging from electron- and astro-physics, heat-mass transfer and solid–fluid–gas dynamics to mechanical and chemical engineering as well as diffusive, radiative and biological processes—and has appeared in numerous historic works in the bibliography. More specifically, we perform a rigorous and thorough investigation of asymptotic properties, at large times and/or at distant positions, of the solutions to fully inhomogeneous, parametric IBVP (Dirichlet, Neumann and Robin), posed in the spatiotemporal quarter-plane, with arbitrary initial values, boundary conditions and forcing terms. The current asymptotic analysis is a sequel to a recent work where novel integral representations (defined along complex contours in the spectral Fourier plane) for the unique solution of such IBVP were derived for the first time, via rigorous extension of the well-known Fokas’ unified transform method to mixed-derivative problems. The important question of whether the solution eventually becomes periodic, at any fixed position, under the effect of (eventually) time-periodic boundary data is, in our setting, likewise positively settled for all the Dirichlet and Neumann problems alike, and for certain critical Robin cases. Several worked examples with typical choices of data are offered to provide insight into how the solution is essentially dictated by the boundary conditions. Our in-depth examination of qualitative properties of the solution formulae allows us to uncover several rather counter-intuitive effects that apparently are characteristic to the type of equations at hand. Highlights of our findings include: (i) “mixed-asymptotics” instability, in the sense that there exists a critical-slope ray in the x−t plane which marks a fundamental change in the long-time asymptotics, (ii) “eventual” blow-up, namely, supplying the Robin configuration with compact-support data yields an unbounded solution, and (iii) break-down of asymptotic periodicity for the Robin problem within a special “bad” range of parameters. This is the first treatise on IBVP for linear pseudo-parabolic PDE and the surprising new phenomena that occur on semi-unbounded domains, even though long-time asymptotics for purely non-linear analogues as well as the Cauchy problem on the whole line and bounded-domain cases have already been studied extensively in the pertinent literature. Our rigorous analytical approach is extendable to other significant models of mathematical physics, continuum mechanics, and so forth. It thus becomes evident that the current study opens up a new avenue and provides the way forward to further explorations.