Research Presentations

My presentations

Asymptotic behavior of solutions of du/dt = Au + f(t) and applications

1. Asymptotic (mild) solutions of an evolution equation means a function u(t) that satisfies u'(t)=A(t)u(t)+f(t) + epsilon (t), where epsilon(t) approaches zero as t tends to + infinity. Using spectral theory of functions (or sequences) on the half line we can characterize the behavior of functions (sequences) with property that $lim_{t \to \infty} f(t+1)-f(t) =0.$ in terms of its spectral property that $\sigma (f) \subset \{ 1\}$. This opens a way to study details of structures of bounded solutions and the existence of solutions with property like the one stated in the famous theorem due to Katznelson-Tzafriri. See Proceedings of the A.M.S. 150 (2022), 1105-1114. A Katznelson-Tzafriri type theorem for difference equations and applications. A "continuous version" for evolution equations is to appear soon in Journal of Differential Equations.

2 Traveling wave dispersal in partially sedentary age-structured populations: we present a thorough study on the existence of traveling waves in a mathematical model of dispersal in a partially sedentary age-structured population. This type of model was first proposed by Veit and Lewis in [Am. Nat., 148 (1996), 255-274]. We choose the fecundity function to be the Beverton-Holt type function. We extend the theory of traveling waves in the population genetics model of Weinberger in [SIAM J. Math. Anal., 13 (1982), 353-396] to the case when migration depends on age groups and a fraction of the population does not migrate.

3 . Asymptotic behavior of solutions of discrete systems: An interesting question may be raised as when every bounded solution x(n) to the equation x(n+1)=Bx(n)+y(n) (with n = 1, 2, 3...) inherits the behavior of the control y(n). For example, when y(n) approaches zero, a constant, or an almost periodic sequence.

4. Applications of the results on bounded solutions: The role of Perron Theorem in a rigorous mathematical framework for the monotone iteration method to study the traveling waves in reaction diffusion equations. An example is the Lodka-Volterra equations with diffusion.

5. For equations on the real line when the imaginary spectrum of A is countable, if the geometry of the phase space is good, every bounded solution is of the same profile as f. For equations on the half line, every bounded solution is asymptotically approaches a function of the same profile as f. See the paper in JDE and its corrections.

6. For periodic equations in the infinite dimensional case, in general, there is no Floquet Representation to reduce them to the above autonomous case. A direct treatment of these equations is needed. "Circular spectrum" of a bounded function is a way to do so.

7. Existence and uniqueness of almost periodic and bounded solutions to non-homogeneous equations of the form du/dt = Au + f(t). The main non-resonant conditions say that for the generator A of an analytic semigroup, if the imaginary spectrum of A does not intersect the spectrum of f , then the equation has a unique solution of the same profile as f . In case of weak resonance, the Massera Theorem says that if the equation has a bounded solution, it has a solution of the same profile as f.