| Full text | |
| Author(s): |
Bezerra, Flank D. M.
;
Lopez-Lazaro, Heraclio
;
Takaessu, Carlos R.
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Journal of Dynamics and Differential Equations; v. N/A, p. 22-pg., 2025-03-17. |
| Abstract | |
In this paper we study the third-order abstract ordinary differential equations uttt+aAxutt+bAyut+c2Azu=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{ttt}+aA<^>{x}u_{tt}+bA<^>{y}u_t+c<^>2A<^>zu=0, \end{aligned}$$\end{document}where a,b >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b\geqslant 0$$\end{document}, c not equal 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document}, 0 <= x<y <= z <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant x<y\leqslant z\leqslant 1$$\end{document}, and A:D(A)subset of X -> X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:D(A)\subset X \rightarrow X$$\end{document} is an unbounded, closed, densely defined, self-adjoint linear operator, which is defined on a separable Hilbert space X. We characterized the spectrum of the linear operator associated to the equation and analyzed for which coefficients a, b, c and for which powers x, y, z, this system can generate a strongly continuous semigroup, or an analytic semigroup, or does not generate anything. Since the coefficients have an important role when it comes to differential equations of third-order, we appeal to a tool from Galois theory, namely the discriminant of a polynomial. Finally, we compute the probabilities, in some sense, of this equation having a solution (strongly continuous semigroup) and being regular (analytic semigroup). (AU) | |
| FAPESP's process: | 22/02172-2 - Shadowing and hyperbolicity for quasilinear parabolic problems |
| Grantee: | Carlos Roberto Takaessu Junior |
| Support Opportunities: | Scholarships abroad - Research Internship - Doctorate |
| FAPESP's process: | 21/01931-4 - Pullback attractor for non-linear parabolic equations with subdifferential principal part |
| Grantee: | Heraclio Ledgar López Lázaro |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 22/13001-4 - Finite fractal dimension for non-linear parabolic equations with subdifferential principal part |
| Grantee: | Heraclio Ledgar López Lázaro |
| Support Opportunities: | Scholarships abroad - Research Internship - Post-doctor |
| FAPESP's process: | 20/14353-6 - Hyperbolicity for quasilinear parabolic problems |
| Grantee: | Carlos Roberto Takaessu Junior |
| Support Opportunities: | Scholarships in Brazil - Doctorate |