Questions tagged [finite-duration]

This tag is for questions of Finite Duration Solutions to Differential Equations, which after an ending time by itself becomes zero forever after. For ordinary functions which have a starting and ending time, see (piecewise-continuity), and if time is not the involved variable, search for functions of compact-support. Finite-Duration solution cannot be solutions of Linear ODEs, since as solutions they don´t fulfill Uniqueness. Synonyms: (finite-time), (time-limited), (finite-time-convergence), (finite-extinction-time), (sublinear damping), (endiness constraint)

For scalar autonomous ODEs of first and second order where investigated by V. T. Haimo on his papers Finite Time Differential Equations (1985) and Finite Time Controllers (1986), from where extended to its use in automatic control.

Since uniqueness is not hold, the equation must have a point where is non-Lipschitz, so solutions cannot be analytic in the whole domain, but piecewise combinations of solutions with the zero function could be done, see Singular solutions.

For example, If the differential equation fulfills the following conditions it could stand finite duration solutions:

1. The diff. eq. stands the trivial zero solution
2. The diff. eq. have at least one finite "singular point" in time $T\in(-\infty,\,\infty)$ where happens to be true $y(T)=\dot{y}(T)=0$

These conditions are sufficient for 1st and 2nd order autonomous scalar ODEs, but not nesessarilly required.

As example, the differential equation $$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,\,x(0)=1$$ Can stand the finite duration solution $$x(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$$ which have a finite ending time at $t=2$.

As another example, the simplest system made by a brick sliding on an horizontal plane after an initial push, if the friction is modeled as the Coulomb damping, it will experience a uniform acceleration until it stop moving. The Newton's 2nd law that rules this system takes the form: $$x'' = -k\cdot g\cdot \text{sgn}(x')$$ where $k = \frac{\mu_k}{m} >0$, with $m$ the mass and $\mu_k$ the friction coefficient, and $g = 9.8\,\frac{m}{s^2}$ the Earth's gravity acceleration constant.

As is explained in detail here, the system have the closed-form particular solutions: $$x(t) =x(0)+ \frac{kg\cdot\text{sgn}(x'(0))}{2}\cdot\left[\left(\frac{|x'(0)|}{kg}\right)^2-\left(\frac{|x'(0)|}{kg}-t\right)^2\cdot\theta\!\left(\frac{|x'(0)|}{kg}-t\right)\right]$$

Here is a list of examples of their use on physics:

• "A note on the dynamics of an oscillator in the presence of strong friction" - H.Amann & J.I.Diaz
• "A conservation law with spatially localized sublinear damping" - Christophe Besse & Sylvain Ervedoza
• "Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping" - J. Karsai & J. R. Graef
• "Extinction time for some nonlinear heat equations" - Louis A. Assalé, Théodore K. Boni, Diabate Nabongo
• "Stability of the separable solution for fast diffusion" - James G. Berryman & Charles J. Holland
• "Degenerate parabolic equations with general nonlinearities" - Robert Kersner
• "Nonlinear Heat Conduction with Absorption: Space Localization and Extinction in Finite Time" - Robert Kersner
• "Fast diffusion flow on manifolds of nonpositive curvature" - M. Bonforte, G. Grillo, J. Vázquez
• "Finite extinction time for a class of non-linear parabolic equations" - Gregorio Diaz & Ildefonso Diaz
• "Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation" - Rémi Carles, Clément Gallo
• "Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption" - Razvan Gabriel Iagar, Philippe Laurençot