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In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in $L_{1}$ -space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

Traces and fine properties of a $B D$ class of vector fields and applications

Luigi Ambrosio, Gianluca Crippa, Stefania Maniglia (2005)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Traces and the F. and M. Riesz theorem for vector fields

Shiferaw Berhanu, Jorge Hounie (2003)

Annales de l’institut Fourier

This work studies conditions that insure the existence of weak boundary values for solutions of a complex, planar, smooth vector field $L$ . Applications to the F. and M. Riesz property for vector fields are discussed.

Transport equation and Cauchy problem for $B V$ vector fields and applications

Luigi Ambrosio (2004)

Journées Équations aux dérivées partielles

Transport equations with partially $B V$ velocities

Nicolas Lerner (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem $\partial_{t} u + X u = f, u_{|_{t = 0}} = g,$ where $X$ is the vector fieldwith a boundedness condition on the divergence of each vector field $a_{1}, a_{2}$ . This model was studied in the paper [LL] with a $W^{1, 1}$ regularity assumption replacing our $B V$ hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of...

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Régis Monneau, Jean-Michel Roquejoffre, Violaine Roussier-Michon (2013)

Annales scientifiques de l'École Normale Supérieure

We construct travelling wave graphs of the form $z = - c t + φ (x)$ , $φ : x \in ℝ^{N - 1} \mapsto φ (x) \in ℝ$ , $N \geq 2$ , solutions to the $N$ -dimensional forced mean curvature motion $V_{n} = - c_{0} + κ$ ( $c \geq c_{0}$ ) with prescribed asymptotics. For any $1$ -homogeneous function $φ_{\infty}$ , viscosity solution to the eikonal equation $| D φ_{\infty} | = \sqrt{{(c / c_{0})}^{2} - 1}$ , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $φ_{\infty}$ . We also describe $φ_{\infty}$ in terms of a probability measure on $§^{N - 2}$ .

Über die Menge erster Integrale gewisser partieller Differetialgleichungen erster Ordnung

Jindřich Palát (1973)

Sborník prací Přírodovědecké fakulty University Palackého v Olomouci. Matematika

Über ein System partieller Differentialgleichungen und einen zugehörigen Bergman-Operator.

Erwin Kreyszig (1974)

Elemente der Mathematik

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