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Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems
Well-posedness of a model for the growth of tree stems and vines
Department of Mathematics, Penn State University, University Park, PA, 16802, USA |
The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles.
The main theorem shows that the evolution problem is well posed, until a specific "breakdown configuration" is reached. A formula is proved, characterizing the reaction produced by unilateral constraints. At a.e. time $t$, this is determined by the minimization of an elastic energy functional under suitable constraints.
References:
[1] |
A. Bressan, M. Palladino and W. Shen,
Growth models for tree stems and vines, J. Differential Equations, 263 (2017), 2280-2316.
doi: 10.1016/j.jde.2017.03.047. |
[2] |
L. Cesari,
Optimization -Theory and Applications, Springer-Verlag, 1983. |
[3] |
G. Colombo and V. Goncharov,
The sweeping processes without convexity, Set-Valued Anal., 7 (1999), 357-374.
doi: 10.1023/A:1008774529556. |
[4] |
G. Colombo and M. Monteiro Marques,
Sweeping by a continuous prox-regular set, J. Differential Equations, 187 (2003), 46-62.
doi: 10.1016/S0022-0396(02)00021-9. |
[5] |
O. Leyser and S. Day,
Mechanisms in Plant Development, Blackwell Publishing, 2003. |
[6] |
J. J. Moreau,
Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[7] |
R. Rossi and U. Stefanelli,
An order approach to a class of quasivariational sweeping processes, Adv. Diff. Equat., 10 (2005), 527-552.
|
[8] |
show all references
References:
[1] |
A. Bressan, M. Palladino and W. Shen,
Growth models for tree stems and vines, J. Differential Equations, 263 (2017), 2280-2316.
doi: 10.1016/j.jde.2017.03.047. |
[2] |
L. Cesari,
Optimization -Theory and Applications, Springer-Verlag, 1983. |
[3] |
G. Colombo and V. Goncharov,
The sweeping processes without convexity, Set-Valued Anal., 7 (1999), 357-374.
doi: 10.1023/A:1008774529556. |
[4] |
G. Colombo and M. Monteiro Marques,
Sweeping by a continuous prox-regular set, J. Differential Equations, 187 (2003), 46-62.
doi: 10.1016/S0022-0396(02)00021-9. |
[5] |
O. Leyser and S. Day,
Mechanisms in Plant Development, Blackwell Publishing, 2003. |
[6] |
J. J. Moreau,
Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[7] |
R. Rossi and U. Stefanelli,
An order approach to a class of quasivariational sweeping processes, Adv. Diff. Equat., 10 (2005), 527-552.
|
[8] |



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