Abstract
We study the asymptotic behaviour of the following linear growth-fragmentation equation$$\dfrac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{\partial x} \big(x u(t,x)\big) + B(x) u(t,x) =4 B(2x)u(t,2x),$$ and prove that under fairly general assumptions on the division rate $B(x),$ its solution converges towards an oscillatory function,explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypo-coercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted $L^2$ space, where well-posedness is obtained via semigroup analysis. We also propose a non-dissipative numerical scheme, able to capture the oscillations.
Highlights
The mathematical study of the growth-fragmentation equation and its linear or nonlinear variants has led to a wide literature
Our method relies on General Relative Entropy inequalities (Section 1), which unexpectedly may be adapted to this case and which are the key ingredient for an explicit convergence result (Theorem 2 in Section 2, which is the main result of our study)
We studied here the asymptotic behaviour of a non-hypocoercive case of the growth-fragmentation equation
Summary
The mathematical study of the growth-fragmentation equation and its linear or nonlinear variants has led to a wide literature. A stronger “memory” of the initial behaviour may be observed, contrary to the main case, where the only memory of the initial state which remains asymptotically is a weighted average Among these results, the case when the growth rate is linear, i.e. gpxq “ x, and the mother cell divides into two equal offspring, i.e. Our method relies on General Relative Entropy inequalities (Section 1), which unexpectedly may be adapted to this case and which are the key ingredient for an explicit convergence result (Theorem 2, which is the main result of our study).
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