Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
The large time behaviour of measure-valued solutions to the growth-fragmentation equation with equal mitosis, \[ \partial_t u(t,x) + \partial_x(x u(t,x)) + B(x) u(t,x) = 4 B(2x) u(t,2x) , \qquad (t,x)\in (0,\infty)^2, \] is investigated when \(B: (0,\infty)\to [0,\infty)\) is a continuous function which satisfies suitable conditions for small and large values of \(x\). It is already known that the spectrum of the associated linear operator is contained in \(\{z\in\mathbb{C}\ :\ \Re{z}\ge 1\}\) and that there is a countable family \((\lambda_k,\mathcal{U}_k)_{k\in\mathbb{Z}}\) of eigenelements with \(\lambda_k = 1 + \frac{2ik\pi}{\log{2}}\) and \(\mathcal{U}_k(x)=x^{-\frac{2ik\pi}{\log{2}}}\mathcal{U}_0(x)\). In particular, the convergence of \(e^{-t} u(t)\) to a stationary function as \(t\to\infty\) is not true in general. Instead, \(e^{-t} u(t)\) converges to a time-dependent \(\log{2}\)-periodic function \((\rho_t)_{t\ge 0}\) solely determined by the initial condition. This result is extended in two directions here. On the one hand, existence and uniqueness of measure-valued solutions \((\mu_t)_{t\ge 0}\) to the growth-fragmentation equation with equal mitosis are obtained when the initial condition \(\mu_0\) ranges in the set of measures \(\dot{\mathcal{M}}\) defined by \[ \dot{\mathcal{M}} = \left\{ \mu \text{ is a Radon measure satisfying } \int_{(0,\infty)} x \mathrm{d}|\mu| < \infty \right\}. \] Also, convergence of \(e^{-t} \mu_t\) to a time-dependent \(\log{2}\)-periodic family \((\rho_t)_{t\ge 0}\) of measures in \(\dot{\mathcal{M}}\) solely determined by the initial condition is proved. On the other hand, an exponential decay of the total variation of \(e^{-t}\mu_t -\rho_t\) in an appropriate weighted space is derived.
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