Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
It is by now a well-known fact that, under fairly general assumptions on the data, there is \(\lambda>0\) such that the growth-fragmentation equation \[ \partial_t g + \partial_x (a g) + \lambda g = \mathcal{L}[g]\,, \qquad (t,x)\in (0,\infty)\times (0,\infty)\,, \] with boundary condition \(g(t,0)=0\), \(t>0\), has a unique nonnegative and integrable stationary solution \(G\) satisfying \(\|G\|_{L^1}=1\). Here, the growth rate \(a\) is a nonnegative function of \(x\) and the fragmentation operator \(\mathcal{L}\) is given by \[ \mathcal{L}[g](x) = \int_x^\infty b(y,x) g(y)\;dy - B(x) g(x)\,, \quad B(x) =\int_0^x \frac{y}{x} b(x,y)\;dy\,, \quad x>0\,. \] It is then expected that, given a nonnegative and integrable initial condition \(g_{in}\) satisfying \(\|g_{in}\|_{L^1}=1\), the corresponding solution \(g(t)\) to the growth-fragmentation equation converges to \(G\) as \(t\to\infty\). This has been shown to be true in some cases, and the paper under review focuses on establishing a rate of convergence to zero of \((g(t)-G)\). Assuming that \(a(x)=x^\alpha\), \(\alpha\in\{0,1\}\), \(\int_0^x b(x,y)\;dy = \kappa B(x)\) for some \(\kappa>1\), and \(2B_m x^{\gamma-1} \leq b(x,y) \leq 2B_M x^{\gamma-1}\), \(0<y<x\), for some \(\gamma>-\alpha\) and \(B_M\geq B_m>0\), it is proved that \((g(t)-G)\) decays exponentially fast to zero as \(t\to 0\) in a suitable topology provided \(\gamma\in (0,2)\) or \((\alpha,\gamma)=(1,2)\) or \((\alpha,\gamma)=(0,0)\) and \(B_m=B_M\). The cornerstone of the proof is to establish a so-called entropy/entropy dissipation inequality of the form \(H_2[g|G] \leq C\;D^b[g|G]\) for some \(C>0\), where \(H_2[g|G]=\int_0^\infty (g-G)^2 (\phi/G)\;dx\) is a Liapunov functional for the growth-fragmentation equation for a suitably chosen weight function \(\phi\) and \(D^b[g|G]\) is the associated dissipation (that is, \(dH_2[g|G]/dt = -D^b[g|G]\)).
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