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Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

We study the asymptotic behavior of linear evolution equations of the type \partial_t g = Dg + Lg - λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg + Lg. In the case Dg = -\partial_x g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg = -x \partial_x g, it is known that λ = 2 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation \partial_t f = Lf. By means of entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In both cases mentioned above we show these conditions are met for a wide range of fragmentation coefficients, so the exponential convergence holds.