Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate
91
0
0.0
(
0
)
Authors
Etienne Bernard
Ask ChatGPT about the research
No Arabic abstract
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.
rate research
Read More
We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space is the largest one in which we can expect convergence to the steady size distribution. Although this convergence is known to occur under fairly general conditions on the coefficients of the equation, we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded. First we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the Dyson-Phillips series. Then we extend it to general kernels by using the notion of quasi-compactness and the fact that it is a topological invariant.
We give here an explicit formula for the following critical case of the growth-fragmentation equation $$frac{partial}{partial t} u(t, x) + frac{partial}{partial x} (gxu(t, x)) + bu(t, x) = balpha^2 u(t, alpha x), qquad u(0, x) = u_0 (x),$$ for some constants $g > 0$, $b > 0$ and $alpha > 1$ - the case $alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.
We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.
We consider the fragmentation equation $dfrac{partial}{partial t}f (t, x) = --B(x)f (t, x) + int_{ y=x}^{ y=infty} k(y, x)B(y)f (t, y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size distribution $f (t, $times$)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x) = alpha x^{gamma}$ and a self-similar fragmentation kernel $k(y, x) = frac{1}{y} k_0 (x/ y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(alpha, gamma, k _0)$ and a representation formula for $k_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.
We prove the growth rate of global solutions of the equation $u_t=Delta u-u^{- u}$ in $R^ntimes (0,infty)$, $u(x,0)=u_0>0$ in $R^n$, where $ u>0$ is a constant. More precisely for any $0<u_0in C(R^n)$ satisfying $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+|x|^2)^{alpha_2}$ in $R^n$ for some constants $1/(1+ u)lealpha_1<1$, $alpha_2gealpha_1$ and $A_2ge A_1= (2alpha_1(1-3)(n+2alpha_1-2))^{-1/(1+ u)}$ where $0<3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{alpha_1}le u(x,t)le A_2(1+|x|^2+b_2t)^{alpha_2}$ in $R^ntimes (0,infty)$ where $b_1=2(n+2alpha_1-2)3$ and $b_2=2n$ if $0<alpha_2le 1$ and $b_2=2(n+2alpha_2-2)$ if $alpha_2>1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
