Study of the solution of a semilinear evolution equation of a prion proliferation model in the presence of chaperone in a product space

Abstract

A mathematical model for the dynamics of prion proliferation in the presence of chaperone involving a coupled system consisting of an ordinary differential equation and a partial integro-differential equation is analyzed. For bounded reaction rates, we prove the existence and uniqueness of positive classical solutions with the help of the theory of evolution system. In the case of unbounded reaction rates, the model is set up into a semilinear evolution equation form in the product Banach space R × L 1 ((z 0 , ∞); (q + z)dz) and the existence of a unique positive local mild solution is established by using C 0-semigroups theory of operators. KEYWORDS classical and mild solutions, C 0-semigroups, prions proliferation, semilinear evolution equations MSC CLASSIFICATION 45K05; 47H07 • S(t) = Population of PrP C monomers at time t. • u(t, z) = Population of PrP Sc polymers of length z at time t, where z > z 0. • = Constant rate of production of normal PrP C in the system.

Analysis of a prion proliferation model with polymer coagulation in the presence of chaperone

“…In the case of $$$ \eta =0,\rho =0 $$$ and $$$ \tau \equiv \mathrm{constant}, $$$ problems ()–() are studied in Choudhary et al [14] and the existence of classical and weak solutions are proved for bounded and unbounded kernels, respectively. We investigated the mild and classical solutions, see Kumar et al [15], of a partial integro‐differential equation () together with chaperone under assumption () and $$$ \beta, \mu \in {L}_{\infty, \kern.5em \mathrm{loc}}^{+}(Z) $$$, respectively. In earlier studies [1, 6], the stability of equilibria for disease‐free and disease state are examined by converting the equations into a system of ODEs.…”

Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

“…Further, they described the effect of the chaperone numerically. Recently, in [9], we have investigated the mild and classical solutions of the partial integro-differential equation (2) together with chaperone for different kernels.…”