Well-posedness for a model of prion proliferation dynamics

Laurençot, Philippe; Walker, Christoph

doi:10.1007/s00028-006-0279-2

Cited by 30 publications

(51 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an example that has recently received some attention, we can point to models of growth and proliferation of prions (i.e., of proteins with transmissible pathological conformations) responsible for the Bovine Spongiform Encephalopathy ("Mad Cow Disease") [102,187] and several math-ematical models have already been object of a rigorous analysis [136,198,216]. According to the contemporary biological understanding, there are two basic prion forms, a normal, non-infectious, monomeric one (denoted by PrP C in the literature) and an infectious polymeric form (PrP Sc ) formed by the polymerization of the monomeric form.…”

Section: Y/ny; T/dymentioning

confidence: 99%

See 1 more Smart Citation

Mathematical Aspects of Coagulation-Fragmentation Equations

Costa

2015

CIM Series in Mathematical Sciences

View full text Add to dashboard Cite

We give an overview of the mathematical literature on the coagulationlike equations, from an analytic deterministic perspective. In Sect. 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Sect. 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the function spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sects. 3 and 4 we are concerned with several aspects of the solutions behaviour. We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof.

show abstract

Section: Y/ny; T/dymentioning

confidence: 99%

“…Continuous mass versions of these equations were also considered in the literature [102,136,198,216]. …”

Section: Y/ny; T/dymentioning

confidence: 99%

Mathematical Aspects of Coagulation-Fragmentation Equations

Costa

2015

CIM Series in Mathematical Sciences

View full text Add to dashboard Cite

show abstract

“…(Note that in [13] the problem is completed with the boundary condition U (t,x 0 ) = 0 while x 0 > 0, τ (x 0 ) > 0.) According to [8,13] we adopt the following definition. Definition 2.2.…”

Section: Basic Properties Of the Equationsmentioning

confidence: 99%

“…Even when τ (x 0 ) = 0, difficulties might arise when x → τ (x) is not regular enough to define the associated characteristics. However, according to the analysis of [8,13], we have seen that the notion of "monomer preserving solution" appears naturally, inserting (2.9) as a constraint. It leads to the question of deciding how this condition is related to (1.2) and (1.3) and to determine the corresponding boundary condition to be used at x = x 0 .…”

Section: Boundary Condition For the Continuous Systemmentioning

confidence: 99%

See 1 more Smart Citation

Scaling limit of a discrete prion dynamics model

Doumic¹,

Goudon

Lepoutre³

2009

Communications in Mathematical Sciences

View full text Add to dashboard Cite

show abstract

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

Kumar

Choudhary

Kumar

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

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.

show abstract

Well-posedness for a model of prion proliferation dynamics

Cited by 30 publications

References 11 publications

Mathematical Aspects of Coagulation-Fragmentation Equations

Mathematical Aspects of Coagulation-Fragmentation Equations

Scaling limit of a discrete prion dynamics model

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

Contact Info

Product

Resources

About