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doi: 10.3934/eect.2021039
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Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani-333031, Rajasthan, India

* Corresponding author: Kapil Kumar Choudhary

Received  October 2020 Revised  April 2021 Early access August 2021

The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space $ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $ Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.

Citation: Kapil Kumar Choudhary, Rajiv Kumar, Rajesh Kumar. Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space. Evolution Equations & Control Theory, doi: 10.3934/eect.2021039
References:
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E. Leis and C. Walker, Existence of global classical and weak solutions to a prion equation with polymer joining, J. Evolution Equations, 17 (2017), 1227-1258.  doi: 10.1007/s00028-016-0379-6.  Google Scholar

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show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems: Abstract Linear Theory, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[3]

R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995.  Google Scholar

[4]

H. EnglerJ. Pruss and G. F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl., 324 (2006), 98-117.  doi: 10.1016/j.jmaa.2005.11.021.  Google Scholar

[5]

P. Gabriel, Global stability for the prion equation with general incidence, Math. Biosc. Eng., 12 (2015), 789-801.  doi: 10.3934/mbe.2015.12.789.  Google Scholar

[6]

M. L. GreerP. van den DriesscheL. Wang and G. F. Webb, Effects of general incidence and polymer joining on nucleated polymerzation in a model of prion proliferation, SIAM J. Appl. Math., 68 (2007), 154-170.  doi: 10.1137/06066076X.  Google Scholar

[7]

M. L. GreerL. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoretial Biology, 242 (2006), 598-606.  doi: 10.1016/j.jtbi.2006.04.010.  Google Scholar

[8]

R. Kumar and P. Murali, Modeling and analysis of prion dynamics in the presence of a chaperone, Mathematical Biosciences, 213 (2008), 50-55.  doi: 10.1016/j.mbs.2008.02.002.  Google Scholar

[9]

R. KumarK. K. Choudhary and R. Kumar, Study of the solution of a semilinear evolution equation of a prion proliferation model in the presence of chaperone in a product space, Math. Meth. Appl. Sci., 44 (2021), 1942-1955.  doi: 10.1002/mma.6894.  Google Scholar

[10]

M. LachowiczP. Laurençot and D. Wrzosek, On the Oort–Hulst–Safronov coagulation equation and its relation to the Smoluchowski equation, SIAM J. Math. Anal., 34 (2003), 1399-1421.  doi: 10.1137/S0036141002414470.  Google Scholar

[11]

P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evolution Equations, 7 (2007), 241-264.  doi: 10.1007/s00028-006-0279-2.  Google Scholar

[12]

E. Leis and C. Walker, Existence of global classical and weak solutions to a prion equation with polymer joining, J. Evolution Equations, 17 (2017), 1227-1258.  doi: 10.1007/s00028-016-0379-6.  Google Scholar

[13]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[14]

J. PrussL. Pujo-MenjouetG. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235.  doi: 10.3934/dcdsb.2006.6.225.  Google Scholar

[15]

G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl., 324 (2006), 580-603.  doi: 10.1016/j.jmaa.2005.12.036.  Google Scholar

[16]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, With a foreword by A. Pazy. Second edition. Pitman Monographs and Surveys in Pure and Applied Mathematics, 75. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995.  Google Scholar

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