# American Institute of Mathematical Sciences

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 and Control Theory, doi: 10.3934/eect.2021039
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems: Abstract Linear Theory, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. [2] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006. [3] R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995. [4] H. Engler, J. 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. [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. [6] M. L. Greer, P. van den Driessche, L. 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. [7] M. L. Greer, L. 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. [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. [9] R. Kumar, K. 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. [10] M. Lachowicz, P. 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. [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. [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. [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. [14] J. Pruss, L. Pujo-Menjouet, G. 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. [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. [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.

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##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems: Abstract Linear Theory, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. [2] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006. [3] R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995. [4] H. Engler, J. 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. [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. [6] M. L. Greer, P. van den Driessche, L. 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. [7] M. L. Greer, L. 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. [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. [9] R. Kumar, K. 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. [10] M. Lachowicz, P. 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. [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. [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. [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. [14] J. Pruss, L. Pujo-Menjouet, G. 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. [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. [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.
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