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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kapil Kumar Choudhary ◽  
Rajiv Kumar ◽  
Rajesh Kumar
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Banach Space ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Weak Compactness ◽  
Evolution System ◽  
Global Classical Solution ◽  
Global Weak Solutions ◽  
Degradation Rates

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>

scholarly journals A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

scholarly journals Existence, uniqueness, and regularity of solutions to semilinear retarded differential equations

2004 ◽  
Vol 2004 (3) ◽  
pp. 213-219 ◽  
Author(s):  
D. Bahuguna
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Regularity Of Solutions ◽  
Retarded Differential Equation ◽  
Retarded Differential Equations

In the present work, we consider a semilinear retarded differential equation in a Banach space. We first establish the existence and uniqueness of a mild solution and then prove its regularity under different additional conditions. Finally, we consider some applications of the abstract results.

scholarly journals Exact Controllability of an Impulsive Semilinear System with Deviated Argument in a Banach Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Sanjukta Das ◽  
Dwijendra N. Pandey ◽  
N. Sukavanam
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Semigroup Theory ◽  
Exact Controllability ◽  
Impulsive Conditions ◽  
Banach Contraction ◽  
Deviated Argument ◽  
Functional Differential

A functional differential equation with deviated argument coupled with impulsive conditions is studied for the existence and uniqueness of the mild solution and exact controllability of the system. The results are obtained by using Banach contraction principle and C0 semigroup theory without imposing additional assumptions such as analyticity and compactness conditions on the generated semigroup and the nonlinear term. An example is provided to illustrate the presented theory.

scholarly journals Uniqueness of weak solutions to a prion equation with polymer joining

Analysis ◽  
2017 ◽  
Vol 37 (2) ◽  
Author(s):  
Elena Leis ◽  
Christoph Walker
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Differential Equation ◽  
Weak Solutions ◽  
Reaction Rates ◽  
Global Weak Solutions ◽  
Polymer Joining

AbstractWe consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [

scholarly journals On the correct formulation of a nonlinear differential equations in Banach space

2001 ◽  
Vol 26 (7) ◽  
pp. 437-444
Author(s):  
Mahmoud M. El-Borai ◽  
Osama L. Moustafa ◽  
Fayez H. Michael
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Nonlinear Differential Equations ◽  
Initial Value ◽  
Correct Formulation

We study, the existence and uniqueness of the initial value problems in a Banach spaceEfor the abstract nonlinear differential equation(dn−1/dtn−1)(du/dt+Au)=B(t)u+f(t,W(t)), and consider the correct solution of this problem. We also give an application of the theory of partial differential equations.

scholarly journals Global Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

2021 ◽  
pp. 91-108
Author(s):  
Shiyu Li
Keyword(s):  
Initial Data ◽  
Surface Waves ◽  
Shallow Water ◽  
Strong Solution ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Water Regime ◽  
Global Weak Solutions ◽  
Holm Equation ◽  
Sign Conditions

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime:                                        ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.

scholarly journals Weighted Pseudo Almost Automorphic Solutions to Nonautonomous Semilinear Differential Equations

2017 ◽  
Vol 1 (1) ◽  
pp. 27-32
Author(s):  
Saud M. Alsulami
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Evolution Family ◽  
Almost Automorphic ◽  
Exponentially Stable ◽  
Semilinear Differential Equation ◽  
Pseudo Almost Automorphic

We consider the existence and uniqueness of Weighted Pseudo almost automorphic solutionsto the non-autonomous semilinear differential equation in a Banach space X :( ) = ( ) ( ) ( , ( )), ' u t A t u t f t u t t Rwhere A(t), t R, generates an exponentially stable evolution family {U(t, s)} andf :R X X satisfies a Lipschitz condition with respect to the second argument.MSC 2010: 43A60; 34G20, 47Dxx

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