scholarly journals Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

2018 ◽  
Vol 23 (10) ◽  
pp. 4397-4431
Author(s):  
Jan Giesselmann ◽  
◽  
Niklas Kolbe ◽  
Lukacova-MedvidovaMaria ◽  
Nikolaos Sfakianakis ◽  
...  
Keyword(s):  
Existence And Uniqueness ◽  
Cancer Invasion ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Global Classical Solutions

scholarly journals On a global classical solution of a quasilinear hyperbolic equation

1989 ◽  
Vol 12 (1) ◽  
pp. 29-37 ◽  
Author(s):  
Y. Ebihara ◽  
D. C. Pereira
Keyword(s):  
Hyperbolic Equation ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Torsional Vibrations ◽  
Global Classical Solution ◽  
Quasilinear Hyperbolic Equation ◽  
Global Classical Solutions

In this paper we establish the existence and uniqueness of global classical solutions for the equation which arises in the study of the extensional vibrations of thin rod, or torsional vibrations of thin rod.

scholarly journals Optimal control of the two-dimensional Vlasov-Maxwell system

2020 ◽  
Author(s):  
Jörg Weber
Keyword(s):  
Adjoint Equation ◽  
Collisionless Plasma ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Maxwell System ◽  
Global Classical Solutions ◽  
Global Existence Of Solutions ◽  
Dimensional Version ◽  
The One

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

scholarly journals Global classical solutions to the Cauchy problem for a nonlinear wave equation

1998 ◽  
Vol 21 (3) ◽  
pp. 533-548 ◽  
Author(s):  
Haroldo R. Clark
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Classical Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Global Classical Solutions ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem{u″+M(|A12u|2)Au=0   in   ]0,T[u(0)=u0,       u′(0)=u1,whereu′is the derivative in the sense of distributions and|A12u|is theH-norm ofA12u. We prove the existence and uniqueness of global classical solution.

scholarly journals Strongly damped semilinear equations

1995 ◽  
Vol 8 (4) ◽  
pp. 397-404 ◽  
Author(s):  
D. Bahuguna
Keyword(s):  
Existence And Uniqueness ◽  
Analytic Semigroups ◽  
Classical Solutions ◽  
Semilinear Equations ◽  
Forcing Term ◽  
Global Classical Solutions ◽  
Strongly Damped

A class of strongly damped semilinear equations is studied by using the theory of analytic semigroups. Conditions (on the nonlinear forcing term) are given under which the existence and uniqueness of local and global classical solutions are ensured.

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