Displaying similar documents to “On the existence of classical solutions for differential-functional IBVP.”

Existence of classical solutions for parabolic functional differential equations with initial boundary conditions of Robin type

Milena Matusik (2012)

Annales Polonici Mathematici

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The paper deals with the initial boundary value problem of Robin type for parabolic functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in the classical sense. A theorem on the existence of a classical solution is proved. Our formulation and results cover differential equations with deviated variables and differential integral problems.

Monotone iteration for infinite systems of parabolic equations with functional dependence

Anna Pudełko (2007)

Annales Polonici Mathematici

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We consider the initial value problem for an infinite system of differential-functional equations of parabolic type. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. The solutions are obtained by the monotone iterative method. We prove theorems on weak partial differential-functional inequalities as well the existence and uniqueness theorems in the class of continuous bounded functions and in the class of...

Global existence of solutions to a chemotaxis system with volume filling effect

Tomasz Cieślak (2008)

Colloquium Mathematicae

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A system of quasilinear parabolic equations modelling chemotaxis and taking into account the volume filling effect is studied under no-flux boundary conditions. The resulting system is non-uniformly parabolic. A Lyapunov functional for the system is constructed. The proof of existence and uniqueness of regular global-in-time solutions is given in cases when either the Lyapunov functional is bounded from below or chemotactic forces are suitably weakened. In the first case solutions are...

Parabolic differential-functional inequalities in viscosity sense

Krzysztof Topolski (1998)

Annales Polonici Mathematici

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We consider viscosity solutions for second order differential-functional equations of parabolic type. Initial value and mixed problems are studied. Comparison theorems for subsolutions, supersolutions and solutions are considered.

The solutions of the quasilinear Keller-Segel system with the volume filling effect do not blow up whenever the Lyapunov functional is bounded from below

Tomasz Cieślak (2006)

Banach Center Publications

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In [2] we proved two kinds of mechanisms of preventing the blow up in a quasilinear non-uniformly parabolic Keller-Segel systems. One of them was a priori boundedness from below of the Lyapunov functional. In fact, we were able to present a condition under which the Lyapunov functional is bounded from below and a solution exists globally. In the present paper we prove that whenever the Lyapunov functional is bounded from below the solution exists globally.

Instantaneous shrinking of the support for solutions to certain parabolic equations and systems

Anatolii S. Kalashnikov (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The paper contains conditions ensuring instantaneous shrinking of the support for solutions to semilinear parabolic equations with compactly supported coefficients of nonlinear terms and reaction-diffusion systems.

Existence of solutions for infinite systems of parabolic equations with functional dependence

Anna Pudełko (2005)

Annales Polonici Mathematici

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The Cauchy problem for an infinite system of parabolic type equations is studied. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. We prove the existence and uniqueness of a bounded solution under Carathéodory type conditions and its differentiability, as well as the existence and uniqueness in the class of functions satisfying a natural growth condition. Both results are obtained by the fixed point method. ...

Existence and uniqueness of solutions of nonlinear infinite systems of parabolic differential-functional equations

Stanisław Brzychczy (2001)

Annales Polonici Mathematici

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We consider the Fourier first initial-boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations of parabolic type. The right-hand sides of the system are functionals of unknown functions. The existence and uniqueness of the solution are proved by the Banach fixed point theorem.

On some nonlocal systems containing a parabolic PDE and a first order ODE

Ádám Besenyei (2010)

Mathematica Bohemica

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Two models of reaction-diffusion are presented: a non-Fickian diffusion model described by a system of a parabolic PDE and a first order ODE, further, porosity-mineralogy changes in porous medium which is modelled by a system consisting of an ODE, a parabolic and an elliptic equation. Existence of weak solutions is shown by the Schauder fixed point theorem combined with the theory of monotone type operators.