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Rarefaction waves in nonlocal convection-diffusion equations

100%
Pudełko A.
Colloquium Mathematicum
|
2014
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tom 137
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nr 1
27-42
EN
We consider a nonlocal convection-diffusion equation $u_t = J*u - u - uu_x$, where J is a probability density. We supplement this equation with step-like initial conditions and prove the convergence of the corresponding solutions towards a rarefaction wave, i.e. a unique entropy solution of the Riemann problem for the inviscid Burgers equation.
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Existence of solutions for infinite systems of parabolic equations with functional dependence

100%
Pudełko A.
Annales Polonici Mathematici
|
2005
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tom 86
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nr 2
123-135
EN
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.
3
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Monotone iteration for infinite systems of parabolic equations with functional dependence

100%
Pudełko A.
Annales Polonici Mathematici
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2007
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tom 90
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nr 1
1-19
EN
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 functions satisfying a certain growth condition.
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