Geometrical properties of the solutions of one-dimensional nonlinear parabolic equations

1995 ◽  
Vol 303 (1) ◽  
pp. 741-769 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Juan Luis Vazquez
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
One Dimensional ◽  
Geometrical Properties ◽  
Nonlinear Parabolic

scholarly journals Oscillation of the solutions of systems of nonlinear parabolic equations with functional arguments

2007 ◽  
Vol 38 (4) ◽  
pp. 367-379
Author(s):  
Yutaka Shoukaku
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Differential Inequalities ◽  
Dimensional Problem ◽  
Limit Condition ◽  
Nonlinear Functional ◽  
One Dimensional ◽  
Nonlinear Parabolic ◽  
Functional Differential ◽  
Systems Of Parabolic Equations

In the present paper the oscillatory properties of the solutions of systems of parabolic equations are investigated and oscillation criteria is derived for every solution of boundary value problems to be oscillatory or satisfies some limit condition. Our approach is to reduce the multi-dimensional problem to a one-dimensional problem for nonlinear functional differential inequalities.

An Efficient Algorithm Based on Extrapolation for the Solution of Nonlinear Parabolic Equations

2019 ◽  
Vol 20 (5) ◽  
pp. 527-541
Author(s):  
M. Ghasemi
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Spatial Direction ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Nonlinear Parabolic ◽  
End Conditions ◽  
Finite Difference Discretization ◽  
Difference Discretization ◽  
Diffusion Problems

AbstractTwo numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite difference discretization is used to approximate the solution at each time level. The convergence analysis is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equation with applications in fluid mechanics as well as convection–diffusion problems with applications in transport are solved to show the applicability and good performance of the procedures.

scholarly journals OSCILLATIONS OF SOLUTIONS TO PARABOLIC EQUATIONS WITH DEVIATING ARGUMENTS

1997 ◽  
Vol 28 (3) ◽  
pp. 169-181
Author(s):  
SATOSHI TANAKA ◽  
NORIO YOSHIDA
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Differential Inequalities ◽  
Dimensional Problem ◽  
Deviating Arguments ◽  
One Dimensional ◽  
Nonlinear Parabolic

Nonlinear parabolic equations with deviating arguments arc studied and sufficient conditions are derived for every solution of boundary value problems to be oscillatory in a cylindrical domam. Two kinds of boundary conditions are considered. Our approach is to reduce the multi-dimensional problem to a one-dimensional problem for differential inequalities of neutral type.

BIFURCATIONAL FEATURES IN SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS WITH WEAK DIFFUSION

2005 ◽  
Vol 15 (11) ◽  
pp. 3595-3606 ◽  
Author(s):  
S. A. KASCHENKO
Keyword(s):  
Parabolic Equations ◽  
Diffusion Coefficients ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Boundary ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Spatial Systems ◽  
Nonlinear Parabolic ◽  
Parabolic Boundary Value Problems ◽  
Weak Diffusion

Asymptotic solutions of parabolic boundary value problems are studied in a neighborhood of both an equilibrium state and a cycle in near-critical cases which can be considered as infinite-dimensional due to small values of the diffusion coefficients. Algorithms are developed to construct normalized equations in such situations. Principle difference between bifurcations in two-dimensional and one-dimensional spatial systems is demonstrated.

The form of blow-up for nonlinear parabolic equations

1984 ◽  
Vol 98 (1-2) ◽  
pp. 183-202 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Upper And Lower Solutions ◽  
Nonlinear Parabolic Equations ◽  
One Dimensional ◽  
Nonlinear Parabolic ◽  
Infinite Region ◽  
Limiting Behaviour ◽  
Semilinear Parabolic ◽  
Modified Equation

SynopsisSemilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

Kinetics of oxidation of ammonia on cobalt catalyst I model of the reactor with catalytically active wall

1982 ◽  
Vol 47 (8) ◽  
pp. 2087-2096 ◽  
Author(s):  
Bohumil Bernauer ◽  
Antonín Šimeček ◽  
Jan Vosolsobě
Keyword(s):  
Parabolic Equations ◽  
Cobalt Catalyst ◽  
Nonlinear Parabolic Equations ◽  
Catalytic Reactions ◽  
Temperature Profiles ◽  
Dimensional Model ◽  
Kinetics Of Oxidation ◽  
Nonlinear Parabolic ◽  
Catalytically Active ◽  
Kinetics Of

A two dimensional model of a tabular reactor with the catalytically active wall has been proposed in which several exothermic catalytic reactions take place. The derived dimensionless equations enable evaluation of concentration and temperature profiles on the surface of the active component. The resulting nonlinear parabolic equations have been solved by the method of orthogonal collocations.

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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