scholarly journals Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach

1991 ◽  
Vol 100 (2) ◽  
pp. 435-462 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Juan L. Vazquez
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Behaviour ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Systems Approach ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Parabolic

Asymptotic convergence to dipole solutions in nonlinear parabolic equations

1995 ◽  
Vol 125 (5) ◽  
pp. 877-900 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Sergey A. Posashkov ◽  
Juan Luis Vazquez
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Asymptotic Convergence ◽  
Lateral Boundary ◽  
Compactly Supported Function ◽  
Nonlinear Parabolic ◽  
General Power

We study the asymptotic behaviour as t → ∞ of the solution u = u(x, t) ≧ 0 to the quasilinear heat equation with absorption ut = (um)xx − f(u) posed for t > 0 in a half-line I = { 0 < x < ∞}. For definiteness, we take f(u) = up but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p >m. We impose u = 0 on the lateral boundary {x = 0, t > 0}, and consider a non-negative, integrable and compactly supported function uo(x) as initial data. This problem is equivalent to solving the corresponding equation in the whole line with antisymmetric initial data, uo(−x) = −uo(x).

On the asymptotic behaviour of solutions of nonlinear parabolic equations

2006 ◽  
Vol 136 (2) ◽  
pp. 365-384 ◽  
Author(s):  
A. Kon'kov
Keyword(s):  
Asymptotic Behaviour ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Asymptotic Behaviour Of Solutions ◽  
Nonlinear Parabolic

For a class of nonlinear parabolic equations and inequalities, we establish conditions guaranteeing that any solution tends to 0 as t → ∞.

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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