On the asymptotic behaviour of solutions of nonlinear parabolic equations

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).

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

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19822087 ◽

1982 ◽

Vol 47 (8) ◽

pp. 2087-2096 ◽

Cited By ~ 3

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.

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

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02044-z ◽

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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Entropy Solutions for Nonlinear Parabolic Equations with Nonstandard Growth in Non-reflexive Orlicz Spaces

Mediterranean Journal of Mathematics ◽

10.1007/s00009-020-01668-3 ◽

2021 ◽

Vol 18 (1) ◽

Author(s):

M. Bourahma ◽

A. Benkirane ◽

J. Bennouna

Keyword(s):

Parabolic Equations ◽

Orlicz Spaces ◽

Nonlinear Parabolic Equations ◽

Entropy Solutions ◽

Nonstandard Growth ◽

Nonlinear Parabolic

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Quasilinearization Methods for Nonlinear Parabolic Equations with Functional Dependence

Georgian Mathematical Journal ◽

10.1515/gmj.2002.431 ◽

2002 ◽

Vol 9 (3) ◽

pp. 431-448

Author(s):

A. Bychowska

Keyword(s):

Cauchy Problem ◽

Parabolic Equations ◽

Unknown Function ◽

Functional Dependence ◽

Nonlinear Parabolic Equations ◽

Quasilinearization Method ◽

Convergence Theorems ◽

Nonlinear Parabolic ◽

Quasilinearization Methods ◽

Derivatives Of

Abstract We consider a Cauchy problem for nonlinear parabolic equations with functional dependence. We prove convergence theorems for a general quasilinearization method in two cases: (i) the Hale functional acting only on the unknown function, (ii) including partial derivatives of the unknown function.

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