Asymptotic behaviour and source-type solutions for a diffusion-convection equation. (English) Zbl 0807.35059

Escobedo, Miguel; Vazquez, Juan Luis; Zuazua, Enrike

Arch. Ration. Mech. Anal. 124, No. 1, 43-65 (1993).

The paper deals with the equation \(u_ t = u_{xx} - | u |^{q - 1} u_ x\), \(t > 0\), \(x \in \mathbb{R}\). The asymptotic behaviour of the solutions of this equation is studied in the subcritical exponent range, that is, in the case \(1<q<2\). It is proved that as \(t \to \infty\) the solutions with integrable data look like the entropy solutions of the equation \(u_ t + (| u |^{q - 1} u/q)_ x = 0\) with initial data \(M \delta (x)\). Also the authors investigate the question of existence of fundamental solutions of the equation under consideration and they construct fundamental solutions for all exponents \(q>1\).

Reviewer: E.Minchev (Sofia)

MSC:

35K55	Nonlinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35E05	Fundamental solutions to PDEs and systems of PDEs with constant coefficients

Keywords:

source-type solutions; diffusion-convection equation; subcritical exponent range; entropy solutions

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